Monotonicity Formulas for Harmonic Functions in $\textrm{RCD}(0,N)$ Spaces

Abstract

We generalize to the $\textrm{RCD}(0,N)$ setting a family of monotonicity formulas by Colding and Minicozzi for positive harmonic functions in Riemannian manifolds with nonnegative Ricci curvature. Rigidity and almost rigidity statements are also proven, the second appearing to be new even in the smooth setting. Motivated by the recent work in Agostiniani et al. (Invent. Math. 222(3):1033–1101, 2020), we also introduce the notion of electrostatic potential in $\textrm{RCD}$ spaces, which also satisfies our monotonicity formulas. Our arguments are mainly based on new estimates for harmonic functions in $\textrm{RCD}(K,N)$ spaces and on a new functional version of the ‘(almost) outer volume cone implies (almost) outer metric cone’ theorem.

1 Introduction and Main Results

Monotonicity formulas are in general an important tool in analysis and geometry and have proven to be crucial in the study of functional inequalities, regularity of PDEs and minimal surfaces, one example being the celebrated Almgren frequency function [1]. We refer also to [36] for an overview on this topic and further references. Recently these formulas also found application in the theory of metric geometry with particular interest in the regularity of singular spaces. It is, however, important to recall that, especially on this context, monotonicity formulas are often coupled with rigidity and almost rigidity properties. In other words, besides knowing that a quantity is monotone it is often important to know that, if it happens to be constant or almost constant, then some extra regularity or almost regularity of the objects involved is present. An example of this, in the context of singular metric spaces, is the Bishop-Gromov volume ratio, where this said (almost) regularity is realized as (almost) local conical structure of the space. This feature was exploited first in [30] to noncollapsed Ricci-limit spaces and subsequently in [41] to $\textsf {ncRCD}$ spaces in order show that blow-up limits are cones and was recently developed in [4] to deduce volume bounds for the singular set, extending the analogous result for Ricci-limit spaces ( [31]). These works fall in the more general theory of quantitative differentiation which, roughly said, allows to pass from a monotone quantity with almost rigidity properties (also called coerciveness) to nontrivial results about effective regularity of the space (or a particular function). We refer to [28] for a detailed overview on this topic.

All of this motivates the study of new monotonicity formulas in $\textrm{RCD}$ spaces, where the Bishop-Gromov volume ratio and the Perelman ${{\mathcal {W}}}$ functional (see [69]) were up to now essentially the unique examples.

Our interest will be in extending a class of monotonicity formulas for harmonic functions in Riemannian manifolds with nonnegative Ricci curvature, to the $\textrm{RCD}$ setting. These types of formulas were first introduced by a series of works by Colding and Minicozzi [35,36,37] and were used by the same authors to prove the uniqueness of tangent cones for Einstein manifolds [38]. Recently the same monotone quantities were reinterpreted in [6] in the study of the electrostatic potential of a set and lead to the proof of new Willmore-type geometrical inequalities for Riemannian manifolds (see also [13] for the Euclidean setting).

The Monotonicity Formula

Our first main result is the extension to the nonsmooth setting of the whole class of monotonicity formulas derived in [6]. More precisely,we show that in a nonparabolic $\textrm{RCD}(0,N)$ space (see Definition 3.1), if $u\in L^\infty (\Omega )\cap C(\Omega )$ solves

$$\begin{aligned} {\left\{ \begin{array}{ll} {\varvec{\Delta }}|_{\Omega } u=0,\\ \liminf _{y\rightarrow x} u(y)\ge 1, &{} \text { for every }x \in \partial \Omega ,\\ u(x) \rightarrow 0 &{} \text {as }\textsf {d} (x,\partial \Omega )\rightarrow +\infty ,\\ \end{array}\right. } \end{aligned}$$

(P)

where $\Omega$ is open, unbounded and with $\partial \Omega$ bounded, then the function

$$\begin{aligned} (0,1)\ni t \mapsto U_{\beta }(t):=\frac{1}{t^{\beta \frac{N-1}{N-2}}}\int |\nabla u|^{\beta +1}\, {\mathrm d}\textrm{Per}(\{u<t\}), \, \end{aligned}$$

with $\beta \ge \frac{N-2}{N-1}$, is (continuous) and nondecreasing (see the beginning of Sect. 5.2 for a discussion on the well definition of the function $U_{\beta }$).

Rigidity and Almost Rigidity

As said above,we are also interested in the rigidity and almost rigidity properties of the functions $U_\beta$ and it turns out that, similarly to the Bishop-Gromov inequality, the special configurations connected to $U_\beta$ are cones. This is not a coincidence, indeed as we will see the reason behind the conical structure arising from these two quantities is the same, i.e. the existence of a function satisfying a precise PDE: $\Delta v=N$ and $|\nabla \sqrt{2v}|^2=1$. We refer to the introduction of [35] for a more detailed parallelism with the Bishop-Gromov inequality and the interpretation of $U_{\beta }$ as an area functional.

We will prove that

• if the derivative of $U_{\beta }$ vanishes at $t_0$, then $\{u<t_0\}$ is isometric to a truncated $\mathrm{RCD(0,N)}$ cone,

• if the derivative $U_{\beta }$ at $t_0$ is sufficiently small, then $\{u<t_0\}$ is close, in a suitable sense, to a truncated $\mathrm{RCD(0,N)}$ cone.

The first result is a generalization of the rigidity result in [6], while the second is new even in the smooth setting. It has to be said that almost rigidity results were already present in the work of Colding-Minicozzi (cf. with [38, Sect. 1.7] and [36, Sect. 2.2]). However, one of the novelty of our analysis is that, similarly to what happens for the splitting theorem [45], we are able to show that $\{u<t_0\}$ is close to a cone which is itself an $\mathrm{RCD(0,N)}$ space, while they only prove closeness to a ‘generic’ cone, moreover we prove that almost rigidity holds for all the functions $U_{\beta }$, while they only consider the case $\beta =3.$

Let us point out that, as it is now well understood, nonsmooth ($\textrm{RCD}$) spaces enter into play naturally in almost rigidity statements under Ricci curvature lower bounds even in the smooth case. We recall for example the existence of Riemannian manifolds having tangent cones at infinity with nonsmooth cross-section (see [39]).

Existence of Solutions

To justify the interest on the function $U_\beta$, it is clearly important to have many example of solutions to (P). The main examples in the smooth setting are the Green function (mainly explored in the papers of Colding-Minicozzi) and the electrostatic potential, which was the object of interest in [6]. In $\textrm{RCD}$ spaces the Green function was already built and studied in [25], while little or nothing was know about the electrostatic potential. One of our results will be the existence of an electrostatic (or capacitary) potential for a bounded open set E, under some mild regularity assumptions on its boundary. That is, we will prove the existence of a solution to (P) with $\Omega =\textrm{X}\setminus {{\bar{E}}}$, continuous up to the boundary of $\Omega$ with $u=1$ in $\partial \Omega$. Moreover, we will also prove a relation that links the Cheeger energy of u to the Capacity of E (see (8.2)).

New Functional Version of the “(Almost) Outer Volume Cone Implies (Almost) Outer Metric Cone” Theorem

The rigidity and almost rigidity results for $U_{\beta }$ that we described above will follow from a new functional and “outer” version of the volume cone to metric cone theorem for $\textrm{RCD}(0,N)$ spaces proved in [40]. In particular we prove that in an $\textrm{RCD}(0,N)$ space if a function u satisfies $\Delta u=N$ and $|\nabla \sqrt{2u}|^2=1$ then its superlevel sets are isometric to truncated cones (see Theorem 6.1).

Moreover,we will prove an effective almost version of the above result that appears to be new also in the smooth setting. More precisely,we will show that if a function u satisfies $\Delta u=N|\nabla \sqrt{2u}|^2$ and $|\nabla \sqrt{2u}|$ is almost constant, i.e. $|\nabla |\nabla \sqrt{u}||$ is small (in an integral sense), then the whole space $\textrm{X}$ is close in the pmGH topology to an $\textrm{RCD}(0,N)$ space $\textrm{X}'$, that is a truncated cone outside a bounded set (see Theorem 6.7).

We will also provide a second version of the almost rigidity result in Theorem 6.7, which is more in the spirit of the “almost volume annulus implies metric annulus” theorems of Cheeger-Colding ( [29]). Roughly said under the same hypotheses of Theorem 6.7, we prove that the sets $\{t_2<u<t_1\}$, when endowed with their intrinsic metrics, are close to annuli of an $\mathrm{RCD(0,N)}$ cone, also endowed with their intrinsic metrics (see Theorem 6.9).

Let us remark that a main portion of the proof of Theorem 6.1 is a repetition of the arguments in [40]; however, in writing this note, the authors realized that some steps in [40] were overlooked. For this reason along our exposition,we will also take the chance to fix and adjust some of the original arguments.

New Estimates for Harmonic Functions

As a by-product of our argument we prove some regularity results and estimates for harmonic functions in $\textrm{RCD}(K,N)$ spaces, which appear to be new in the nonsmooth setting and interesting on their own. In particular we prove that if u is harmonic, then the function $|\nabla u|^{\beta }$ has $W ^{1,2}$-regularity for every $\beta \ge \frac{1}{2} \frac{N-2}{N-1}$, together with an explicit bound for $|\nabla |\nabla u|^{\beta }|$ (see Theorem 4.4 for the precise statement). It is important to observe that the exponent $\beta$ is allowed to be strictly smaller than 1, which makes the result nontrivial. The estimates that we obtain are similar and strongly inspired by the ones for Riemannian manifolds obtained in [32].

Future Applications

This work is a part of a project that we are pursuing and whose objective is to investigate the possibility of developing a second-order analysis on “codimension-1” sets in $\textrm{RCD}$ spaces (recall that a “first-order” theory in this setting has already been extensively studied, see eg. [3, 22] and [42]). This is motivated by the recent [6] where, in the context of smooth Riemannian manifolds, it is proved that the monotonicity of $U_\beta$ (see above) implies a family of inequalities related to the mean curvature of hypersurfaces (more precisely to a lower bounds on their p-Willmore energy).

In particular in a forthcoming work [61], we will propose a notion of mean curvature and Willmore energy for the boundary of subsets of $\textrm{RCD}$ spaces. Our definitions will be tailored to the monotonicity results obtained in this note (in particular formula (5.7) and the estimate (5.6)) to obtain lower bounds for the Willmore energy, in the spirit of [6].

Plan of the Paper

The exposition will be organized as follows. In Sect. 2,we will introduce the needed tools and fix some notations. In Sect. 3,we introduce the notion of nonparabolic $\textrm{RCD}$ space and derive its main features. In Sect. 4,we will introduce a class of vector fields with nonnegative divergence (see Corollary 4.5), which are at the core of the proof of the monotonicity formula. In such section, we will also deduce new estimate for harmonic functions. In Sect. 5.1, we will prove some key decay estimates for solution of (P) and subsequently in Sect. 5.2, we will prove the main monotonicity result. In Sect. 6.1, we will state a new functional version of the “from outer functional cone to outer metric cone” theorem and prove the main new ingredients for its proof. The rest of the argument, being analogous to the one in [40], will be postponed to Appendix A. The ‘almost’ version of the previous theorem will be proved in Sect. 6.2. Then we will use these functional rigidity and almost rigidity results to deduce in Sect. 7 the main rigidity and almost rigidity statements from the monotonicity formula. Finally,the existence for the electrostatic potential will be given in Sect. 8. This argument relies on results mainly taken from [19], however,to make the proof self-contained and more readable, in Appendix B we will redo (and simplify) the proofs of all the results we need in the setting of $\textrm{RCD}$ spaces.

2 Preliminaries and Notations

2.1 Calculus Tools

Throughout all this note,a metric measure space (abbreviated in m.m.s.) is a triple $(\textrm{X},\textsf {d} ,{\mathfrak {m}})$ where

$$\begin{aligned}&(\textrm{X},\textsf {d} ) \text { is a complete and separable metric space and } {\mathfrak {m}}\text { is a nonnegative and nonzero}\\&\text {Borel measure on } \textrm{X},\text { finite on bounded sets and such that } \mathop {\textrm{supp}}\nolimits {\mathfrak {m}}=\textrm{X}. \end{aligned}$$

We will also use the notion of pointed metric measure space (abbreviated in p.m.m.s.), which is a quadruple $(\textrm{X},\textsf {d} ,{\mathfrak {m}},{{\bar{x}}})$, where $(\textrm{X},\textsf {d} ,{\mathfrak {m}})$ is a metric measure space and ${{\bar{x}}} \in \textrm{X}.$

Two metric measure spaces $(\textrm{X}_i,\textsf {d} _i,{\mathfrak {m}}_i)_{i=1,2}$ are said to be isomorphic if there exists an isometry $\iota : \textrm{X}_1\rightarrow \textrm{X}_2$ as metric spaces such that $\iota _*{\mathfrak {m}}_1={\mathfrak {m}}_2.$

We will denote by $\textsf {LIP} (\textrm{X}),\textsf {LIP} _{\textsf {loc} }(\textrm{X}),\textsf {LIP} _{b}(\textrm{X}),\textsf {LIP} _{bs}(\textrm{X})$ and $C_{bs}(\textrm{X}),$ respectively, the spaces of Lipschitz functions, locally Lipschitz functions, bounded Lipschitz functions, Lipschitz functions with bounded support and bounded continuous functions with bounded support in $(\textrm{X},\textsf {d} )$. For a function $f \in \textsf {LIP} _{\textsf {loc} }(\textrm{X})$,we denote by $\textrm{lip}f: \textrm{X}\rightarrow [0,+\infty )$ its local Lipschitz constant defined by

$$\begin{aligned} \textrm{lip}f (x):=\limsup _{y \rightarrow x} \frac{|f(x)-f(y)|}{\textsf {d} (x,y)}. \end{aligned}$$

(2.1)

2.1.1 Sobolev Spaces Via Test Plans

We will adopt the notion of Sobolev spaces on metric measures spaces via test plans introduced in [7]. This approach turns out to be equivalent [7] both to the notion of Sobolev space firstly introduced by Cheeger [27] and to the one introduced by Shanmugalingam [77].

A curve $\gamma \in C([0,1],\textrm{X})$ belongs to the space of absolutely continuous curves $AC([0,1],\textrm{X})$ if there exists $f \in L^1(0,1)$ such that $\textsf {d} (\gamma _t,\gamma _s)\le \int _s^t f(r)\, {\mathrm d}r,$ for every $0\le s<t\le 1.$ In this case it holds that the limit $|{{\dot{\gamma }}}_t|:=\lim _{h\rightarrow 0}h^{-1}\textsf {d} (\gamma _{t+h},\gamma _t)$ exists for a.e. $t \in (0,1)$ and is called metric speed at time t. The length $L(\gamma )$ of an absolutely continuous curve $\gamma$ is defined by

$$\begin{aligned} L(\gamma ):=\int _0^1|{{\dot{\gamma }}}_t|\, {\mathrm d}t. \end{aligned}$$

We recall also the evaluation map $\mathrm{{e}}_t : C([0,1],\textrm{X})\rightarrow \textrm{X}$ defined by $e_t(t):=\gamma _t.$

A Borel probability measure ${\varvec{\pi }}$ on $AC([0,1],\textrm{X})$ is said to be a test plan if

$$\begin{aligned}&\exists C>0 \, : \, {\mathrm{{e}}_t}_*{\varvec{\pi }}\le C{\mathfrak {m}}, \quad \forall t \in [0,1],\\&\int \int _0^1 |{{\dot{\gamma }}}_t|^2\, {\mathrm d}t\, {\mathrm d}{\varvec{\pi }}<+\infty . \end{aligned}$$

Definition 2.1

(Sobolev class) The Sobolev class $\textrm{S}^2(\textrm{X})$ is the space of all functions $f \in L^0({\mathfrak {m}})$ such that there exists a nonnegative $G\in L^2({\mathfrak {m}})$ for which

$$\begin{aligned} \int |f(\gamma _1)-f(\gamma _0)|\, {\mathrm d}{\varvec{\pi }}\le \int \int _0^1 |\dot{\gamma }_t|G(\gamma _t)\, {\mathrm d}t \, {\mathrm d}{\varvec{\pi }}, \quad \forall {\varvec{\pi }}\text { test plan.} \end{aligned}$$

(2.2)

For every $f \in \textrm{S}^2(\textrm{X})$, there exists a unique function G with minimal $L^2({\mathfrak {m}})$ norm such that (2.2) holds, called minimal weak upper gradient and denoted by |Df|. The minimal weak upper gradient has the following lower semicontinuity property: suppose that $(f_n)\subset \textrm{S}^2(\textrm{X})$ converges to $f\in L^0({\mathfrak {m}})$ ${\mathfrak {m}}$-a.e. and $|Df_n|$ converges weakly in $L^2({\mathfrak {m}})$ to a function $G \in L^2({\mathfrak {m}})$, then

$$\begin{aligned} f\in \textrm{S}^2(\textrm{X}) \quad \text { and } \quad |Df|\le G, \, {\mathfrak {m}}\text {-a.e..} \end{aligned}$$

(2.3)

We define the Sobolev space $W ^{1,2}(\textrm{X}):=L^2({\mathfrak {m}})\cap \textrm{S}^2(\textrm{X})$. $W ^{1,2}(\textrm{X})$ becomes a Banach space, when endowed with the norm

$$\begin{aligned} \Vert f\Vert _{ W ^{1,2}(\textrm{X})}:=\sqrt{\Vert f\Vert _{L^2({\mathfrak {m}})}^2+\Vert |Df|\Vert _{L^2({\mathfrak {m}})}^2}. \end{aligned}$$

The Cheeger energy functional $\textrm{Ch}: L^2({\mathfrak {m}})\rightarrow [0,+\infty ]$ is defined by

$$\begin{aligned} \textrm{Ch}(f):={\left\{ \begin{array}{ll} \frac{1}{2}\int |Df|^2\, {\mathrm d}{\mathfrak {m}}, &{} f \in W ^{1,2}(\textrm{X}),\\ +\infty \, &{} \text {otherwise}. \end{array}\right. } \end{aligned}$$

It follows from (2.3) that $\textrm{Ch}$ is a convex and lower semicontinuous functional on $L^2({\mathfrak {m}})$.

A metric measure space is said to be infinitesimally Hilbertian if $W ^{1,2}(\textrm{X})$ is a Hilbert space or equivalently if $\textrm{Ch}$ is a quadratic form (see [48]).

2.1.2 Tangent Module

We assume the reader to be familiar with theory of $L^0$-normed modules on a m.m.s. $(\textrm{X},\textsf {d} ,{\mathfrak {m}})$ and we refer to [47] for a detailed account on this theory.

Definition 2.2

($L^0$-normed $L^0$-module) An $L^0$-normed $L^0$-module is a triple $({\mathscr {M}},|.|,\tau )$, where ${\mathscr {M}}$ is a module over the commutative ring $L^0({\mathfrak {m}})$, $({\mathscr {M}},\tau )$ is a topological vector space, $|.|: {\mathscr {M}} \rightarrow L^0({\mathfrak {m}})$ is a map satisfying (in the ${\mathfrak {m}}$-a.e. sense)

$$\begin{aligned} \begin{aligned} |v|\ge 0&\quad \text { for every }v\in {\mathscr {M}}, \text { with equality if and only if }v=0,\\ |v+w|\le |v|+|w|&\quad \text { for every }v,w\in {\mathscr {M}},\\ |fv|=|f||v|&\quad \text { for every }f\in L^0({\mathfrak {m}})\text { and }v\in {\mathscr {M}}, \end{aligned} \end{aligned}$$

and such that $\tau$ is induced by the distance $\textsf {d} _{0}(v,w):=\int |v-w|\wedge 1\, {\mathrm d}{\mathfrak {m}}'$ (where ${\mathfrak {m}}'$ is a probability measure on $\textrm{X}$ such that ${\mathfrak {m}}\ll {\mathfrak {m}}'\ll {\mathfrak {m}}$), which is also assumed to be complete.

An $L^0$-normed module ${\mathscr {M}}$ is a Hilbert module if for every $v,w \in {\mathscr {M}}$ it holds

$$\begin{aligned} |v+w|^2+|v-w|^2=2|v|^2+2|w|^2, \quad {\mathfrak {m}}\text {-a.e.}. \end{aligned}$$

If this is the case, by polarization, we can define a scalar product $\langle ,\rangle : {\mathscr {M}}^0\times {\mathscr {M}}^0\rightarrow L^0({\mathfrak {m}})$ that is symmetric, $L^0({\mathfrak {m}})$-bilinear and satisfies $\langle v,v\rangle =|v|^2$, $|\langle v,w\rangle |\le |v||w|$, ${\mathfrak {m}}$-a.e. for every $v,w \in {\mathscr {M}}^0.$

Given an $L^0$-normed module ${\mathscr {M}}$ and E a Borel subset of $\textrm{X}$, we define the localized module

$$\begin{aligned} {\mathscr {M}}{_{|_{E}}}:=\{ {\chi }_Ev \ : \ v \in {\mathscr {M}}\}. \end{aligned}$$

${\mathscr {M}}{_{|_{E}}}$ inherits naturally a structure of $L^0$-normed module and if ${\mathscr {M}}$ is a Hilbert module then ${\mathscr {M}}{_{|_{E}}}$ is Hilbert as well with $\langle .,.\rangle _{{\mathscr {M}}{_{|_{E}}}}={\chi }_E\langle .,.\rangle _{{\mathscr {M}}}$. ${\mathscr {M}}{_{|_{E}}}$ can also be seen as the quotient of ${\mathscr {M}}$ by the equivalence relation ‘$v \sim w$ if and only if $|v-w|=0$ ${\mathfrak {m}}$-a.e. in E’. This identification will be used in the rest of the note without further notice.

Definition 2.3

(Tangent module) Suppose that $(\textrm{X},\textsf {d} ,{\mathfrak {m}})$ is an infinitesimally Hilbertian m.m.s.. Then there exists a (unique) couple $(L^0(T\textrm{X}),\nabla )$, where $L^0(T\textrm{X})$ is an $L^0$-normed module and $\nabla : W ^{1,2}(\textrm{X})\rightarrow L^0(T\textrm{X})$, called gradient operator, is a linear and continuous map such that

$$\begin{aligned}&|\nabla f| \text { coincides with the minimal weak upper gradient of } f\\&\left\{ \sum _{i=1}^n {\chi }_{E_i}\nabla f_i \ : \{E_i\}_{i=1}^n \text { Borel partition of } \textrm{X}, \{f_i\}_{i=1}^n \subset W ^{1,2}(\textrm{X}) \right\} \\&\quad \text { is dense in } L^0(T\textrm{X}). \end{aligned}$$

The gradient operator has the following properties

$$\begin{aligned}&Locality: \, \nabla f=\nabla g,\, {\mathfrak {m}}\text {-a.e. in } \{f=g\},\\&Leibniz rule: \, \nabla (fg)=g\nabla f+f\nabla g, \text { for every } f,g \in W ^{1,2}(\textrm{X})\cap L^\infty ({\mathfrak {m}}),\\&Chain rule: \, \nabla (\varphi (f))=\varphi '(f)\nabla f, \text { for every } f\in W ^{1,2}(\textrm{X}),\, \varphi \in C^1({{\,\mathrm{{\mathbb {R}}}\,}})\cap \textsf {LIP} ({{\,\mathrm{{\mathbb {R}}}\,}}). \end{aligned}$$

Finally, we denote by $L^2(T\textrm{X})$ the subset of $L^0(T\textrm{X})$ containing the elements with square integrable pointwise norm.

2.1.3 Local Sobolev Spaces

Let $(\textrm{X},\textsf {d} ,{\mathfrak {m}})$ be a proper and infinitesimally Hilbertian m.m.s. and let $\Omega$ be an open subset of $\textrm{X}$. We define

$$\begin{aligned} W ^{1,2}_\textsf {loc} (\Omega ):=\{u \in L_\textsf {loc} ^2(\Omega ) \ | \ u\eta \in W ^{1,2}(X), \text { for every }\eta \in \textsf {LIP} _{\mathrm c}(\Omega )\} , \end{aligned}$$

which is actually equivalent to ask that for every Borel set $\Omega '\subset \subset \Omega$ there exists $u' \in W ^{1,2}(\textrm{X})$ such that $u=u'$, ${\mathfrak {m}}$-a.e. in $\Omega '.$

For any $u\in W ^{1,2}_\textsf {loc} (\Omega )$, we define its gradient $\nabla u$ as the unique element of $L^0(TX){_{|_{\Omega }}}$ such that

$$\begin{aligned} \nabla u:=\nabla (\eta u),{\mathfrak {m}}\text { -a.e. in } \{\eta =1\}, \quad \forall \eta \in \textsf {LIP} _{\mathrm c}(\Omega ), \end{aligned}$$

which is well defined thanks to the locality property of the gradient. In particular $|\nabla u|\in L^2_{\textsf {loc} }(\Omega )$. It is straightforward to check that $\nabla u$ satisfies the expected locality property, Leibniz rule and chain rule. We only state explicitly a version of the chain rule that we will need: if $u \in W ^{1,2}_{\textsf {loc} }(\Omega )$ and $\varphi \in C^1(I)$, with I open interval, are such that

$$\begin{aligned} u(\Omega ')\subset \subset I \text { (up to a } {\mathfrak {m}}\text {-negligible set), for every }\Omega '\subset \subset \Omega , \end{aligned}$$

(2.4)

then $\varphi (u)\in W ^{1,2}_{\textsf {loc} }(\Omega )$ and $\nabla \varphi (u)=\varphi '(u)\nabla u.$

Observe that, since $\nabla u\in L^0(T\textrm{X}){_{|_{\Omega }}}$, it makes sense to compute the scalar product $\langle \nabla u,v\rangle \in L^0(\Omega ,{\mathfrak {m}})$, for every $v \in L^0(T\textrm{X}){_{|_{\Omega }}}$; moreover, this scalar product also satisfies $|\langle \nabla u,v\rangle |\le |\nabla u||v|,$ ${\mathfrak {m}}$-a.e. in $\Omega$ (recall the discussion in Sect. 2.1.2).

We also define the spaces

$$\begin{aligned}&W ^{1,2}(\Omega ):=\{f \in W ^{1,2}_\textsf {loc} (\Omega ) \ | \ f, |\nabla f|\in L^2(\Omega )\},\\&W ^{1,2}_0(\Omega ):=\overline{\textsf {LIP} _c(\Omega )}^{ W ^{1,2}(X)}\subset W ^{1,2}(\textrm{X}) . \end{aligned}$$

We end this subsection with the following technical lemma.

Lemma 2.4

Let $u \in W ^{1,2}_{\textsf {loc} }(\Omega )$ be nonnegative and $\alpha \in (0,1)$ be such that

$$\begin{aligned} {\chi }_{\{u>0\}} u^{\alpha -1}|\nabla u| \in L^{2}_{\textsf {loc} }(\Omega ). \end{aligned}$$

Then $u^{\alpha }\in W ^{1,2}_{\textsf {loc} }(\Omega )$ and $\nabla u=\alpha ^{-1} u^{1-\alpha }\,\nabla u^{\alpha }$, ${\mathfrak {m}}$-a.e. in $\Omega .$

Proof

For the first part, it is enough to show that $f:=\eta u^{\alpha }\in W ^{1,2}(\textrm{X})$ for every $\eta \in \textsf {LIP} _{c}(\Omega )$ with $|\eta |\le 1$. Fix $\varepsilon \in (0,1)$ arbitrary and define $f_{\varepsilon }=\eta (u+\varepsilon )^{\alpha }.$ Then from the nonnegativity of u, we have $f_{\varepsilon }\in W ^{1,2}(\textrm{X})$ and recalling that $|\nabla u|=0$ ${\mathfrak {m}}$-a.e. in $\{u=0\}$ we have

$$\begin{aligned} |\nabla f_{\varepsilon }|\le \textrm{Lip}\eta \, C_{\alpha }(u+2)+\alpha {\chi }_{u>0} u^{\alpha -1}|\nabla u|,\quad {\mathfrak {m}}\text {-a.e..} \end{aligned}$$

It follows that the family $\{f_{\varepsilon }\}_{\varepsilon \in (0,1)}$ is bounded in $W ^{1,2}(\textrm{X})$. Moreover $f_{\varepsilon }\rightarrow f$ in $L^2$, therefore from the lower semicontinuity of the Cheeger energy, it follows that $f \in W ^{1,2}(\textrm{X})$. For the second part, we observe that $u\wedge n=(u^{\alpha }\wedge n^\alpha )^{1/\alpha }$ ${\mathfrak {m}}$-a.e. in $\Omega$ and that $(t\wedge n^{\alpha })^{1/\alpha }\in \textsf {LIP} ({{\,\mathrm{{\mathbb {R}}}\,}})$. Therefore from the locality and the chain rule for the gradient

$$\begin{aligned} \nabla u=\alpha ^{-1} u^{1-\alpha }\,\nabla u^{\alpha }, \quad {\mathfrak {m}}\text {-a.e. in } \{u\le n\} \end{aligned}$$

and we conclude from the arbitrariness of n. $\square$

2.1.4 Laplacian and Divergence Operators

In this section, we assume $(\textrm{X},\textsf {d} ,{\mathfrak {m}})$ to be a proper and infinitesimally Hilbertian m.m.s.

Definition 2.5

(Measure-valued Laplacian [48]) Let $\Omega \subset \textrm{X}$ open. We say that and $u \in W ^{1,2}_\textsf {loc} (\Omega )$ belongs to the domain of the measure-valued Laplacian $D({\varvec{\Delta }},\Omega )$ if there exists a Radon measure ${\varvec{\Delta }}{_{|_{\Omega }}} u$ in $\Omega$ such that

$$\begin{aligned} -\int _{\Omega } \langle \nabla f, \nabla u\rangle {\mathrm d}{\mathfrak {m}}=\int _{\Omega } f {\mathrm d}{\varvec{\Delta }}{_{|_{\Omega }}} u, \end{aligned}$$

(2.5)

for every $f \in \textsf {LIP} _c(\Omega ).$

Let us remark that in the above definition with the term Radon measure, we denote a set function $\mu : \{ \text {Borel sets relatively compact in} \Omega \}\rightarrow {{\,\mathrm{{\mathbb {R}}}\,}}$ which can be written as $\mu (B)=\mu ^+(B)-\mu ^-(B)$, for some positive Radon measures $\mu ^+,\mu ^-$. In particular we do not require $\mu$ to be a Borel measure on the whole $\Omega$, this weaker assumption is needed for example in Proposition 2.6 and to write the Laplacian of the distance function (see also the discussion in [26]).

When no confusion can occur, we will drop the subscript $\Omega$ and simply write ${\varvec{\Delta }}u.$ Moreover, we will write $D({\varvec{\Delta }})$ in place of $D({\varvec{\Delta }},\textrm{X})$ and whenever ${\varvec{\Delta }}\ll {\mathfrak {m}}$, we will use the nonbold notation $\Delta .$

A function $u \in D({\varvec{\Delta }},\Omega )$ is said to be subharmonic if ${\varvec{\Delta }}u\ge 0$, superharmonic if ${\varvec{\Delta }}u\le 0$ and harmonic if ${\varvec{\Delta }}u=0$.

Finally, it easily follows from the definition that the Laplacian operator is linear and satisfies the following locality property:

$$\begin{aligned}{} & {} \text {if } u,v \in D({\varvec{\Delta }},\Omega ) \text { and } u=v\, {\mathfrak {m}}\text {-a.e. in } U,\\{} & {} \quad \text { with } U \text { open and relatively compact in } \Omega ,\text { then } {\varvec{\Delta }}u{_{|_{U}}}={\varvec{\Delta }}v{_{|_{U}}}. \end{aligned}$$

The following existence and comparison result is proven in [48, Prop 4.13].

Proposition 2.6

Let $u \in W ^{1,2}_{\textsf {loc} }(\Omega )$ and suppose that there exists $g \in L^1_{\textsf {loc} }(\Omega )$ such that

$$\begin{aligned} -\int _{\Omega } \langle \nabla f, \nabla u\rangle \, {\mathrm d}{\mathfrak {m}}\ge \int _{\Omega } gf \, {\mathrm d}{\mathfrak {m}}, \quad \forall f \in \textsf {LIP} _c(\Omega ), \text { with } f \ge 0, \end{aligned}$$

Then $u \in D({\varvec{\Delta }},\Omega )$ and ${\varvec{\Delta }}u\ge g{\mathfrak {m}}{_{|_{\Omega }}}.$

Remark 2.7

Let $u \in D({\varvec{\Delta }},\Omega )$. Suppose that ${\varvec{\Delta }}u \ll {\mathfrak {m}}$ (resp. ${\varvec{\Delta }}u\ge g {\mathfrak {m}}$) with $\frac{{\mathrm d}{\varvec{\Delta }}u }{{\mathrm d}{\mathfrak {m}}}\in L^2_{\textsf {loc} }(\Omega )$ (resp. $g\in L^1_{\textsf {loc} }(\Omega )$). Then, recalling that in an infinitesimally Hilbertian m.m.s. Lipschitz functions are dense in $W ^{1,2}(\textrm{X})$ (see [7]), by a truncation and cut-off argument it follows that (2.5) (resp. $-\int \langle \nabla f, \nabla u \rangle {\mathrm d}{\mathfrak {m}}\ge \int gf{\mathrm d}{\mathfrak {m}}$) holds also for every $f \in W ^{1,2}(X)$ (resp. $f \in W ^{1,2}(X)\cap L^\infty ({\mathfrak {m}})$, $f \ge 0$) with support compact in $\Omega .$ $\square$

Definition 2.8

(Measure-valued divergence [53]) Let $v \in L^0(TX){_{|_{\Omega }}}$ be such that $|v|\in L^2_{\textsf {loc} }(\Omega )$, we say that $v \in D({{\textbf {div}}},\Omega )$ if there exists a Radon measure ${{\textbf {div}}}{_{|_{\Omega }}}(v)$ in $\Omega$ such that

$$\begin{aligned} \int _{\Omega } \langle \nabla f,v\rangle {\mathrm d}{\mathfrak {m}}=-\int _{\Omega } f \, {\mathrm d}\, {{\textbf {div}}}{_{|_{\Omega }}}(v), \end{aligned}$$

(2.6)

for every $f \in \textsf {LIP} _c(\Omega ).$

It is clear from the definition that given $u \in W ^{1,2}_{\textsf {loc} }(\Omega )$, we have $\nabla f \in D({{\textbf {div}}},\Omega )$ if and only if $u \in D({\varvec{\Delta }},\Omega )$ and in this case ${{\textbf {div}}}{_{|_{\Omega }}}(\nabla u)={\varvec{\Delta }}{_{|_{\Omega }}}u.$ As for the Laplacian, we will often write simply ${\textbf {div}}(v)$ instead of ${{\textbf {div}}}{_{|_{\Omega }}}(v)$.

Remark 2.9

Analogously to the measure-valued Laplacian, we have that if ${{\textbf {div}}}(v) \in L^2_{\textsf {loc} }(\Omega )$, then (2.6) holds also for every $f \in W ^{1,2}(X)$ with support compact in $\Omega .$ $\square$

2.2 RCD Spaces

We assume the reader to be familiar with the definition and theory of $\textrm{RCD}(K,N)$ spaces (see [8, 48]). We limit ourselves to recall some of their main properties that will be needed.

The Sobolev-to-Lipschitz property holds (the definition we recall comes from [45] and so does the argument that we adopt to prove the ‘local version’ below - the validity of this property on $\textrm{RCD }(K,\infty )$ spaces was known from [9]): for every $f \in W ^{1,2}(\textrm{X})$ such that $|Df|\in L^\infty ({\mathfrak {m}})$, f has a Lipschitz representative and $\textrm{Lip}f\le \Vert |D f|\Vert _{L^\infty }$.

The local variant we will actually use is the following:

Proposition 2.10

(Local Sobolev-to-Lipschitz property) Let $\textrm{X}$ be an $\textrm{RCD}(K,N)$ space, $K \in {{\,\mathrm{{\mathbb {R}}}\,}}$ and $N\in [1,+\infty )$ and let $\Omega \subset \textrm{X}$ be open. Suppose $f \in W ^{1,2}_{\textsf {loc} }(\Omega )$ is such that $\Vert |\nabla f|\Vert _{L^{\infty }(\Omega )}<+\infty$, then f has a locally Lipschitz representative. Moreover for such representative it holds

$$\begin{aligned} |f(x)-f(y)|\le \Vert |\nabla f|\Vert _{L^{\infty }(\Omega )}\, \textsf {d} (x,y), \end{aligned}$$

(2.7)

for every $x,y \in \Omega$ such that $\textsf {d} (x,y)\le \textsf {d} (x,\partial \Omega ).$

Proof

The fact that f has a locally Lipschitz representative it follows from the Sobolev-to-Lipschitz property and a cut-off argument.

For the second part, we observe that it is sufficient to consider the case $\textsf {d} (x,y)<\textsf {d} (x,\partial \Omega ),$ since the equality case follows by continuity. Hence for some $r>0$, we have $\textsf {d} (x,y)<r<\textsf {d} (x,\partial \Omega )$ and we can consider ${\tilde{f}} \in W ^{1,2}(\textrm{X})$ such that ${\tilde{f}} =f$ ${\mathfrak {m}}$-a.e. in $B_{r}(x)$. Then for $\varepsilon < (r-\textsf {d} (x,y))/4$, we define $\mu ^0_{\varepsilon }:={\mathfrak {m}}{_{|_{B_{\varepsilon }(x)}}}{\mathfrak {m}}(B_{\varepsilon }(x))^{-1}$, $\mu ^1_{\varepsilon }:={\mathfrak {m}}{_{|_{B_{\varepsilon }(y)}}}{\mathfrak {m}}(B_{\varepsilon }(y))^{-1}$. thanks to the results in [74] and [75] there exists a unique ${\varvec{\pi }}^{\varepsilon }\in \textsf {OptGeo} (\mu ^0_{\varepsilon },\mu ^1_{\varepsilon })$ such that ${\mathrm{{e}}_t}_*{\varvec{\pi }}^{\varepsilon }\le C{\mathfrak {m}}$, $\forall t \in [0,1]$, for some constant C depending on ${\varepsilon }$. In particular, ${\varvec{\pi }}^{\varepsilon }$ is a test plan. Moreover from the triangle inequality, it follows that ${\varvec{\pi }}^{\varepsilon }$ is concentrated on curves $\gamma$ with support contained in $B_{r}(x)$. Therefore from (2.2)

$$\begin{aligned} \left| \int f \,{\mathrm d}\mu ^1_{\varepsilon }- \int f \, {\mathrm d}\mu ^0_{\varepsilon }\right|&\le \int |{\tilde{f}}(\gamma _1)-\tilde{f}(\gamma _0)|\, {\mathrm d}{\varvec{\pi }}^{\varepsilon }(\gamma )\\&\le \Vert |\nabla \tilde{f}|\Vert _{L^{\infty }(B_{r}(x))} \int \int _0^1|{{\dot{\gamma }}}_t| {\mathrm d}t d{\varvec{\pi }}^{\varepsilon }\le \Vert |\nabla f|\Vert _{L^{\infty }}W_2(\mu ^0_{\varepsilon },\mu ^1_{\varepsilon }). \end{aligned}$$

Letting $r\rightarrow 0^+$ from the continuity of f, we obtain (2.7). $\square$

From Proposition 2.10 it also follows that

$$\begin{aligned} \text {if } \Omega \text { is connected, }u \in W ^{1,2}_{\textsf {loc} }(\Omega ) \text { and } |\nabla u|=0,\, {\mathfrak {m}}\text {-a.e., then }u \text { is constant in } \Omega . \end{aligned}$$

(2.8)

The Bishop-Gromov inequality holds (see [81]), i.e.

$$\begin{aligned} \frac{{\mathfrak {m}}(B_R(x))}{v_{K,N}(R)}\le \frac{{\mathfrak {m}}(B_r(x))}{v_{K,N}(r)}, \quad \text {for any } 0<r<R \text { and any } x \in \textrm{X}, \end{aligned}$$

where for the quantities $v_{K,N}(r)$ coincides, for $N \in {\mathbb {N}}$, with the volume of the ball with radius r in the model space of dimension N and Ricci curvature K (see [81] for the definition of $v_{K,N}(r)$ for arbitrary $N\in [1,\infty )$). In particular $(\textrm{X},\textsf {d} ,{\mathfrak {m}})$ is proper and uniformly locally doubling. We also note that in the case $K=0$, this implies that the limit

$$\begin{aligned} \textsf {AVR} (\textrm{X}):=\lim _{r\rightarrow +\infty }\frac{{\mathfrak {m}}(B_r(x))}{r^N} \end{aligned}$$

exists finite and does not depend on the point $x \in \ X$. We call the quantity $\textsf { AVR} (\textrm{X})$ asymptotic volume ratio of $\textrm{X}$ and if $\textsf { AVR} (\textrm{X})>0$ we say that $\textrm{X}$ has Euclidean volume growth.

We will need the following Laplacian comparison for $\textrm{RCD}(0,N)$ spaces (see [48, Corollary 5.15]):

$$\begin{aligned} \textsf {d} (x_0,.)^2 \in D({\varvec{\Delta }}) \text { and } {\varvec{\Delta }}\textsf {d} (x_0,.)^2 \le 2N{\mathfrak {m}}, \quad \text { for every }x_0\in \textrm{X}, \end{aligned}$$

(2.9)

moreover in any $\textrm{RCD}(K,N)$ space with $N<+\infty$ it holds that

$$\begin{aligned} |\nabla \textsf {d} (x_0,.)|=1, \quad {\mathfrak {m}}\text {-a.e.}. \end{aligned}$$

(2.10)

From to the results in [27] and the fact that $\textrm{lip}\textsf {d} (x_0,.)\equiv 1$ it follows that (2.10) actually holds in the general setting of doubling m.m.s. satisfying a Poincaré inequality; however, a more direct proof in the setting of $\textrm{RCD}(K,N)$ spaces is also available (see for example [55, Prop. 3.1]).

Recall that, since $\textrm{RCD}(K,N)$ spaces are infinitesimally Hilbertian, the heat flow $h_t: L^2({\mathfrak {m}})\rightarrow L^2({\mathfrak {m}})$, $t \ge 0$, defined as the gradient flow of $\textrm{Ch}$ on $L^2({\mathfrak {m}})$ is linear, continuous, self-adjoint and satisfies $h_t(f)\in D(\Delta )\cap W ^{1,2}(\textrm{X})$, moreover the curve $(0,\infty )\ni t \mapsto h_t f \in L^2({\mathfrak {m}})$ is locally absolutely continuous for every $f \in L^2({\mathfrak {m}})$ and

$$\begin{aligned} \frac{{\mathrm d}}{{\mathrm d}t}h_t(f)=\Delta h_t(f)\in L^2({\mathfrak {m}}), \quad \text {for a.e. } t>0, \end{aligned}$$

(see [47] for further details). Moreover $h_t$ has the so-called $L^\infty$-to Lipschitz regularization property (see [9]), i.e. there exists a constant $C(K)>0$ such that for every $f \in L^\infty \cap L^2({\mathfrak {m}})$ it holds that $|\nabla h_t f|\in L^\infty ({\mathfrak {m}})$ and

$$\begin{aligned} \Vert |\nabla h_t f|\Vert _{L^\infty ({\mathfrak {m}})}\le \frac{C(K)}{\sqrt{t}} \Vert f\Vert _{L^\infty ({\mathfrak {m}})}, \quad \, \forall \, t \in (0,1). \end{aligned}$$

(2.11)

In an $\textrm{RCD}(K,N)$ space, it can be given also a notion of heat kernel (see [9]) $p_t: \textrm{X}\times \textrm{X}\rightarrow [0,+\infty ]$ which has a locally Hölder-continuous representative (see [78, 79]), which satisfies the following pointwise bounds [67], generalizing the classical estimates of Li and Yau in the smooth case [70]:

$$\begin{aligned} \begin{aligned}&\frac{1}{C_1{\mathfrak {m}}(B(x,\sqrt{t}))}\exp \left\{ -\frac{\textsf {d} ^2(x,y)}{3t}-ct\right\} \le p_t(x,y)\\&\quad \le \frac{C_1}{{\mathfrak {m}}(B(x,\sqrt{t}))}\exp \left\{ -\frac{\textsf {d} ^2(x,y)}{5t}+ct \right\} ,\\&\left| {\nabla p_t(x,\cdot )}(y)\right| \le \frac{C_1}{\sqrt{t}{\mathfrak {m}}(B(x,\sqrt{t}))}\exp \left\{ -\frac{\textsf {d} ^2(x,y)}{5t}+ct\right\} \quad \text {for } {\mathfrak {m}}\text {-a.e. } y\in X, \end{aligned} \end{aligned}$$

(2.12)

for any $x,y\in X$, for any $t>0$ and where $c,C_1$ are positive constants depending only on K, N such that $c=0$ if $K=0$.

We introduce the algebra of test functions $\textsf {Test} (\textrm{X})$ [76] defined as

$$\begin{aligned} \textsf {Test} (\textrm{X}):=\{f \in \ L^\infty ({\mathfrak {m}}) \cap \textsf {LIP} (\textrm{X})\cap D(\Delta ) \ | \ \Delta f \in W ^{1,2}(\textrm{X})\}. \end{aligned}$$

It turns out that $\textsf {Test} (\textrm{X})$ is dense in $W ^{1,2}(\textrm{X})$ and $\langle \nabla f,\nabla g \rangle \in W ^{1,2}(\textrm{X})$ for every $f,g \in W ^{1,2}(\textrm{X}).$

We recall the notion of Hessian for a test function as constructed in [47]: for any $f \in \textsf {Test} (\textrm{X})$ there exists $\textrm{Hess}(f) : [L^0(T\textrm{X})]^2 \rightarrow L^0({\mathfrak {m}})$, $L^0$-bilinear symmetric and continuous in the sense that $|\textrm{Hess}(f)(v,w)|\le |\textrm{Hess}(f)|_{OP} |v||w|$, for some (minimal) function $|\textrm{Hess}(f)|_{OP}\in L^2({\mathfrak {m}})$. Moreover $\textrm{Hess}(f)$ is characterized by the identity

$$\begin{aligned}&2\int h\textrm{Hess}(f)(\nabla f_1, \nabla f_2) {\mathrm d}{\mathfrak {m}}\\&\quad =- \int \langle \nabla f,\nabla f_1\rangle \text {div}(h\nabla f_1)+\nabla f,\nabla f_2 \text {div}(h\nabla f_2)- \text {div}(h\nabla f) \langle \nabla f_1, \nabla f_2\rangle {\mathrm d}{\mathfrak {m}}, \end{aligned}$$

for any choice of $h,f_1,f_2 \in \textsf {Test} (\textrm{X})$. The Hessian also induces an $L^0$-linear and continuous map $\textrm{Hess}(f) : L^0(T\textrm{X})\rightarrow L^0(T\textrm{X})$ characterized by

$$\begin{aligned} \langle \textrm{Hess}(f) (v),w\rangle =\textrm{Hess}(f)(v,w), \quad {\mathfrak {m}}\text {-a.e.}, \quad \forall w \in L^0(T\textrm{X}) \end{aligned}$$

and which satisfies $|\textrm{Hess}(f)(v)|\le |\textrm{Hess}(f)|_{OP}|v|.$ We recall also the following identity, which essentially is contained in [47, Prop. 3.3.22],

$$\begin{aligned} 2\textrm{Hess}(f) (\nabla f)=\nabla |\nabla f|^2, \quad \forall f \in \textsf {Test} (\textrm{X}). \end{aligned}$$

(2.13)

Combining [56, Theorem 5.1] with the recent [25], we have the following result (we refer to [47] for the definition of dimension and local base for a normed module).

Theorem 2.11

(Constancy of the dimension) Let $\textrm{X}$ be any $\textrm{RCD}(K,N)$ space with $K\in {{\,\mathrm{{\mathbb {R}}}\,}}$ and $N \in [1,\infty )$. Then there exists an integer $\dim (\textrm{X})\in [1,N]$ such that the tangent module $L^0(T\textrm{X})$ has constant dimension equal to $\dim (\textrm{X})$.

A corollary of this result is the existence of a global orthonormal base: $\{e_1,...,e_{\dim (\textrm{X})}\}\subset L^0(T\textrm{X})$ such that $\langle e_i,e_j\rangle =\delta _{i,j}$ ${\mathfrak {m}}$-a.e. for every $i,j=1,...,\dim (\textrm{X})$. In particular for every $v \in L^0(T\textrm{X})$ it holds that $v=\sum _{i=1}^{\dim (\textrm{X})}v_i e_i$, where $v_i:=\langle v,e_i\rangle$. Then, denoted by $(Hf)_{i,j}$ the functions $\textrm{Hess}(f) (e_i, e_j)\in L^0({\mathfrak {m}})$, for $i,j \in \{1,...,\dim (\textrm{X})\}$, we can write

$$\begin{aligned} \textrm{Hess}(f) (v)= \sum _{1\le i,j\le \dim (\textrm{X})}(Hf)_{i,j}v_j \,e_i,\quad \forall v \in L^0(T\textrm{X}). \end{aligned}$$

(2.14)

Moreover, we define the trace and Hilbert-Schmidt norm $\textsf { tr} \textrm{Hess}(f),|\textrm{Hess}(f)|_{HS} \in L^2({\mathfrak {m}})$ as

$$\begin{aligned}&|\textrm{Hess}(f)|_{HS}^2 :=\sum _{1\le i,j\le \dim (\textrm{X})}(Hf)_{i,j}^2, , \end{aligned}$$

(2.15)

$$\begin{aligned}&\textsf { tr} \textrm{Hess}(f):=\sum _{1\le i\le \dim (\textrm{X})}(Hf)_{i,i}, \end{aligned}$$

(2.16)

which are well defined in the sense that they do not depend on the choice of the base, as can be easily verified by a direct computation. It always holds that $|\textrm{Hess}(f)|_{OP}\le |\textrm{Hess}(f)|_{HS}$ ${\mathfrak {m}}$-a.e. (see [47, Sect. 3.2]).

In view of the above, we can restate Theorem 3.3 in [63] as follows (see also [14, 44] for the “basic version” of the Bochner inequality.)

Theorem 2.12

(Improved Bochner inequality) Let $\textrm{X}$ be any $\textrm{RCD}(K,N)$ space with $K\in {{\,\mathrm{{\mathbb {R}}}\,}}$ and $N \in [1,\infty )$. Then for any $f \in \textsf {Test} (\textrm{X})$, it holds that $|\nabla f|^2 \in D({\varvec{\Delta }})$ and

$$\begin{aligned} {\varvec{\Delta }}\left( \frac{|\nabla f|^2}{2}\right)\ge & {} \left( |\textrm{Hess}(f)|_{HS}^2+K|\nabla f|^2+\langle \nabla f,\nabla \Delta f\rangle \right. \nonumber \\{} & {} \left. + \frac{(\Delta f-\textsf {tr} \textrm{Hess}(f))^2}{N-\dim (\textrm{X})}\right) {\mathfrak {m}}, \end{aligned}$$

(2.17)

where $\frac{(\Delta f-\textsf {tr} \textrm{Hess}(f))^2}{N-\dim (\textrm{X})}$ is taken to be 0 in the case $\dim (\textrm{X})=N.$

See [73] for a proof of the following result.

Proposition 2.13

(Good cut-off functions) Let $\textrm{X}$ be an $\textrm{RCD}(K,N)$ space, $K \in {{\,\mathrm{{\mathbb {R}}}\,}}$ and $N\in (1,\infty )$. Then for every $0<r<R<+\infty$, every compact set P and every open set U containing P, such that $\text {diam}(U)<R$ and $\textsf {d} (P,U^c)>r$, there exists a function $\eta \in \textsf {Test} (\textrm{X})$ such that

• 1. $0\le \eta \le 1$, $\eta =1$ in P and $\mathop {\textrm{supp}}\nolimits \eta \subset U$,

• 2. $|\nabla \eta |+|\Delta \eta |\le C(R,r,N,K)$,

moreover C does not depend on R in the case $K=0.$

We recall that for any $N \in [1,\infty )$ the Euclidean N cone over a m.m.s. $(Z,{\mathfrak {m}}_Z,\textsf {d} _Z)$ is defined to be the space $C(Z):=[0,\infty )\times Z/ (\{0\}\times Z)$ endowed with the following distance and measure

$$\begin{aligned}&\textsf {d} _{C(Z)}((t,z),(s,z')):=\sqrt{s^2+t^2-2st\cos (\textsf {d} _Z(z,z')\wedge \pi )},\\&{\mathfrak {m}}_{C(Z)}:=t^{N-1}{\mathrm d}t\,\otimes \, {\mathfrak {m}}_Z. \end{aligned}$$

It was proven in [68] that

$$\begin{aligned} \begin{aligned}&\text {if the Euclidan } N\text {cone } (N\ge 2) \text { over a m.m.s. } Z \text { is an } \textrm{RCD}(0,N) \text { space, }\\&\text {then } \text {diam}(Z)\le \pi \text { and } Z \text { is an } \textrm{RCD}(N-2,N-1) \text { space.} \end{aligned} \end{aligned}$$

(2.18)

2.2.1 Localized Bochner Inequality

Along this subsection, $\Omega$ is an open subset of $\textrm{X}$. We define

This definition is motivated by the following observation:

(2.19)

This follows from the fact that for any $f \in \textsf {LIP} (\textrm{X})\cap D(\Delta )$, it holds that $|\nabla f|^2 \in W ^{1,2}(\Omega )$, which is essentially a consequence of the following inequality (see [47, Corollary 3.3.9])

$$\begin{aligned} \int |\textrm{Hess}(f)|_{HS}^2\, {\mathrm d}{\mathfrak {m}}\le \int (\Delta f)^2\, {\mathrm d}{\mathfrak {m}}-K\int |\nabla f|^2\, {\mathrm d}{\mathfrak {m}}, \quad \forall f \in \textsf {Test} \textrm{X}. \end{aligned}$$

(2.19) allows us to define for every $u \in \textsf {Test} _{\textsf {loc} }(\Omega )$ the functions $|\textrm{Hess}(u)|_{HS}, \textsf {tr} \textrm{Hess}(u)\in L^2_{\textsf {loc} }(\Omega )$ as

$$\begin{aligned}&|\textrm{Hess}(u)|_{HS}:=|\textrm{Hess}(\eta u)|_{HS},\, {\mathfrak {m}}\text {-a.e. in } \Omega ',\\&\textsf {tr} \textrm{Hess}(u):=\textsf {tr} \textrm{Hess}(\eta u),\, {\mathfrak {m}}\text {-a.e. in } \Omega ', \end{aligned}$$

for every $\eta \in \textsf {Test} (\textrm{X})$ with compact support in $\Omega$ and such that $\eta =1$ in $\Omega '\subset \subset \Omega$. This definition is well posed thanks to the locality property of the Hessian (see [47, Prop. 3.3.24]). Recall also from Proposition 2.13 that many such functions $\eta$ exist. The following follows directly from (2.17) and the above definitions.

Proposition 2.14

(Local improved Bochner Inequality) Let $u \in \textsf {Test} _\textsf {loc} (\Omega )$, then $|\nabla u|^2 \in D({\varvec{\Delta }},\Omega )$ and

$$\begin{aligned} {\varvec{\Delta }}{_{|_{\Omega }}} (|\nabla u|^2)\ge & {} 2\left( |\textrm{Hess}(u)|_{HS}^2+\frac{(\Delta u-\textsf {tr} \textrm{Hess}( u))^2}{N-\dim (\textrm{X})}\right. \nonumber \\{} & {} \left. +\langle \nabla u, \nabla \Delta u\rangle +K|\nabla u|^2 \right) {\mathfrak {m}}|_{\Omega }, \end{aligned}$$

(2.20)

where $\frac{(\Delta u-\textsf {tr} \textrm{Hess}(u))^2}{N-\dim (\textrm{X})}$ is taken to be 0 in the case $\dim (\textrm{X})=N.$

We conclude this subsection observing that

$$\begin{aligned} \text {if } u\in {\textsf {Test}}_\textsf {loc} (\Omega ), \text { then } |\nabla u|\in W ^{1,2}_{\textsf {loc} }(\Omega ) \text { and } \nabla |\nabla u|^2=2|\nabla u| \nabla |\nabla u|, \end{aligned}$$

(2.21)

indeed from (2.13) it follows that $|\nabla |\nabla u|^2||\nabla u|^{-1}\le 2|\textrm{Hess}(u)|_{OP},$ ${\mathfrak {m}}$-a.e. in $\{|\nabla u|>0\}$ and the claim follows applying Lemma 2.4 with $\alpha =1/2$ and $|\nabla u|^2$ in place of u.

2.2.2 (Sub)harmonic Functions in RCD Spaces

In this subsection,$(\textrm{X},\textsf {d} ,{\mathfrak {m}})$ is an $\textrm{RCD}(K,N)$ m.m.s. with $N \in [1,\infty )$.

Proposition 2.15

Let $\Omega$ be an open and bounded subset of $\textrm{X}$ and let $u \in D({\varvec{\Delta }}, \Omega )$ be such that ${\varvec{\Delta }}u \ge 0$. Then

• weak maximum principle: if u is upper semicontinuous in ${\bar{\Omega }}$, then

  $$\begin{aligned} \textrm{ess} \sup _{\Omega } u \le \sup _{\partial \Omega } u, \end{aligned}$$

• strong maximum principle:

  $$\begin{aligned} \text {if } u \in C(\Omega ),\, \Omega \text { is connected and } u(x)=\sup _{\Omega }u \text { for some } x \in \Omega , \text { then } u \text { is constant.} \end{aligned}$$

Proof

The result is a direct consequence of the maximum principle for harmonic functions, which holds in more general doubling metric measure spaces supporting a Poincaré inequality (see [19]). We report here a short justification which uses more direct arguments available in the $\textrm{RCD}$ from [59].

For the first part suppose by contradiction that $\text {ess} \sup _{\Omega } u > \sup _{\partial \Omega } u$. Then there exists $c \in {{\,\mathrm{{\mathbb {R}}}\,}}$ such that $\sup _{\partial \Omega } u<c<\text {ess} \sup _{\Omega }$, in particular from the upper semicontinuity and compactness of $\partial \Omega$ we have that $f-\min (f,c)\in W ^{1,2}(\textrm{X})$ with compact support in $\Omega$. From this point the proof continue exactly as in [59, Thm. 2.3].

For the second part,we apply the strong maximum principle in [59, Thm. 2.8] to a ball $B_r(x)\subset \Omega$, obtaining that u is constantly equal to $\sup _{\Omega }u$ in $B_r(x).$ In particular,the set $\{u=\sup _{\Omega }u\}\cap \Omega$ is open (and closed in $\Omega$) and thus must coincide with $\Omega$. $\square$

It is known that in $\textrm{RCD}(K,N)$ spaces harmonic functions are continuous. It follows for example from the existence of Harnack inequalities (see Appendix B) that they are locally Hölder. It turns out that they are actually locally Lipschitz and that the following gradient bound, analogous to the one due to Cheng and Yau [34] in the smooth setting, holds [66, Theorem 1.2]. We state it only in the case $N=0$ (see also [65] for an analogous result).

Theorem 2.16

(Gradient estimate) For every $N \in [1,\infty )$,there exists a positive constant $C=C(N)$ such that the following holds. Let $\textrm{X}$ be an $\textrm{RCD}(0,N)$ space and let $u\in D({\varvec{\Delta }},B_{2R}(x))$ be positive and harmonic in $B_{2R}(x)$, then

$$\begin{aligned} \left\| \frac{|Du|}{u} \right\| _{L^{\infty }(B_R(x))} \le \frac{C}{R}. \end{aligned}$$

(2.22)

Lemma 2.17

Let $\textrm{X}$ be an $\textrm{RCD}(0,N)$ space and let $\{u_i\}$ be a sequence of harmonic and continuous functions in $\Omega$. Suppose that

$$\begin{aligned} \sup _\Omega |u_i|<C, \end{aligned}$$

then there exists a subsequence $u_{i_k}$ that converges locally uniformly to a function u harmonic in $\Omega$.

Proof

The existence of a (non-relabelled) subsequence $u_i$ converging locally uniformly to a continuous function u follows from the gradient estimate (2.22) and Ascoli-Arzelà. It remains to prove that u is harmonic in $\Omega .$ Fix $\Omega ' \subset \subset \Omega$ and $\eta \in \textrm{Lip}_c(\Omega )$ with $\eta =1$ in $\Omega '$. Then $\eta u_i\in W ^{1,2}(\textrm{X})$ converges to $u\eta$ in $L^2(\Omega ').$ Moreover, again by (2.22), we have

$$\begin{aligned} \sup _i \Vert |\nabla (\eta u_i)|\Vert _{L^2({\mathfrak {m}})} <+\infty . \end{aligned}$$

In particular (recall that $W ^{1,2}(\textrm{X})$ is Hilbert) up to a subsequence $u_i\eta \rightharpoonup \eta u$ in $W ^{1,2}(\textrm{X})$ and therefore (from the locality of the gradient) $\int _{\Omega } \langle \nabla u, \nabla f\rangle \, {\mathrm d}{\mathfrak {m}}=0$ for every $f \in \textsf {LIP} _{c}(\Omega ')$ (see also [48, Prop 5.19] for a similar limiting argument). From the arbitrariness of $\Omega '$, we deduce both that $u\in W ^{1,2}_{\textsf {loc} }(\Omega )$ and that u is harmonic in $\Omega .$ $\square$

2.2.3 pmGH-Convergence

We will adopt the following definition of pointed-measure Gromov Hausdorff convergence of pointed metric measure spaces, which is equivalent to the usual one in the case when $\mathop {\textrm{supp}}\nolimits {\mathfrak {m}}_n=X_n$ and the measures ${\mathfrak {m}}_n$ are uniformly locally doubling. We refer to [54] for a discussion on the various notions of convergence of p.m.m. spaces and their relations.

Definition 2.18

(pointed-measure Gromov Hausdorff convergence) We say that the sequence $(\textrm{X}_n,\textsf {d} _n,{\mathfrak {m}}_n,x_n)$ of p.m.m.s. pointed-measure Gromov Hausdorff-converges (pmGH-converges in short) to the p.m.m.s. $(\textrm{X}_\infty ,\textsf {d} _\infty ,{\mathfrak {m}}_\infty ,x_\infty )$, if there are sequences $R_n\uparrow +\infty$, $\varepsilon _n\downarrow 0$ and Borel maps $f_n:X_n\rightarrow X_\infty$ such that

• (1) $f_n( x_n)=x_\infty$,

• (2) $\sup _{x,y\in B_{R_n}(x_n)}|\textsf {d} _n(x,y)-\textsf {d} _\infty (f_n(x),f_n(y))|\le \varepsilon _n$,

• (3) the $\varepsilon _n$-neighbourhood of $f_n(B_{R_n}( x_n))$ contains $B_{R_n-\varepsilon _n}( x_\infty )$,

• (4) for any $\varphi \in C_{bs}(X_\infty )$ it holds $\lim \limits _{n\rightarrow \infty }\int \varphi \circ f_n\,{\mathrm d}{\mathfrak {m}}_n=\int \varphi \,{\mathrm d}{\mathfrak {m}}_\infty$.

It is proven in [54] that there exists a distance $\textsf {d} _{pmGH}$ that metrizes the pmGH-convergence for the class of $\textrm{RCD}(K,N)$ spaces with $K\in {{\,\mathrm{{\mathbb {R}}}\,}}$ and $N<+\infty$ fixed, and more generally for every family of uniformly locally doubling metric measure spaces.

It will be also useful to recall the so-called extrinsic approach (see [54]) to pmGH-convergence, valid in the case of uniformly doubling and geodesic m.m.s.: the pmGH-convergence can be realized by a proper metric space $(Y,\textsf {d} )$ where $\textrm{X}_i,\textrm{X}_\infty$ are subsets of Y such that $\textsf {d} _Y{_{|_{\textrm{X}_i\times \textrm{X}_i}}}=\textsf {d} _i,\textsf {d} _Y{_{|_{\textrm{X}_\infty \times \textrm{X}_\infty }}}=\textsf {d} _\infty$, $\textsf {d} _{Y}(x_i,x_\infty )\rightarrow 0$, ${\mathfrak {m}}_i\rightharpoonup {\mathfrak {m}}_\infty$ in duality with $C_{bs}(Y)$ and $\textsf {d} ^Y_H(B^{\textrm{X}_n}_R(x_n),B^{\textrm{X}_\infty }_R(x_\infty ))\rightarrow 0$ for every $R>0.$

After the works in [9, 54, 71, 80, 81] and in view of the Gromov compactness theorem [57, Sect. 5.A], the following fundamental compactness result for RCD spaces is known.

Proposition 2.19

Suppose $(\textrm{X}_n,\textsf {d} _n,{\mathfrak {m}}_n,x_n)$ are $\textrm{RCD}(K_n,N)$ spaces with $N\in [1,\infty )$, $K_n\rightarrow K\in {{\,\mathrm{{\mathbb {R}}}\,}}$ and ${\mathfrak {m}}(B_1(x_n))\in [v^{-1},v]$ for some $v>1$. Then there exists a subsequence $(\textrm{X}_{n_k},\textsf {d} _{n_k},{\mathfrak {m}}_{n_k},x_{n_k})$ that pmGH-converges to an $\textrm{RCD}(K,N)$ space $(X_\infty ,\textsf {d} _\infty ,{\mathfrak {m}}_\infty ,x_\infty )$.

2.2.4 Stability Results Under pmGH-Convergence

In this subsection, $(X_i,\textsf {d} _i,{\mathfrak {m}}_i,x_i)$ is a pmGH-converging sequence of $\textrm{RCD}(K,N)$ spaces, $N<+\infty$, and $(Y,\textsf {d} )$ is a proper metric space which realizes such convergence through the extrinsic approach (see the previous subsection).

Definition 2.20

(Locally uniform / uniform convergence) Let $f_i: \textrm{X}_i \rightarrow {{\,\mathrm{{\mathbb {R}}}\,}}$, $f_\infty : \textrm{X}_\infty \rightarrow {{\,\mathrm{{\mathbb {R}}}\,}}$. We say that $f_i$ converges locally uniformly to $f_\infty$ if for every $y \in \textrm{X}_\infty$ and every sequence $y_i\in \textrm{X}_i$ such that $\textsf {d} _Y(y_i,y)\rightarrow 0$ it holds that $\lim _i f_i(y_i)=f_\infty (y)$.

We say that $f_i$ converges uniformly to $f_\infty$ if for every $\varepsilon >0$ there exists $\delta >0$ such that $|f_i(y_i)-f_\infty (y)|<\varepsilon$ for every $i \ge \delta ^{-1}$ and every $y_i$ such that $\textsf {d} _Y(y_i,y)<\delta .$

We point out that in the case of a fixed proper metric space the two notions of convergence in the above definition coincide, respectively, with the usual locally uniform and uniform convergence.

The following is a version of the Ascoli-Arzelà theorem for varying metric spaces (see also [83, Prop. 27.20]). The proof can be achieved arguing as in case of a fixed (proper) metric space and we will skip it.

Proposition 2.21

Let $f_i: \textrm{X}_i \rightarrow {{\,\mathrm{{\mathbb {R}}}\,}}$ be equi-Lipschitz, equibounded functions with $\mathop {\textrm{supp}}\nolimits f_i \subset B_R(x_i)$, then there exists a subsequence that converges uniformly to a Lipschitz function $f:\textrm{X}_\infty \rightarrow {{\,\mathrm{{\mathbb {R}}}\,}}.$

Definition 2.22

(Weak/strong $L^2$-convergence) A sequence of functions $f_i \in L^2({\mathfrak {m}}_i)$ converges weakly in $L^2$ to a function $f\in L^2({\mathfrak {m}}_\infty )$ if $f_i{\mathfrak {m}}_i\rightharpoonup f{\mathfrak {m}}_\infty$ in duality with $C_{bs}(Y)$ and $\sup _i \Vert f_i\Vert _{L^2({\mathfrak {m}}_i)}<+\infty .$

A sequence of functions $f_i \in L^2({\mathfrak {m}}_i)$ converges strongly in $L^2$ to a function $f\in L^2({\mathfrak {m}}_\infty )$ if it converges weakly in $L^2$ to f and $\lim _i \Vert f_i\Vert _{L^2({\mathfrak {m}}_i)}=\Vert f\Vert _{L^2({\mathfrak {m}}_\infty )}.$

In the following proposition we collect some basic facts about strong and weak $L^2$ convergence.

Proposition 2.23

• (a) If $f_i \in L^2({\mathfrak {m}}_i)$ converge strongly in $L^2$ to $f \in L^2({\mathfrak {m}}_\infty )$ and $f_i$ have uniformly bounded support, then $\varphi \circ f_i$ converge strongly in $L^2$ to $\varphi \circ f$, for every $\varphi \in C({{\,\mathrm{{\mathbb {R}}}\,}})$ such that $\varphi (0)=0$ and $|\varphi (t)|\le C(1+|t|)$ for some positive constant $C>0$.

• (b) If $f_i,g_i \in L^2({\mathfrak {m}}_i)$ are uniformly bounded in $L^\infty ({\mathfrak {m}}_i)$ and converge strongly in $L^2$, respectively, to $f,g \in L^2({\mathfrak {m}}_\infty )$, then $f_ig_i$ converge strongly in $L^2$ to fg.

• (c) If $f_i : \textrm{X}_i \rightarrow {{\,\mathrm{{\mathbb {R}}}\,}}$, with $\mathop {\textrm{supp}}\nolimits f_i \subset B^{\textrm{X}_i}_{R}(x_i)$, converge uniformly to a bounded function $f: \textrm{X}_\infty \rightarrow {{\,\mathrm{{\mathbb {R}}}\,}}$ then $f_i$ converge strongly in $L^2$ to f.

Proof

(a) follows from the characterization of $L^2$-strong convergence as weak convergence of the graphs (see [8, Sect. 5.2], [54] and also [10, Remark 5.2]). Indeed from [54, (6.6)] $L^2$-convergence is equivalent to weak convergence of $(\textsf {id} \times f_i)_*{\mathfrak {m}}_i$ to $(\textsf {id} \times f)_*{\mathfrak {m}}_\infty$ in duality with $\zeta \in C(Y\times {{\,\mathrm{{\mathbb {R}}}\,}})$ satisfying $|\zeta (y,t)|\le \psi (y)+C|t|^2$. Then the claim follows observing that: under our assumptions (since $\varphi (0)=0$), testing convergence against such $\zeta$’s is equivalent to test against $\eta \zeta$, where $\eta \in C_{bs}(Y)$ is such that $\eta \equiv 1$ in the supports of $f_i,f_\infty$ and moreover $|\eta (y)\zeta (y,\varphi (t))|\le |\eta (y)|(\psi (y)+C(1+|t|)^2)$ for every $\varphi \in C({{\,\mathrm{{\mathbb {R}}}\,}})$ as in the hypotheses.

The proof of (b) can be found in [10, Proposition 3.3].

(c) is a consequence of [12, Prop 3.2]; however, we include an alternative direct argument. We first observe that from the properness of Y it follows that the functions $f_i$ are equibounded, moreover they have uniform bounded support by hypothesis. Therefore we can apply the generalized version of Fatou lemma for varying measure (see e.g. [41, Lemma 2.5]), first to $f_i \varphi$ and then to $-f_i\varphi$ to obtain that $\int f_i\varphi \,{\mathrm d}{\mathfrak {m}}_i\rightarrow \int f \varphi \, {\mathrm d}{\mathfrak {m}},$ where $\varphi$ is an arbitrary function in $C_{bs}(Y).$ Applying the same lemma also the functions $f_i^2,-f_i^2$ we deduce that $\int f_i^2\, {\mathrm d}{\mathfrak {m}}_i\rightarrow \int f^2\,{\mathrm d}{\mathfrak {m}}$, concluding the proof. $\square$

We will also need the following result about stability of Laplacian and gradient with respect to strong $L^2$ convergence. Here $\Delta _i$ (resp. $\Delta _\infty$) represents the Laplacian operator in $\textrm{X}_i$ (resp. $\textrm{X}_\infty$) and $\nabla _i$ (resp. $\nabla _\infty$) represents the gradient operator in $\textrm{X}_i$ (resp. $\textrm{X}_\infty$).

Theorem 2.24

[11, Theorems 2.7, 2.8] Let $f_i \in D(\Delta _i)$ be such that

$$\begin{aligned} \sup _i \Vert f_i\Vert _{L^2({\mathfrak {m}}_i)}+\Vert \Delta _i f_i\Vert _{L^2({\mathfrak {m}}_i)}<+\infty \end{aligned}$$

and assume that $f_i$ converge strongly in $L^2$ to f. Then $f \in D(\Delta _\infty )$, $\Delta _i f\rightarrow \Delta _\infty f$ weakly in $L^2$ and $|\nabla _i f_i|\rightarrow |\nabla _\infty f|$ strongly in $L^2$.

We conclude this subsection with the following technical result.

Lemma 2.25

[10, Lemma 5.8] Let $f_i \in W ^{1,2}(\textrm{X}_i)$ be such that $\sup _i \Vert |\nabla f_i|\Vert _{L^2({\mathfrak {m}}_i)}<+\infty$ and converging strongly in $L^2$ to $f \in W ^{1,2}(\textrm{X}_\infty )$, then for any $A\subset Y$ open it holds

$$\begin{aligned} \int _A |\nabla _\infty f|^2\, {\mathrm d}{\mathfrak {m}}_\infty \le \liminf _{i} \int _A |\nabla _i f |^2\, {\mathrm d}{\mathfrak {m}}_i. \end{aligned}$$

2.3 Regular Lagrangian Flows

In the work of Ambrosio and Trevisan [16],it was extended the theory of flows for Sobolev vector fields (recall [43] and [2]) to very general metric measure spaces and in particular for $\textrm{RCD}(K,\infty )$ spaces.

We restate here, using the language we introduced in the previous sections, their main results of existence and uniqueness.

Definition 2.26

Let $v_t:[0,T]\rightarrow L^2(T\textrm{X})$ be Borel, we say that a map $F: [0,T]\times X \rightarrow X$ is a Regular Lagrangian Flow associated to $v_t$ if the following are satisfied:

• 1. There exists $C>0$ such that

  $$\begin{aligned} {F_t}_*{\mathfrak {m}}\le C {\mathfrak {m}}, \quad \text { for every } t \in [0,T], \end{aligned}$$

  (2.23)

• 2. for ${\mathfrak {m}}$-a.e. $x\in X$ the function $[0,T]\ni t \mapsto F_t(x)$ is continuous and satisfies $F_0(x)=x.$

• 3. for every $f \in \textsf {Test} (X)$ it holds that for ${\mathfrak {m}}$-a.e. $x\in X$ the function $(0,T)\ni t \mapsto f\circ F_t(x)$ is absolutely continuous and

  $$\begin{aligned} \frac{{\mathrm d}}{{\mathrm d}t} f\circ F_t(x)=\langle \nabla f, v_t\rangle \circ F_t(x), \quad \text { for a.e. } t \in (0,T). \end{aligned}$$

  (2.24)

Notice that in (2.24),we are implicitly choosing for every $t \in (0,T)$ a Borel representative of $\langle \nabla f,v_t\rangle$, however,(2.23) ensures that the validity of item 3 in Definition 2.26 is independent of this choice.

Observe also that in Definition 2.26, we are assuming that the map F is pointwise defined;however, the definition is stable under modification in a negligible set of trajectories in the following sense. If $F_t(x)$ is a Regular Lagrangian Flow for $v_t$ (as in Definition 2.26) and for ${\mathfrak {m}}$-a.e. x, $\tilde{F}_t(x)=F_t(x)$ holds for every $t\in [0,T]$, for some map ${\tilde{F}} : [0,T]\times X \rightarrow X$ then ${\tilde{F}}$ is also a Regular Lagrangian Flow for $v_t$. In any case to avoid technical issues, in our discussion we prefer to fix a pointwise defined representative for the flow map F.

Remark 2.27

If $F_t$ is a regular Lagrangian flow for a vector field $v_t$, then for ${\mathfrak {m}}$-a.e. x it holds that the curve $[0,T] \ni t \mapsto F_t(x)$ is absolutely continuous and its metric speed is given by

$$\begin{aligned} |\dot{F_t(x)}|=|v_t|\circ F_t(x), \quad \text { a.e. } t\in [0,T]. \end{aligned}$$

This follows from [16, Lemmas 7.4 and 9.2]. Observe that this statement is independent of the chosen representative of $|v_t|$, thanks to (2.23).$\square$

We will see below in Theorem 2.29 that the existence and uniqueness of a Regular Lagrangian Flow is linked to the existence and uniqueness of a solution to the continuity equation [50]:

Definition 2.28

Let $v_t:[0,T]\rightarrow L^2(T\textrm{X})$ be Borel and $t \mapsto \mu _t \in {\mathscr {P}}(\textrm{X})$, $t \in [0,T]$ be also a Borel map. Suppose also that $\Vert |v_t|\Vert _{L^2({\mathfrak {m}})}\in L^{1}(0,T)$ and $\mu _t \le C {\mathfrak {m}}$ for every $t \in [0,T]$ and some positive constant C. We say that $\mu _t$ is a weak solution of the continuity equation

$$\begin{aligned} \frac{{\mathrm d}}{{\mathrm d}t}\mu _t+\textrm{div}(v_t \mu _t)=0, \end{aligned}$$

with initial datum $\mu _0,$ if for every $f\in \textrm{Lip}_{bs}(\textrm{X})$ the function $[0,T]\ni t \mapsto \int f \,{\mathrm d}\mu _t$ is absolutely continuous and

$$\begin{aligned} \frac{{\mathrm d}}{{\mathrm d}t}\int f\, {\mathrm d}\mu _t = \int \langle \nabla f,v_t\rangle \, {\mathrm d}\mu _t\, \quad \quad \text { for } a.e.\,t\in (0,T). \end{aligned}$$

We refer to [47, Sect. 3.4] for the definition of the space $W ^{1,2}_{C}(T\textrm{X})$ and the object $\nabla _{sym}v\in L^2(T^{2\otimes }\textrm{X})$. For our purposes it sufficient to know that for any $f \in \textsf {Test} (\textrm{X})$ we have $\nabla f \in W ^{1,2}_C(T\textrm{X}),$ with $\nabla _{sym}\nabla f=\nabla (\nabla f)= \textrm{Hess}(f)^{\sharp }$ (see [47, Sect. 3.2]).

Theorem 2.29

[16] Let $v_t:[0,T]\rightarrow L^2(T\textrm{X})$ be Borel and such that $v_t \in D(\textrm{div})$ for every $t\in [0,T]$. Assume furthermore that $\Vert |v_t|\Vert _{L^2({\mathfrak {m}})}\in L^{1}(0,T)$, $\Vert \textrm{div}(v_t)^-\Vert _{L^\infty }\in L^\infty (0,T)$ and $\Vert \nabla _{sym} v_t\Vert _{L^2(T^{\otimes 2}X)}\in L^1(0,T)$. Then

• 1. there exists a unique Regular Lagrangian flow $F_t$ associated to $v_t$,

• 2. for every initial datum $\mu _0 \in {\mathscr {P}}(\textrm{X})$ with $\mu _0\le C {\mathfrak {m}}$ there exists a unique weak solution $\mu _t$ to the continuity equation and it is given by $\mu _t:={F_t}_*{\mu _0}.$

We remark that the uniqueness of the Regular Lagrangian Flow in Theorem 2.29 has to be intended up to modification in a negligible set of trajectories, as discussed above.

Remark 2.30

Let $v_t$ be as in Theorem 2.29 and autonomous, i.e. $v_t\equiv v$ for some $v \in D(\textrm{div})\cap W^{1,2}_{C}(TX)$ with $\textrm{div}(v)^{-}\in L^\infty ({\mathfrak {m}}).$ Then, thanks to the uniqueness given by Theorem 2.29, the Lagrangian flow $F_t$ relative to v can be extended uniquely (up to a set of negligible trajectories) to a map $F: [0,\infty )\times X \rightarrow X$ which satisfies the following group property

$$\begin{aligned} F_{s}\circ F_t= F_{s+t}, \quad {\mathfrak {m}}\text {-a.e}. \end{aligned}$$

(2.25)

for every $s,t \in [0,\infty ).$

Moreover if also $\textrm{div}(v)\in L^\infty ({\mathfrak {m}})$, it can be shown (see for example [58, Lemma 3.18]) that, denoting by $F^{-v}_{t}$ the Lagrangian flow relative to $-v$ (which exists unique for all times $t\ge 0$, thanks to Theorem 2.29 and the previous observation)

$$\begin{aligned} F^{-v}_t\circ F_t = \textsf {id} , \quad {\mathfrak {m}}\text {-a.e}, \end{aligned}$$

for every $t\ge 0.$ Hence setting $F_{-t}:=F^{-v}_t$ we can extend F to $F:(-\infty ,+\infty )\times X\rightarrow X$, for which (2.25) is satisfied for every $s,t \in {{\,\mathrm{{\mathbb {R}}}\,}}.$ $\square$

2.3.1 Functions of Bounded Variation

We recall the definition of function of bounded variation on a metric measure space. For a detailed treatment of this topic see for example [72] and [5].

Definition 2.31

(Functions of bounded variation) We say that function $f \in L^1({\mathfrak {m}})$ belongs to the space $\textsf {BV} (\textrm{X})$ of functions of bounded variation if there exists a sequence $f_n\in \textrm{Lip}_{\textsf {loc} }(\textrm{X})$ such that $f_n \rightarrow f$ in $L^1({\mathfrak {m}})$ and

$$\begin{aligned} \limsup _{n \rightarrow +\infty } \int \textrm{lip}f_n \,{\mathrm d}{\mathfrak {m}}<+\infty , \end{aligned}$$

(where $\textrm{lip}f_n$ was defined in (2.1)). By localizing this construction we also define

$$\begin{aligned} \Vert Df\Vert (A):=\inf \left\{ \liminf _{n} \int _A \textrm{lip}f_n \,{\mathrm d}{\mathfrak {m}}\ : \ f_n\in \textrm{Lip}_{\textsf {loc} }(A), \, f_n \rightarrow f \text { in } L^1(A) \right\} , \end{aligned}$$

for any $A\subset \textrm{X}$ open. It is proven in [72] (at least for doubling m.m.s.) that this set function is the restriction to open sets of a finite and positive Borel measure on $\textrm{X}$ that we call total variation of f and still denote by $\Vert Df\Vert .$

It is proven in [49, Remark 3.5], in the case of proper $\mathrm{RCD(K,\infty )}$ spaces (and thus also for any $\textrm{RCD}(K,N)$, with $N<+\infty$), the equivalence between the total variation and the weak upper gradient, meaning that if $f \in \textsf {BV} (\textrm{X})\cap \textrm{Lip}_{\textsf {loc} }(\textrm{X})$, then

$$\begin{aligned} \Vert Df\Vert =|\nabla f|{\mathfrak {m}}. \end{aligned}$$

(2.26)

Definition 2.32

(Sets of finite perimeter) Given a Borel set $E\subset \textrm{X}$ and any open set $A\subset \textrm{X}$ we define the perimeter $\textrm{Per}(A,E)$ as

$$\begin{aligned} \textrm{Per}(A,E):=\inf \left\{ \liminf _{n} \int _A \textrm{lip}f_n {\mathrm d}\,{\mathfrak {m}}\ : \ f_n \in \textrm{Lip}_{\textsf {loc} }(A), \, f_n \rightarrow {\chi }_E \text { in } L^1_{\textsf {loc} }(A) \right\} . \end{aligned}$$

We say that E has finite perimeter if $\textrm{Per}(E,\textrm{X})<+\infty$. In this case, it can be proved that the set function $\textrm{Per}(E,A)$ is the restriction to open sets of a finite and positive Borel measure on $\textrm{X}$ that we still denote by $\textrm{Per}(E,.)$.

We will need the following variant of the coarea formula, which follows from (2.26), the standard coarea formula for m.m.s. (see [72, Remark 4.3]) and a simple truncation argument.

Proposition 2.33

(Coarea formula) Let $\textrm{X}$ be an $\textrm{RCD}(K,N)$ m.m.s with $N<+\infty$ and let $\Omega \subset \textrm{X}$ open. Let $u \in \textsf {LIP} _{\textsf {loc} }(\Omega )$ be positive and such that $u^{-1}([a,b])$ is compact in $\Omega$, for every $[a,b] \subset (0,1)$. Then $\{u<t\}$ has finite perimeter for a.e. $t \in (0,1)$ and for any $f : \Omega \rightarrow [-\infty ,+\infty ]$ Borel and in $L^1_{\textsf {loc} }(\Omega ,|\nabla u|{\mathfrak {m}}{_{|_{\Omega }}})$ it holds that

$$\begin{aligned} \int _{\Omega } \varphi (u) f\, |\nabla u| \,{\mathrm d}{\mathfrak {m}}= & {} \int _{\Omega } \varphi (t) \int f \,{\mathrm d}\textrm{Per}(\{u<t\},.)\, {\mathrm d}t,\nonumber \\{} & {} \quad \forall \varphi :[0,1] \rightarrow {{\,\mathrm{{\mathbb {R}}}\,}}\text { Borel, with } \mathop {\textrm{supp}}\nolimits \varphi \subset (0,1). \end{aligned}$$

(2.27)

3 Nonparabolic RCD Spaces

In this section, we introduce the notion of nonparabolic $\textrm{RCD}(0,N)$ space: this is the natural setting to study problem (P), indeed already for smooth manifolds the existence of a solution to (P) implies that the manifold is nonparabolic (see for example Theorem 2.3 in [6]).

We recall that a (non-compact) Riemannian manifold is said to be nonparabolic if it admits a positive global Green function. It has been proved by Varopouls [82] that in the case $\textsf {Ric} \ge 0$ the nonparabolicity is equivalent to (3.1). This motivates the following.

Definition 3.1

(Nonparabolic RCD space) Let $(\textrm{X},\textsf {d} ,{\mathfrak {m}})$ be an $\textrm{RCD}(0,N)$ space with $N<+\infty .$ We say that $\textrm{X}$ is nonparabolic if

$$\begin{aligned} \int _1^{+\infty } \frac{s}{{\mathfrak {m}}(B_s(x))}\, {\mathrm d}s<+\infty , \quad \text { for every } x \in X. \end{aligned}$$

(3.1)

Observe that the above quantity is finite for one $x\in X$ if and only if it is finite for all $x\in X.$

We point out that condition (3.1) in the context of $\textrm{RCD}$ spaces was already introduced in [25].

Remark 3.2

It follows immediately from the Bishop-Gromov inequality that if $\textrm{X}$ is a nonparabolic $\textrm{RCD}(0,N)$ space then it is non-compact and $N>2.$ $\square$

In the following two subsections,we develop the two main features of a nonparabolic $\textrm{RCD}$ space that we will need in this note: the first is the existence of a Green function, which also provides an explicit solution to (P); the second is that the number of ends is equal to one (see Definition 3.6).

3.1 The Green Function

It turns out that on a nonparabolic RCD space, it can be given a notion of positive global Green function. Following [25] we define the Green function $G :\textrm{X}\times \textrm{X}\rightarrow [0,+\infty ]$ as

$$\begin{aligned} G(x,y):=\int _0^\infty p_t(x,y)\, {\mathrm d}t. \end{aligned}$$

We also set $G_x(y):=G(x,y).$ For any $\varepsilon >0$ we also define the quasi Green function $G^{\varepsilon } :\textrm{X}\times \textrm{X}\rightarrow [0,+\infty ]$ as

$$\begin{aligned} G^{\varepsilon }(x,y):=\int _{\varepsilon }^\infty p_t(x,y)\, {\mathrm d}t \end{aligned}$$

(3.2)

and as above we set $G^{\varepsilon }_x(y):=G^{\varepsilon }(x,y).$ It is proved in [25, Lemma 2.5] that $G^{\varepsilon }_x\in \textsf {LIP} (\textrm{X})\cap D({\varvec{\Delta }})$ with ${\varvec{\Delta }}G_x^{\varepsilon }=-p_{\varepsilon }(x,y){\mathfrak {m}}$, in particular $G_x^{\varepsilon }$ is superharmonic in the whole $\textrm{X}$.

Proposition 3.3

(Estimates for the Green functions, [25, Prop. 2.3], see also [62]) Let $(\textrm{X},\textsf {d} ,{\mathfrak {m}})$ be a nonparabolic $\textrm{RCD}(0,N)$ m.m.s. Then there exists a constant $C=C(N)>1$ such that

$$\begin{aligned} \frac{1}{C}\int _{\textsf {d} (x,y)}^\infty \frac{s}{{\mathfrak {m}}(B_s(x))} \,{\mathrm d}s \le G(x,y)\le C\int _{\textsf {d} (x,y)}^\infty \frac{s}{{\mathfrak {m}}(B_s(x))} \,{\mathrm d}s, \quad \forall x,y \in \textrm{X}. \end{aligned}$$

(3.3)

Proposition 3.4

Let $\textrm{X}$ be a nonparabolic $\textrm{RCD}(0,N)$ m.m.s.. Then $G_x$ is positive, continuous and harmonic in $\textrm{X}\setminus \{x\},$ for every $x\in \textrm{X}.$

Proof

Fix $R,\delta >0$ with $R>\delta$ and define $A_{\delta ,R}=B_R(x)\setminus {{\bar{B}}}_{\delta }(x).$ It enough to prove that $G_x$ is harmonic on $A_{\delta ,R}$. Recall that $G^{\varepsilon }_x\in \textsf {LIP} (\textrm{X})\cap D({\varvec{\Delta }})$ with ${\varvec{\Delta }}G_x^{\varepsilon }=-p_{\varepsilon }(x,y){\mathfrak {m}}$ and that $G_x^{\varepsilon }\rightarrow G_x$ in $L^1_{\textsf {loc} }(\textrm{X})$. We now observe that from the heat kernel bounds (2.12) we have $\sup _{t\in (0,1)} \Vert p_t(x,.)+|\nabla p_t(x,.)|\Vert _{L^\infty (A_{\delta ,R})}<+\infty$ and that $\sup _{\varepsilon >0} \Vert G^{\varepsilon }_x\Vert _{L^\infty (A_{\delta ,R})} <+\infty .$ From this, following the arguments in the proof of [25, Lemma 2.5], we can prove that the sequence $G_x^{\varepsilon }$ is Cauchy in $W ^{1,2}(A_{\delta ,R}).$ In particular $G_x^{\varepsilon }\rightarrow G_x$ in $W ^{1,2}(A_{\delta ,R})$ and, since $p_t(x,.)\rightarrow 0$ uniformly in $A_{\delta ,R}$, we deduce that $G_x$ is harmonic in $A_{\delta ,R}$ (cf. with Lemma 4.7).

We pass to the continuity of $G_x$ in $\textrm{X}\setminus \{x\}$. We first observe that from (2.12) we have

$$\begin{aligned} \Vert |\nabla p_t(x,.)|\Vert _{L^\infty (\textrm{X}\setminus B_{\delta }(x))}\le C(N)t^{-1/2} {\mathfrak {m}}(B_{\sqrt{t}}(x))^{-1}e^{\frac{-\delta ^2}{5t}}=:\beta (t,\delta ) \end{aligned}$$

and that, thanks to the Bishop-Gromov inequality and the nonparabolicity assumption, $\int _0^\infty \beta (t,\delta )\, {\mathrm d}t<+\infty ,$ for every $\delta >0.$ Therefore from the continuity of $p_t$ and Proposition 2.10 we deduce that

$$\begin{aligned} \limsup _{z\rightarrow y}|G_x(z)-G_x(y)|\le \left( \int _0^\infty \beta (t,\delta )\, {\mathrm d}t\right) \limsup _{z\rightarrow y} \textsf {d} (z,y)=0, \quad \forall y \in \textrm{X}\setminus {{\bar{B}}}_{\delta }(x_0), \end{aligned}$$

from which the claimed continuity follows. $\square$

As anticipated, the Green function provides a solution to (P), in particular we have the following:

Corollary 3.5

Let $\textrm{X}$ be a nonparabolic $\textrm{RCD}(0,N)$ and let $\Omega \subset \textrm{X}$ be open, unbounded with $\partial \Omega$ bounded. Then for every $x_0 \in \Omega ^c$ there exists $\lambda >0$ such that $\lambda G_{x_0}$ is a solution to (P).

Proof

From Proposition 3.4,it follows that $G_{x_0}$ is harmonic and continuous in $\textrm{X}\setminus \{x_0\}$, moreover from (3.3), we have that $G_{x_0}(x)\rightarrow 0$ as $\textsf {d} (x,x_0)\rightarrow +\infty$. Therefore, since $\partial \Omega$ is bounded and $G_{x_0}$ is positive, we can simply take $\lambda =(\min _{\partial \Omega } G_{x_0})^{-1}.$ $\square$

3.2 Number of Ends

Let us introduce the notion of ends for a metric space. The definition is usually given for manifolds;however, since the definition is purely metric, it carries over verbatim to metric spaces.

Definition 3.6

(Number of ends of a metric space) Let $(X,\textsf {d} )$ be a metric space and $k \in {\mathbb {N}}$. We say that $\textrm{X}$ has k ends if both the following are true:

• 1. for any K compact, $\textrm{X}\setminus K$ has at most k unbounded (topological) connected components,

• 2. there exists $K'$ compact such that $\textrm{X}\setminus K'$ has exactly k unbounded (topological) connected components.

Remark 3.7

Recall that in a length metric space, the notions of topological/ path/Lipschitz path connectedness are all equivalent (the standard argument used for manifolds carries over).

The following result generalizes to the nonsmooth setting a well-known result for Riemannian manifolds.

Proposition 3.8

Suppose $(\textrm{X},\textsf {d} ,{\mathfrak {m}})$ is a noncompact $\textrm{RCD}(0,N)$ space, $N\in [1,\infty )$. Then exactly one of the following holds:

• (i) $\textrm{X}$ is a cylinder, meaning that X is isomorphic to the product $({{\,\mathrm{{\mathbb {R}}}\,}}\times \textrm{X}',\textsf {d} _{Eucl}\times \textsf {d} ',{\mathcal {L}}^1\otimes {\mathfrak {m}}')$, where $(X',\textsf {d} ',{\mathfrak {m}}')$ is a compact $\textrm{RCD}(0,N-1)$ m.m.s. if $N\ge 2$ and a single point if $N\in [1,2)$,

• (ii) for every C bounded subset of X, there exists $R>0$ such that the following holds: for every couple of points $x,y \in X$ satisfying $\textsf {d} (x,C),\textsf {d} (y,C)>R$ there exists $\gamma \in \textrm{Lip}([0,1],\textrm{X})$ such that $\gamma (0)=x,\, \gamma (1)=y$, $\gamma \subset X\setminus C$ and $\textrm{Lip}(\gamma )\le 5\textsf {d} (x,y).$ In particular $\textrm{X}$ has one end.

Proof

We closely follow [6, Prop. 2.10]. Suppose that ii) does not hold, it follows that there exists a bounded set C, two sequences of points $(x_k),(y_k) \subset X$ and geodesics $(\gamma _k)$ between $x_k$ and $y_k$ such that $\textsf {d} (x_k,C),\textsf {d} (y_k,C)\rightarrow +\infty$ and $\gamma _k$ intersects C for all k. Since $\textrm{X}$ is proper and C is compact, with a compactness argument (assuming all the $\gamma _k$ parametrized by arc length) we deduce that $\textrm{X}$ contains a line. In particular, begin $\textrm{X}$ an $\textrm{RCD}(0,N)$ space, by the splitting theorem [45], we infer that $\textrm{X}$ is isomorphic to the product $({{\,\mathrm{{\mathbb {R}}}\,}}\times \textrm{X}',\textsf {d} _{Eucl}\times \textsf {d} ',{\mathcal {L}}^1\otimes {\mathfrak {m}}')$, where $(X',\textsf {d} ',{\mathfrak {m}}')$ is an $\textrm{RCD}(0,N-1)$ space if $N\ge 2$ and a single point if $N\in [1,2)$. It remains to prove that $\textrm{X}'$ is bounded.

Suppose it is not. We claim that this would imply the validity of (ii) and thus a contradiction. Indeed suppose that $C\subset X$ is bounded, then $C\subset B_R(p)$ for some $R>0$ and $p\in \textrm{X}$. It is enough to show that for every couple of points $x_0,x_1 \in B_{2R}(p)^c$ there exists $\gamma \in \textrm{Lip}([0,1],\textrm{X})$ joining them and with image contained in $B_R(p)^c.$ In the case $\textsf {d} (x_0,x_1)\le R$,we conclude immediately by taking a geodesic between $x_0$ and $x_1$, hence we can suppose that $\textsf {d} (x_0,x_1)>R$. Identifying $\textrm{X}$ with ${{\,\mathrm{{\mathbb {R}}}\,}}\times \textrm{X}'$ we have that $p=({{\bar{t}}},x'),$ $x_i=( t_i,x'_i)$ $i=0,1,$ for some $\bar{t},t_0,t_1 \in {{\,\mathrm{{\mathbb {R}}}\,}}$ and $x', x_0', x_1'\in \textrm{X}'$. Hence $I \times B'\subset B_{2R}(p)$, where $I:=[{{\bar{t}}}-R,{{\bar{t}}}+R]$ and $B':=B_R(x').$ In particular $x_i \in (I \times B')^c$, $i=0,1$, i.e. for every $i=0,1$ either $t_i \notin I$ or $x_i \notin B'$. We claim that it is sufficient to deal with the case $t_0\in {{\,\mathrm{{\mathbb {R}}}\,}}\setminus I$ and $x_1'\in \textrm{X}'\setminus B',$ indeed the other cases follow from this one by concatenating two paths of this type as follows: if $t_0,t_1\notin I$ and $x_0',x_1'\in B'$ we choose $y' \notin B'$ (which exists since $\textrm{X}'$ is unbounded) and we join $(t_0,x_0')$ to $(t_0,y')$ and then $(t_0,y')$ to $(t_1,x_1')$; if $t_0,t_1\in I$ and $x_0',x_1' \notin B'$ we pick $s \notin I$ and we join $(t_0,x_0')$ to $(s,x_0')$ and then $(s,x_0')$ to $(t_1,x_1')$.

Hence we can assume that $t_0\in {{\,\mathrm{{\mathbb {R}}}\,}}\setminus I$ and $x_1'\in \textrm{X}'\setminus B'.$ To build the required path, consider a geodesic $\eta :[0,1]\rightarrow X'$ going from $x_0'$ to $x_1'$ and define the function $s:[0,1]\rightarrow {{\,\mathrm{{\mathbb {R}}}\,}}$ as $s(t)=t_1\,t+(1-t)\,t_0$. Then the curve

$$\begin{aligned} \gamma (t)={\left\{ \begin{array}{ll} (t_0,\eta (t)), &{} t \in [0,1),\\ (s(t),x_1'), &{} t \in [1,2], \end{array}\right. } \end{aligned}$$

is Lipschitz and has image contained in $(I \times B')^c\subset B_R(p)^c$, hence (up to a reparametrization) satisfies all the requirements.

To estimate $\textrm{Lip}(\gamma )$ we observe that, up to a reparametrization, we can assume that $\textrm{Lip}(\gamma )=L(\gamma )$, hence it is sufficient to bound the length of $\gamma$. In the case $t_0\in {{\,\mathrm{{\mathbb {R}}}\,}}\setminus I$ and $x_1'\in \textrm{X}'\setminus B'$ it is sufficient to observe that $L(\gamma )=\textsf {d} '(x_0',x_1')+|t_0-t_1|\le 2( \textsf {d} _{Eucl}\times \textsf {d} ((t_0,x_0'),(t_1,x_1')))$, where $\gamma$ is the curve constructed above. The general case follows concatenating two paths as described above, where we pick $y'$ and s so that $\textsf {d} '(y',x')<2R$, $|{{\bar{t}}}-s|<2R$. Indeed it can be easily checked these two resulting paths have,respectively, length not grater than 2R and $2R+\textsf {d} (x_0,x_1).$ Since we are assuming $\textsf {d} (x_0,x_1)\ge R$, this concludes the proof. $\square$

Corollary 3.9

If $\textrm{X}$ is a nonparabolic $\textrm{RCD}(0,N)$ space, then it is not a cylinder and in particular item (ii) of Proposition 3.8 holds and $\textrm{X}$ has only one end.

Proof

Suppose by contradiction $\textrm{X}$ is a cylinder ${{\,\mathrm{{\mathbb {R}}}\,}}\times \textrm{X}'$. Then for any $r>0$ and any $(x,t) \in \textrm{X}'\times {{\,\mathrm{{\mathbb {R}}}\,}}$ we have

$$\begin{aligned} {\mathfrak {m}}(B_r((x,t)))=({\mathcal {L}}^1\otimes {\mathfrak {m}}' )(B_r((x,t)))\le ({\mathcal {L}}^1\otimes {\mathfrak {m}}') (\textrm{X}'\times [t-r,t+r] )={\mathfrak {m}}'(\textrm{X}')\, 2r, \end{aligned}$$

which clearly contradicts the fact that $\textrm{X}$ is nonparabolic. $\square$

4 New Estimates for Harmonic Functions

4.1 Preliminary Calculus Rules

In this subsection,we collect and prove some versions of the chain and Leibniz rule for the Laplacian and the divergence operator, that will be used in the following subsection. These results are essentially variants of the ones already contained in [48] and [47]. However, since one of the main obstacles in the proof of Theorem 4.4 and Corollary 4.5 is making the computations rigorous and justifying the derivatives taken, we decided to state and prove in details the results we need.

In everything that follows $\Omega$ is an open subset of a proper and infinitesimally Hilbertian m.m.s. $(\textrm{X},\textsf {d} ,{\mathfrak {m}}).$

Proposition 4.1

(Leibniz rule for ${\varvec{\Delta }}$) Let $u \in D({\varvec{\Delta }},\Omega )$ and let $g\in \textsf {LIP} _\textsf {loc} (\Omega )\cap D({\varvec{\Delta }},\Omega )$ be such that ${\varvec{\Delta }}g \in L^2(\Omega ).$ Then $ug \in D({\varvec{\Delta }},\Omega )$ and

$$\begin{aligned} {\varvec{\Delta }}(ug)=g{\varvec{\Delta }}u+u {\varvec{\Delta }}g+2\langle \nabla u, \nabla g \rangle {\mathfrak {m}}. \end{aligned}$$

Proof

Let $f \in \textsf {LIP} _c(\Omega ).$ Then using the Leibniz rule for the gradient

$$\begin{aligned} \int \langle \nabla (ug), \nabla f\rangle \, {\mathrm d}{\mathfrak {m}}=\int \langle \nabla u, \nabla (fg)\rangle \, {\mathrm d}{\mathfrak {m}}+ \int \langle \nabla g, \nabla (uf)\rangle \, {\mathrm d}{\mathfrak {m}}- 2\int f \langle \nabla u, \nabla g \rangle \, {\mathrm d}{\mathfrak {m}}. \end{aligned}$$

Since $fg\in \textsf {LIP} _c(\Omega )$ and $fu \in W ^{1,2}(X)$ with support compact in $\Omega$, we conclude from Remark 2.7. $\square$

Proposition 4.2

(Leibniz rule for ${{\textbf {div}}}$) Let $v \in D({{\textbf {div}}},\Omega )$. Then

• 1. if $g \in \textsf {LIP} _\textsf {loc} (\Omega )$, then $gv \in D({{\textbf {div}}},\Omega )$ and

  $$\begin{aligned} {{\textbf {div}}}(gv)=\langle \nabla g,v\rangle {\mathfrak {m}}+ g {{\textbf {div}}}(v); \end{aligned}$$

  (4.1)

• 2. if $g \in W ^{1,2}_\textsf {loc} (\Omega )$ and ${{\textbf {div}}}(v) \in L^2_{\textsf {loc} }(\Omega )$, then $gv \in D({{\textbf {div}}},\Omega )$ and (4.1) holds.

Proof

Let $f \in \textsf {LIP} _{\mathrm c}(\Omega ).$ Using the Leibniz rule for the gradient, we get

$$\begin{aligned} -\int \langle \nabla f, gv \rangle {\mathrm d}{\mathfrak {m}}= -\int \langle \nabla (fg),v\rangle {\mathrm d}{\mathfrak {m}}+\int f \langle \nabla g, v \rangle {\mathrm d}{\mathfrak {m}}. \end{aligned}$$

The conclusion follows in the first case observing that $fg\in \textsf {LIP} _c(\Omega )$ and in the second case observing that $fg \in W ^{1,2}(\textrm{X})$ with compact support in $\Omega$ and recalling Remark 2.9. $\square$

Proposition 4.3

(Chain rule for ${\varvec{\Delta }}$) Let $u \in D({\varvec{\Delta }},\Omega )$ and let $\varphi \in C^2(I)$, where I is an open interval such that (2.4) holds. Then

• 1. if $u \in \textsf {LIP} _{\textsf {loc} }(\Omega )$ then $\varphi (u)\in D({\varvec{\Delta }},\Omega )$ and

  $$\begin{aligned} {\varvec{\Delta }}{_{|_{\Omega }}}(\varphi (u))=\varphi '(u){\varvec{\Delta }}{_{|_{\Omega }}} u+\varphi ''(u)|\nabla u|^2{\mathfrak {m}}{_{|_{\Omega }}}. \end{aligned}$$

  (4.2)

• 2. if ${\varvec{\Delta }}u\ge g {\mathfrak {m}}{_{|_{\Omega }}}$ for some $g \in L^1_{\textsf {loc} }(\Omega )$ then $\varphi (u)\in D({\varvec{\Delta }},\Omega )$ and

  $$\begin{aligned} {\varvec{\Delta }}{_{|_{\Omega }}}(\varphi (u))\ge \left( \varphi '(u)g +\varphi ''(u)|\nabla u|^2\right) {\mathfrak {m}}{_{|_{\Omega }}}. \end{aligned}$$

Proof

Let $f \in \textsf {LIP} _c(\Omega ),$ then from the chain rule and Leibniz rule for the gradient

$$\begin{aligned} -\int \langle \nabla (\varphi (u)), \nabla f \rangle {\mathrm d}{\mathfrak {m}}= -\int \langle \nabla (\varphi '(u)f), \nabla u \rangle {\mathrm d}{\mathfrak {m}}+ \int f\langle \nabla \varphi '(u), \nabla u \rangle {\mathrm d}{\mathfrak {m}}. \end{aligned}$$

In the first case, we conclude from the fact that $\varphi '(u)f \in \textrm{Lip}_c(\Omega )$. In the second case,we assume also $f \ge 0$ and observe that $\varphi '(u)f \in W ^{1,2}(X)\cap L^\infty ({\mathfrak {m}})$ is nonnegative with compact support in $\Omega$, hence from Remark 2.7, it follows that

$$\begin{aligned} -\int \langle \nabla (\varphi (u)), \nabla f \rangle {\mathrm d}{\mathfrak {m}}\ge \int \varphi '(u)f \, g{\mathfrak {m}}+ \int f \varphi ''(u)|\nabla u|^2{\mathrm d}{\mathfrak {m}}. \end{aligned}$$

The conclusion follows applying Proposition 2.6. $\square$

4.2 Second-Order Estimates for Harmonic Functions

Before starting the discussion, let us say that this part is independent of all the rest of the note and we only assume that

$$\begin{aligned} \textrm{X}\text { is an } \textrm{RCD}(K,N) \text { with } N \in [2,+\infty ) \text { and } \Omega \text { an open subset of } \textrm{X}. \end{aligned}$$

The goal of this subsection is to prove the following two results, which can be reinterpreted as a generalization of the well-known fact that in a Riemannian manifold with nonnegative Ricci curvature the square norm of the gradient of an harmonic function is subharmonic.

Theorem 4.4

Let $\textrm{X}$ be an $\textrm{RCD}(K,N)$ space with $N \in [2,\infty )$, let u be harmonic in $\Omega$ and $\beta >\frac{N-2}{N-1}$. Then $|\nabla u|^{\beta /2}\in W ^{1,2}_{\textsf {loc} }(\Omega )$, $|\nabla u|^\beta \in D({\varvec{\Delta }},\Omega )$ and

$$\begin{aligned} {\varvec{\Delta }}(|\nabla u|^\beta )\ge C_{\beta ,N}\, |\nabla |\nabla u|^{\frac{\beta }{2}}|^2\, {\mathfrak {m}}{_{|_{\Omega }}}+2\beta K |\nabla u|^2{\mathfrak {m}}{_{|_{\Omega }}}, \end{aligned}$$

(4.3)

where $C_{\beta ,N}=\frac{4}{\beta } \left( \beta -\frac{N-2}{N-1}\right)$. Moreover $|\nabla u|^\beta \in D({\varvec{\Delta }},\Omega )$ also for $\beta =\frac{N-2}{N-1}$ with (4.3) holding without the term containing $C_{\beta ,N}.$

Corollary 4.5

Let $\textrm{X}$ be an $\textrm{RCD}(K,N)$ space with $N\in (2,\infty )$. Suppose u is positive and harmonic in $\Omega$, set $v=u^{\frac{-1}{N-2}}$ and let $\beta > \frac{N-2}{N-1}$. Then $|\nabla v|^{\beta /2}\in W ^{1,2}_{\textsf {loc} }(\Omega )$, $u^2\nabla |\nabla v|^{\beta /2} \in D({{\textbf {div}}},\Omega )$ and

$$\begin{aligned} {\textbf {div}}(u^2\nabla |\nabla v|^{\beta })\ge C_{\beta ,N}\, u^2 |\nabla |\nabla v|^{\frac{\beta }{2}}|^2\, {\mathfrak {m}}{_{|_{\Omega }}}+2\beta K u^2|\nabla v|^\beta \, {\mathfrak {m}}{_{|_{\Omega }}}, \end{aligned}$$

(4.4)

where $C_{\beta ,N}=\frac{4}{\beta } \left( \beta -\frac{N-2}{N-1}\right)$. Moreover $u^2\nabla |\nabla v|^{\beta /2} \in D({{\textbf {div}}},\Omega )$ also for $\beta =\frac{N-2}{N-1}$ with (4.4) holding without the term containing $C_{\beta ,N}.$

Let us remark that the results on the rest of this note rely only on Corollary 4.5. However, we decided to isolate Theorem 4.4, since it contains estimates for general harmonic functions that appear to be useful and interesting on their own.

The main ingredient for the proof of Theorem 4.4 is the following new Kato-type inequality. Observe that if $\dim (\textrm{X})=N$, in which case (from [63]) $\textsf {tr} \textrm{Hess}(u) =\Delta u$, letting $t\rightarrow 0$ we recover the standard refined Kato inequality for harmonic functions.

Lemma 4.6

(Generalized refined Kato inequality) Let $\textrm{X}$ be an $\textrm{RCD}(K,N)$space, with $N\in [1,+\infty )$, and let $n=\dim (\textrm{X}).$ Then for any $u \in \textsf {Test} _{\textsf {loc} } (\Omega )$ (see Sect. 2.2.1) it holds

$$\begin{aligned} \frac{t+n }{t+n-1}|\nabla |\nabla u||^2\le |\textrm{Hess}(u)|^2_{HS}+\frac{(\textsf {tr} \textrm{Hess}(u))^2}{t}, \quad {\mathfrak {m}}\text {-a.e. in } \Omega ,\, \forall t>0. \end{aligned}$$

(4.5)

Proof

Observe that it is enough to prove (4.5) for $u \in \textsf {Test} (\textrm{X})$.

We make the following preliminary observation. Let A be any symmetric $n\times n$ real matrix, then the following inequality holds for every $t>0$

$$\begin{aligned} \frac{t+n }{t+n-1}|A\cdot v|^2\le \frac{|v|^2(\textsf {tr} A)^2}{t} + |v|^2|A|^2, \quad \forall v \in {{\,\mathrm{{\mathbb {R}}}\,}}^n, \quad \forall N\ge n. \end{aligned}$$

(4.6)

where |A| is the Hilbert-Schmidt norm of A. To prove it we can assume that A is diagonal with diagonal entries $\lambda _1,...,\lambda _n$, where $\lambda _n\ge \lambda _i$, $i\le n$ and also that $|v|\le 1.$ Applying twice Cauchy-Schwartz we obtain

$$\begin{aligned}&\frac{(\lambda _1+\cdots +\lambda _n)^2}{t}+\lambda _1^2+\cdots +\lambda _{n-1}^2+\lambda _n^2\\&\quad \ge \frac{(\lambda _1+\cdots +\lambda _n)^2}{t} +\frac{(\lambda _1+\cdots +\lambda _{n-1})^2}{n-1}+\lambda _n^2\\&\quad \ge \frac{\lambda _n^2}{t+n-1}+\lambda _n^2\ge \frac{t+n }{t+n-1} |A\cdot v|^2, \end{aligned}$$

which proves (4.6).

Let $e_1,...,e_n \in L^0(T \textrm{X})$ be a global orthonormal base which exists thanks to Theorem 2.11. Consider the $n\times n$ real matrix $A : \textrm{X}\rightarrow L^0({\mathfrak {m}})^{n^2}$ defined by $\{A_{i,j}(x)\}_{i,j}=\{\textrm{Hess}(u)(e_i,e_j)(x)\}_{i,j}$. Define also the vector $v: \textrm{X}\rightarrow L^0({\mathfrak {m}})^n$ as $v_i(x):=\langle \nabla u, e_i\rangle (x)$. From (2.13) and the formula (2.14) we have that

$$\begin{aligned} 2|\nabla u|\nabla |\nabla u|=\nabla |\nabla u|^2=2\sum _{i=1}^n\sum _{j=1}^n A_{i,j}v_j e_i, \quad \text { in } L^0(T\textrm{X}). \end{aligned}$$

Taking the square pointwise norm on both sides we obtain

$$\begin{aligned} 4|\nabla u|^2 |\nabla |\nabla u||^2= 4 |A\cdot v|^2, \quad {\mathfrak {m}}\text {-a.e..} \end{aligned}$$

Since from (2.15) and (2.16) we have that $|\textrm{Hess}(u)|_{HS}^2=|A|_{HS}^2$ and $\textsf {tr} \textrm{Hess}(u)=\textsf {tr} A$, the conclusion follows combining the above identity with (4.6). $\square$

The second ingredient for the proof is the following simple technical lemma (cf. with [48, Prop. 4.15])

Lemma 4.7

Let $\textrm{X}$ be an $\textrm{RCD}(K,N)$space, with $N<+\infty$. Let $(u_n)\subset D({\varvec{\Delta }},\Omega )$ and $u \in W ^{1,2}_{\textsf {loc} }(\Omega )$ be such that $|\nabla u_n-\nabla u|^2\rightarrow 0$ in $L^1_{\textsf {loc} }(\Omega )$. Moreover assume that ${\varvec{\Delta }}u_n \ge g_n {\mathfrak {m}}$ for some $g_n \in L^1_{\textsf {loc} }(\Omega )$ such that $\int g_nf \,\ {\mathrm d}{\mathfrak {m}}\rightarrow \int gf\, {\mathrm d}{\mathfrak {m}}+ \int hf \, {\mathrm d}{\mathfrak {m}}$, for every $f \in \textsf {LIP} _{c}(\Omega )$ with $f \ge 0$, for some fixed functions $g\in L^0(\Omega ,{\mathfrak {m}})$ and $h \in L^1_{\textsf {loc} }(\Omega )$, with $g \ge 0$ ${\mathfrak {m}}$-a.e.. Then $g \in L^1_{\textsf {loc} }(\Omega )$, $u \in D({\varvec{\Delta }},\Omega )$ and ${\varvec{\Delta }}u \ge (g+h) {\mathfrak {m}}.$

Proof

The assumptions guarantee that $\int \langle \nabla u_n,\nabla f\rangle {\mathrm d}{\mathfrak {m}}\rightarrow \int \langle \nabla u,\nabla f \rangle {\mathrm d}{\mathfrak {m}}$, for every $f \in \textsf {LIP} _{c}(\Omega )$, therefore we can pass to the limit on both sides of $-\int \langle \nabla u_n,\nabla f\rangle {\mathrm d}{\mathfrak {m}}\ge \int g_nf\, {\mathrm d}{\mathfrak {m}}$ to obtain that

$$\begin{aligned} -\int \langle \nabla u,\nabla f\rangle {\mathrm d}{\mathfrak {m}}\ge \int gf\, {\mathrm d}{\mathfrak {m}}+\int hf\, {\mathrm d}{\mathfrak {m}}, \quad \forall f \in \textsf {LIP} _{c}(\Omega ), \text { with } f \ge 0. \end{aligned}$$

From this it follows that $g \in L^1_{\textsf {loc} }(\Omega )$, indeed we can take for any K compact in $\Omega$ a function $f \in \textsf {LIP} _c(\Omega )$ such that $f \ge 0$ and $f=1$ in K and then bring $\int hf\, {\mathrm d}{\mathfrak {m}}$ to the other side of the inequality. The conclusion then follows applying Proposition 2.6. $\square$

Proof of Theorem 4.4

The proof is based on a inductive bootstrap argument. We make the following claim

Observe that, since we already know that $|\nabla u|^{\beta }\in W ^{1,2}_{\textsf {loc} }(\Omega )$ for every $\beta \ge 1$ (recall (2.21)), the first part of the conclusion follows iterating the above statement.

We pass to the proof of the claim, hence we fix $\beta > \frac{N-2}{N-1}$ such that $|\nabla u|^{\beta }\in W ^{1,2}_{\textsf {loc} }(\Omega )$. Since $u \in \textsf {Test} _\textsf {loc} (\Omega )$, from the local Bochner inequality (2.20) combined with the Kato inequality (4.5) (with $t=N-\dim (\textrm{X})$ if $\dim (\textrm{X})<N$ and letting $t\rightarrow 0$ if $N=\dim (\textrm{X})$, since in the latter case $0=\Delta u=\textsf {tr} \textrm{Hess}(u)$) we have $|\nabla u|^2 \in D({\varvec{\Delta }},\Omega )$ and

$$\begin{aligned} {\varvec{\Delta }}(|\nabla u|^2) \ge \left( \frac{2N}{N-1}|\nabla |\nabla u||^2+2K|\nabla u|^2\right) {\mathfrak {m}}{_{|_{\Omega }}}, \end{aligned}$$

Hence from the chain rule for the Laplacian (second version in Proposition 4.3, applied with $\varphi \in C^2({{\,\mathrm{{\mathbb {R}}}\,}})$ as $\varphi (t)=(t+\varepsilon )^{\frac{\beta }{2}}$) we have that $(|\nabla u|^2+\varepsilon )^{\frac{\beta }{2}} \in D({\varvec{\Delta }},\Omega )$ and

$$\begin{aligned} {\varvec{\Delta }}((|\nabla u|^2+\varepsilon )^{\frac{\beta }{2}})\ge & {} \bigg [\beta (|\nabla u|^2+\varepsilon )^{\frac{\beta }{2}-1}|\nabla |\nabla u||^2 \left( \frac{N}{N-1}+\frac{(\beta -2)|\nabla u|^2}{|\nabla u|^2+\varepsilon }\wedge (\beta -2)\right) \nonumber \\{} & {} +2K\beta (|\nabla u|^2+\varepsilon )^{\frac{\beta }{2}} \frac{|\nabla u|^2}{|\nabla u|^2+\varepsilon }\bigg ]{\mathfrak {m}}{_{|_{\Omega }}}, \end{aligned}$$

(4.7)

for every $\varepsilon >0$. Setting $v_{\varepsilon } :=\nabla (|\nabla u|^2+\varepsilon )^{\frac{\beta }{2}}$ it is easy to see using dominated convergence that $|v_{\varepsilon }-\nabla (|\nabla u|^\beta )|^2\rightarrow 0$ in $L^1_{\textsf {loc} }(\Omega )$ as $\varepsilon \rightarrow 0^+$. Moreover for every $\beta > \frac{N-2}{N-1}$, denoting by $g_{\beta ,\varepsilon }\in L^1_{\textsf {loc} }(\Omega )$ the function on the right-hand side of (4.7), we have that $\int _{\Omega } g_{\beta ,\varepsilon } f {\mathrm d}{\mathfrak {m}}\rightarrow \int _{\Omega } g_{\beta } f{\mathrm d}{\mathfrak {m}}+ 2K\beta \int _{\Omega } |\nabla u|^\beta f{\mathrm d}{\mathfrak {m}}$, for every $f \in \textrm{Lip}_c(\Omega )$ with $f \ge 0$, where $g_\beta \in L^0(\Omega ,{\mathfrak {m}})$ is given by

$$\begin{aligned} g_\beta :=\beta \left( \beta -\frac{N-2}{N-1}\right) {\chi }_{|\nabla u|>0}|\nabla u|^{\beta -2}|\nabla |\nabla u||^2.\\ \end{aligned}$$

This can be seen applying dominated convergence for the second term in (4.7) and using,respectively, dominated convergence in the case $\beta \ge 2$ and monotone convergence in the case $\frac{N-2}{N-1}< \beta <2$, to deal with the first term. We are therefore in position to apply Lemma 4.7 and deduce both that $g_\beta \in L^1_{\textsf {loc} }(\Omega )$ and that $|\nabla u|^{\beta }\in D({\varvec{\Delta }},\Omega )$ with ${\varvec{\Delta }}(|\nabla u|^\beta )\ge (g_\beta +2K\beta |\nabla u|^\beta ) {\mathfrak {m}}{_{|_{\Omega }}}$. Moreover the fact that $g_\beta \in L^1_{\textsf {loc} }(\Omega )$ together with Lemma 2.4 implies that $|\nabla u|^{\beta /2}\in W ^{1,2}_{\textsf {loc} }(\Omega )$. This shows the claim and thus concludes the proof of the first part.

We pass to the case $\beta =\frac{N-2}{N-1}$. From the previous part, we know that $|\nabla u|^\beta \in W ^{1,2}_{\textsf {loc} }(\Omega )$, hence we can repeat the above argument and observe that in this case $g_\beta =0,$ from which the conclusions follows. $\square$

Proof of Corollary 4.5

We start observing that from the positivity and local Lipschitzianity of u follows that $u^{-1}\in \textrm{Lip}_{\textsf {loc} }(\Omega )$ and thus $u^{\alpha }\in W ^{1,2}_{\textsf {loc} }(\Omega )$ for every $\alpha \in {{\,\mathrm{{\mathbb {R}}}\,}}.$ Moreover, from the chain rule for the Laplacian (first version in Proposition 4.3, applied with u and $\varphi (t)=t^{\alpha }$, $\alpha \in {{\,\mathrm{{\mathbb {R}}}\,}}$) and by the harmonicity of u, we deduce that $u^{\alpha }\in D({\varvec{\Delta }},\Omega )$ with ${\varvec{\Delta }}(u^{\alpha })=\alpha (\alpha -1)u^{\alpha -2}|\nabla u|^2 {\mathfrak {m}}{_{|_{\Omega }}}$. Hence from the Leibniz rule for the Laplacian (Proposition 4.1) and Theorem 4.4 we have that $|\nabla u|^{\beta }u^\alpha \in D({\varvec{\Delta }},\Omega )$ with

$$\begin{aligned} {\varvec{\Delta }}(|\nabla u|^{\beta }u^\alpha )\ge & {} \left( u^\alpha g_\beta + \alpha (\alpha -1)u^{\alpha -2}|\nabla u|^{\beta +2}+2\alpha u^{\alpha -1} \langle \nabla |\nabla u|^\beta ,\nabla u\rangle \right) {\mathfrak {m}}{_{|_{\Omega }}}\\{} & {} +2K\beta u^\alpha |\nabla u|^\beta \, {\mathfrak {m}}{_{|_{\Omega }}}, \end{aligned}$$

for every $\beta \ge \frac{N-2}{N-1}$ and every $\alpha \in {{\,\mathrm{{\mathbb {R}}}\,}}$, where $g_\beta$ is the same as in the proof of Theorem 4.4.

Applying the Leibniz rule for the divergence (Proposition 4.2), we deduce that $u^2\nabla (|\nabla u|^\beta u^{\alpha }) \in D({\textbf {div}},\Omega )$ with

$$\begin{aligned} \begin{aligned} {\textbf {div}}(u^2\nabla (|\nabla u|^\beta u^{\alpha }))\ge&\bigg (u^{\alpha +2} g_\beta +\alpha (\alpha +1) u^\alpha |\nabla u|^{\beta +2}\\&+2(\alpha +1) u^{\alpha +1} \langle \nabla |\nabla u|^\beta ,\nabla u\rangle \bigg ){\mathfrak {m}}{_{|_{\Omega }}}\\&+2K\beta u^{\alpha +2}|\nabla u|^\beta \, {\mathfrak {m}}{_{|_{\Omega }}}, \end{aligned} \end{aligned}$$

(4.8)

for every $\beta \ge \frac{N-2}{N-1}$ and every $\alpha \in {{\,\mathrm{{\mathbb {R}}}\,}}$.

We now assume that $\beta > \frac{N-2}{N-1}$. Since $|\nabla v|=(N-2)^{-1}u^{\frac{1-N}{N-2}}|\nabla u|$ and since from Theorem 4.4 we have $|\nabla u|^{\beta /2}\in W ^{1,2}_{\textsf {loc} }(\Omega )$, it follows that $|\nabla v|^{\beta /2}\in W ^{1,2}_{\textsf {loc} }(\Omega )$.

To see (4.4),we just take $\alpha =-\beta \frac{N-1}{N-2}$ in (4.8). Then a direct computation gives that the right-hand side of (4.8) equals the right-hand side of (4.4).

Finally choosing $\beta = \frac{N-2}{N-1}$, $\alpha =-1$ in (4.8) and recalling that in this case $g_\beta =0$, shows also the second part of the statement, thus finishing the proof. $\square$

5 The Monotonicity Formula

5.1 Decay Estimates

Throughout this section, $(\textrm{X},\textsf {d} ,{\mathfrak {m}})$ is a nonparabolic $\textrm{RCD}(0,N)$ space (recall from Remark 3.2 that in this case $N>2$), $\Omega \subset \textrm{X}$ is open, unbounded, with $\partial \Omega$ bounded, $x_0\in \Omega ^c$ is fixed and u is a solution to (P). It follows from the maximum principle that

$$\begin{aligned} 0<u<\Vert u\Vert _{L^\infty }, \quad \text {in } \Omega . \end{aligned}$$

Moreover from Corollary 3.9, we must have that $\Omega ^c$ is bounded.

Proposition 5.1

Set $R_0:=3\text {diam}(\Omega ^c)+1,$ then

$$\begin{aligned} \frac{|\nabla u|(x)}{u(x)} \le \frac{C}{\textsf {d} (x,x_0)}, \quad \text { for } {\mathfrak {m}}\text {-a.e. } x \in B_{R_0}(x_0)^c, \end{aligned}$$

where $C=C(N)$ is a positive constant depending only on N.

Proof

Immediate from the gradient estimate (2.22) with $R=\textsf {d} (x,x_0)/4.$ $\square$

Proposition 5.2

For every $D,M>0$, there exists a positive constant $C_2=C_2(N,D,M)$ such that the following holds. Let u, $\Omega$ $x_0 \in \Omega ^c$ as above with $\text {diam}(\Omega ^c)\le D$ and $\Vert u\Vert _{L^\infty }\le M$, then setting $\delta :=\textsf {d} (x_0,\{u\le 1/2\})\wedge 1$ we have

$$\begin{aligned} \frac{\delta ^{N-2}}{2}\textsf {d} (x_0,x)^{2-N}\le u(x)\le C_2 \int _{\textsf {d} (x,x_0)}^{+\infty } s\frac{{\mathfrak {m}}(B_1(x_0))}{{\mathfrak {m}}(B_s(x_0))}\, {\mathrm d}s, \quad \forall x \in B_{R_0}(x_0)^c, \end{aligned}$$

(5.1)

where and $R_0:=3\text {diam}(\Omega ^c)+1$ and the first inequality actually holds in $\Omega \cap B_{\delta }(x_0)^c$.

Before passing to the proof we notice that from the assumption $\liminf _{y\rightarrow \partial \Omega }u(y)\ge 1$ we have $\textsf {d} (x_0,\{u\le 1/2\})>0$, in particular the left inequality in (5.1) is nontrivial.

Proof of Proposition 5.2

We start with the first inequality.

From Laplacian comparison (2.9), we know that $d_{x_0}^2 \in D({\varvec{\Delta }})$ and ${\varvec{\Delta }}d_{x_0}^2\le 2N {\mathfrak {m}}.$ Moreover from (2.10) $|\nabla d_{x_0}^2|^2=4d_{x_0}^2.$ Define now the function $h=d_{x_0}^{2-N}$, then from the chain rule for the Laplacian we have that $h\in D({\varvec{\Delta }},\textrm{X}\setminus \{x_0\})$, and

$$\begin{aligned} {\varvec{\Delta }}h= & {} \frac{2-N}{2} d_{x_0}^{-N}{\varvec{\Delta }}d_{x_0}^{2}+\frac{2-N}{2}\left( \frac{2-N}{2}-1\right) 4d_{x_0}^{2}d_{x_0}^{-N-2}\\= & {} \frac{N-2}{2}d_{x_0}^{-N}(2N{\mathfrak {m}}-{\varvec{\Delta }}d_{x_0}^2)\ge 0. \end{aligned}$$

Hence h is subharmonic in $\textrm{X}\setminus \{x_0\}$. Moreover we have that $\lambda h\le 1/2$ in $B_{\delta }(x_0)^c$, where $\lambda :=\delta ^{N-2}/2$. Finally from the assumption $\textsf {d} (x_0,\{u\le 1/2\})>\delta$, we have $u\ge \lambda h$ in $\Omega \cap B_{\delta }(x_0)^c$. Fix now $r>0$ and define the open set $\Omega ^r:=\{x \in \Omega \ : \ \textsf {d} (x,\Omega ^c)>r\}$. Observe that the function $\lambda h-u$ is subharmonic in $\Omega ^r$. Therefore from the weak maximum principle (see Proposition 2.15), we deduce that

$$\begin{aligned} \sup _{\Omega ^r\cap B_R(x_0)\cap B_{\delta }(x_0)^c} \, (\lambda h-u)&\le \max _{\partial \Omega ^r\cap B_{\delta }(x_0)^c}(\lambda h-u)\vee \max _{ \partial B_{\delta }(x_0)\cap \Omega ^r}(\lambda h-u)\vee \max _{ \partial B_{R}(x_0)}(\lambda h-u)\\&\le \max _{\partial \Omega ^r}(1/2-u)\vee 0\vee \max _{ \partial B_{R}(x_0)}(\lambda h-u)\, , \end{aligned}$$

for every $R>R_0$. Sending R to $+\infty$ and r to 0, recalling that both h and u vanish at infinity (since $N>2)$ and that $\liminf _{x\rightarrow \partial \Omega }u(x)\ge 1,$ we conclude that $\lambda h \le u$ in $\Omega \cap B_{\delta }(x_0)^c$. This proves the first inequality in (5.1).

We now pass to the second inequality in (5.1). We argue by comparison with the quasi Green function $G^1(x):=G^1(x_0,x)$ (recall its definition in (3.2)). Recall that $G^1$ is superharmonic in $\textrm{X}$. Moreover, using the upper bound for the Green function and the estimates of the heat kernel, we deduce that

(5.2)

for every $x \in X\setminus \{x_0\},$ for some positive constant $c_1=c_1(N)>1$. From Bishop-Gromov inequality and using the change of variable $s=t\textsf {d} (x_0,x)^2$, we obtain

$$\begin{aligned} G^1(x)\ge c_1^{-1}\frac{\textsf {d} (x_0,x)^{2-N}}{{\mathfrak {m}}(B_1(x_0))}\int _{\frac{1}{\textsf {d} (x_0,x)^2}}^{+\infty } \frac{e^{-\frac{1}{3t}}}{t^{\frac{N}{2}}}\, {\mathrm d}t \ge C_1 \frac{\textsf {d} (x_0,x)^{2-N}}{{\mathfrak {m}}(B_1(x_0))}, \quad \forall x \in B_1(x_0)^c, \end{aligned}$$

for some constant $C_1$ depending only on N. Therefore, taking $\lambda :=M{\mathfrak {m}}(B_1(x_0))\frac{R_0^{N-2}}{C_1}$, we have $\lambda G^1\ge M\ge u$ in $\partial B_{R_0}(x_0).$ Hence, since $\lambda G^1-u$ is superharmonic in $\Omega$, from the weak maximum principle it follows that for every $R>R_0$

$$\begin{aligned} \inf _{B_{R}(x_0)\cap B_{R_0}(x_0)^c} \, (\lambda G^1-u)= \min _{\partial B_{R}(x_0) \cup \partial B_{R_0}(x_0)} (\lambda G^1-u)\ge \min (0,\min _{ \partial B_{R}(x_0)}(\lambda G^1-u)\, ). \end{aligned}$$

Sending R to $+\infty$ and recalling that both $G^1$ and u go to 0 at infinity, we conclude that $u\le \lambda G^1$ in $B_{R_0}(x_0)^c,$ which combined with the second bound in (5.2) gives the second inequality in (5.1). $\square$

5.2 Monotonicity

As in the previous section $(\textrm{X},\textsf {d} ,{\mathfrak {m}})$ is a nonparabolic $\textrm{RCD}(0,N)$ space, $\Omega \subset \textrm{X}$ is open, unbounded, with $\partial \Omega$ bounded and u is a solution to (P).

We start with the following simple remark, which allows to define $U_\beta$ and will be needed to justify the many applications of the coarea formula along all this section.

Remark 5.3

Since u is locally Lipschitz (recall Proposition 5.1), satisfies $\liminf _{x\rightarrow \partial \Omega }u(x)\ge 1$ and vanishes at infinity, it follows that u satisfies the hypotheses needed to apply the coarea formula (2.27) in $\Omega$. In particular for every $f \in L^1_{\textsf {loc} }(\Omega )$ with $f {\mathfrak {m}}{_{|_{\Omega }}}\ll |\nabla u|{\mathfrak {m}}{_{|_{\Omega }}}$ we have

$$\begin{aligned}{} & {} \int _0^1 \varphi (t)\int g \, {\mathrm d}\textrm{Per}(\{u<r\})\,{\mathrm d}r\nonumber \\{} & {} \quad = \int _{\Omega } \varphi (u) f \, {\mathrm d}{\mathfrak {m}}< +\infty \ , \quad \forall \varphi :[0,1] \rightarrow {{\,\mathrm{{\mathbb {R}}}\,}}\text { Borel, with } \mathop {\textrm{supp}}\nolimits \varphi \subset (0,1),\nonumber \\ \end{aligned}$$

(5.3)

where g is any Borel representative of the function $\frac{{\mathrm d}(f{\mathfrak {m}}{_{|_{\Omega }}})}{{\mathrm d}(|\nabla u|{\mathfrak {m}}{_{|_{\Omega }}})}$. Therefore, by the arbitrariness of $\varphi$, we also deduce that:

$$\begin{aligned} \begin{aligned}&\text {for any } f \in L^1_{\textsf {loc} }(\Omega ) \text { with } f {\mathfrak {m}}{_{|_{\Omega }}}\\&\quad \ll |\nabla u|{\mathfrak {m}}{_{|_{\Omega }}}\text { and for any Borel representative } g \text { of } \frac{{\mathrm d}(f{\mathfrak {m}}{_{|_{\Omega }}})}{{\mathrm d}(|\nabla u|{\mathfrak {m}}{_{|_{\Omega }}})}, \\&\quad \text {the function } (0,1)\ni r\mapsto \int g \, {\mathrm d}\textrm{Per}(\{u<r\}) \text { is in } L^1_{\textsf {loc} }(0,1) \\&\quad \text { and (its a.e. equivalence class)}\\&\quad \text {does not depend on the choice of the representative } g. \end{aligned} \end{aligned}$$

(5.4)

$\square$

Choosing in the above remark $f=\frac{|\nabla u|^{\beta +2}}{u^{\beta \frac{N-1}{N-2}}} \in L^1_{\textsf {loc} }(\Omega )$, with $\beta > -2$, and observing that $\mathop {\textrm{supp}}\nolimits ( \textrm{Per}(\{u<t\})) \subset \{u=t\}$,we deduce that, fixed a Borel representative of $|\nabla u|,$ the function

$$\begin{aligned} U_{\beta }(t):=\frac{1}{t^{\beta \frac{N-1}{N-2}}}\int |\nabla u|^{\beta +1}\, {\mathrm d}\textrm{Per}(\{u<t\}) \, \in L^1_{\textsf {loc} }(0,1), \end{aligned}$$

(5.5)

is well defined and independent of the representative chosen for $|\nabla u|.$ It worth to recall that after the work of [42] a canonical choice for the representative of $|\nabla u|$ could be its quasi-continuous representative (see [42] for the precise definition). An interesting point would be to investigate the relation between the representative of $U_\beta$ given by this canonical choice and the continuous representative of $U_\beta$, which exists thanks to Theorem 5.4. We will not investigate this point in the present paper.

We are ready to state our main result regarding monotonicity.

Theorem 5.4

Let $\textrm{X}$ be a nonparabolic $\textrm{RCD}(0,N)$ space and let $\Omega \subset \textrm{X}$ be open, unbounded, with $\partial \Omega$ bounded. Suppose that u is a solution of (P) and let $U_{\beta }$, with $\beta \ge \frac{N-2}{N-1}$, be the function defined in (5.5). Then $U_{\beta }\in W^{1,1}_{\textsf {loc} }(0,1)$, $U'_{\beta } \in {BV_{\textsf {loc} }}(0,1)$ and

$$\begin{aligned} U^{'-}_{\beta }(t)\ge \frac{C_{\beta ,N}}{t^2}\int _{\{u<t\}} \, u^2|\nabla |\nabla u^{\frac{1}{2-N}}|^{\frac{\beta }{2}}|^2 {\mathrm d}{\mathfrak {m}},\quad \forall \, t \in (0,1], \end{aligned}$$

(5.6)

(recall that $|\nabla u^{\frac{1}{2-N}}|^{\frac{\beta }{2}} \in W ^{1,2}_{loc}(\Omega )$ for every $\beta >\frac{N-2}{N-1}$ by Corollary 4.5) where $C_{\beta ,N}=\frac{4}{\beta }\left( \beta -\frac{N-2}{N-1}\right)$, $U^{'-}_{\beta }$ is the left continuous representative of $U'_{\beta }$ and where the left-hand side is taken to be 0 if $\beta =\frac{N-2}{N-1}$. In particular $U_{\beta }$ is non-decreasing.

To prove Theorem 5.4, we start computing the first derivative of $U_\beta$ (which does not evidently carry a sign).

Proposition 5.5

With the same assumptions as in Theorem 5.4, the function $U_{\beta }$ belongs to $W^{1,1}_{\textsf {loc} }(0,1)$ and its derivative is given by

$$\begin{aligned} U_\beta '(t)= \int \left\langle \frac{\nabla u}{|\nabla u|},\nabla \left( \frac{|\nabla u|^{\beta }}{u^{\beta \frac{N-1}{N-2}}}\right) \right\rangle {\mathrm d}\textrm{Per}(\{u<t\}), \quad \text {a.e. }t\in (0,1), \end{aligned}$$

(5.7)

where the right-hand side has to be intended as in (5.4) with $f=\langle \nabla u,\nabla \left( \frac{|\nabla u|^{\beta }}{u^{\beta \frac{N-1}{N-2}}}\right) \rangle$.

Proof

Consider the vector field $v:=\frac{\nabla u|\nabla u|^{\beta }}{u^{\beta \frac{N-1}{N-2}}}\in L^0(T\textrm{X}){_{|_{\Omega }}}$ for $\beta \ge \frac{N-2}{N-1}$ and observe that from the Leibniz rule for the divergence (second version in Proposition 4.2) $v \in D(\textrm{div},\Omega )$ with

$$\begin{aligned} \textrm{div}(v)=\left\langle \nabla u,\nabla \left( \frac{|\nabla u|^{\beta }}{u^{\beta \frac{N-1}{N-2}}}\right) \right\rangle \in L^1_{\textsf {loc} }(\Omega ), \end{aligned}$$

thanks to the harmonicity of u. In particular $\textrm{div}(v){\mathfrak {m}}\ll |\nabla u|{\mathfrak {m}}$, hence recalling (5.3) and integrating by parts we have

$$\begin{aligned} \int _0^1 \int \frac{\textrm{div}(v)}{|\nabla u|} {\mathrm d}\textrm{Per}(\{u<t\})\varphi (t) {\mathrm d}t&\overset{(4.3)}{=}\int \textrm{div}(v)\varphi (u) {\mathrm d}{\mathfrak {m}}\\&=-\int \frac{|\nabla u|^{\beta +2}}{u^{\beta \frac{N-1}{N-2}}} \varphi '(u){\mathrm d}{\mathfrak {m}}\overset{(1.27)}{=} -\int _0^1 U_{\beta }(t)\varphi '(t) {\mathrm d}t, \end{aligned}$$

for every $\varphi \in C^1_c(0,1)$, where in the last step we used that $\mathop {\textrm{supp}}\nolimits (\textrm{Per}(\{u<t\},.))\subset \{u=t\}$ and with $\frac{\textrm{div}(v)}{|\nabla u|}$ denoting any Borel representative of $\frac{{\mathrm d}(\textrm{div}(v){\mathfrak {m}}{_{|_{\Omega }}})}{{\mathrm d}(|\nabla u|{\mathfrak {m}}{_{|_{\Omega }}}))}$. The conclusion follows. $\square$

To prove that $U'_\beta$ is nonnegative, we need to push our analysis to the second order and in particular to compute the derivative of $U_{\beta }'(t)t^2$. The reason for the term $t^2$ is that the key vector field with nonnegative divergence of Corollary 4.5 presents a term $u^2.$

Proposition 5.6

With the same assumptions as in Theorem 5.4 , the function $U'_{\beta }(t)t^2$ belongs to ${BV}_{\textsf {loc} }(0,1)$ and

$$\begin{aligned} (U_{\beta }'(t)t^2)'\ge C_{\beta ,N} \left( \int \frac{u^2|\nabla |\nabla u^{\frac{1}{2-N}}|^{\frac{\beta }{2}}|^2}{|\nabla u|} {\mathrm d}\textrm{Per}(\{u<t\}) \right) \,{\mathcal {L}}^1{_{|_{(0,1)}}}\ge 0, \end{aligned}$$

(5.8)

where $C_{\beta ,N}=\frac{4}{\beta }\left( \beta -\frac{N-2}{N-1}\right)$ and where the right-hand side has to be intended as in (5.4) with $f=u^2|\nabla |\nabla u^{\frac{1}{2-N}}|^{\frac{\beta }{2}}|^2$ when $\beta >\frac{N-2}{N-1}$ (recall also that from Corollary 4.5$|\nabla u^{\frac{1}{2-N}}|\in W ^{1,2}_{\textsf {loc} }(\Omega )$), and identically 0 in the case $\beta =\frac{N-2}{N-1}$.

Proof

Consider any nonnegative $\varphi \in C^1_c(0,1)$. Applying formula (5.7) and the coarea formula (5.3)

$$\begin{aligned} \int _0^1 (U_{\beta }'(t)t^2)\varphi '(t) {\mathrm d}t&=\int _0^1 \int \left\langle \frac{\nabla u}{|\nabla u|},u^2\nabla \left( \frac{|\nabla u|^{\beta }}{u^{\beta \frac{N-1}{N-2}}}\right) \right\rangle \varphi '(u){\mathrm d}\textrm{Per}(\{u<t\})\\&\overset{(4.3)}{=}\int \langle \nabla (\varphi (u)), u^2\nabla \left( \frac{|\nabla u|^{\beta }}{u^{\beta \frac{N-1}{N-2}}}\right) \rangle \, {\mathrm d}{\mathfrak {m}}, \end{aligned}$$

observing that $\varphi (u)\in \textsf {LIP} _{c}(\Omega )$ and recalling from Corollary 4.5 that $u^2\nabla \left( \frac{|\nabla u|^{\beta }}{u^{\beta \frac{N-1}{N-2}}}\right) \in D({{\textbf {div}}},\Omega )$, we obtain

$$\begin{aligned} -\int _0^1 (U_{\beta }'(t)t^2)\varphi '(t) {\mathrm d}t = \int \varphi (u) \,{\mathrm d}{{\textbf {div}}}\left( u^2\nabla \left( \frac{|Du|^{\beta }}{u^{\beta \frac{N-1}{N-2}}}\right) \right) . \end{aligned}$$

We now plug in (4.4) and (when $\beta >\frac{N-2}{N-1}$) apply the coarea formula (5.3) (observe that $||\nabla |\nabla u^{\frac{1}{2-N}}|^{\frac{\beta }{2}}|^2 |{\mathfrak {m}}\ll |\nabla u|{\mathfrak {m}}$) to obtain

$$\begin{aligned}{} & {} -\int _0^1 (U_{\beta }'(t)t^2)\varphi '(t) {\mathrm d}t \nonumber \\{} & {} \quad \ge {\tilde{C}}_{\beta ,N}\, \int _0^1 \int u^2|\nabla u|^{-1}|\nabla |\nabla u^{\frac{1}{2-N}}|^{\frac{\beta }{2}}|^2\, {\mathfrak {m}}{_{|_{\Omega }}} {\mathrm d}\textrm{Per}(\{u<t\}) \varphi (t) \, {\mathrm d}t \nonumber \\{} & {} \quad \ge 0, \end{aligned}$$

(5.9)

with $u^2|\nabla u|^{-1}|\nabla |\nabla u^{\frac{1}{2-N}}|^{\frac{\beta }{2}}|^2$ denoting any Borel representative of $\frac{{\mathrm d}\left( u^2|\nabla |\nabla u^{\frac{1}{2-N}}|^{\frac{\beta }{2}}|^2{\mathfrak {m}}{_{|_{\Omega }}}\right) }{{\mathrm d}(|\nabla u|{\mathfrak {m}}{_{|_{\Omega }}}))}$. The proof is concluded observing that (5.9) gives at once that the distributional derivative of $U'(t)t^2$ is a locally finite measure (it is positive) and that (5.8) holds. $\square$

Justified by Proposition 5.5, from this point onwards, we will identify $U_{\beta }$ with its continuous representative. Moreover Proposition 5.6 guarantees that $U'_{\beta } \in {BV}_{\textsf {loc} }(0,1)$, thus we will denote by $U^{'-}_{\beta }$ its representative which is left continuous in (0, 1] (notice that $U^{'-}_{\beta }$ might take value $+\infty$ at $t=1$). We observe also that (5.8) implies that

$$\begin{aligned} (0,1] \ni t \mapsto U^{'-}_{\beta }(t)t^2 \text { is a nondecreasing function.} \end{aligned}$$

(5.10)

To prove Theorem 5.4, we aim to integrate (5.8); however,to do so,we still need to know that $U_\beta$ is bounded close to 0. In particular,we prove the following:

Proposition 5.7

With the same assumptions as in Theorem 5.4,

$$\begin{aligned} U_\beta \in L^\infty (0,1/2 ). \end{aligned}$$

(5.11)

Proof

It is enough to show that

$$\begin{aligned} \left| \int _0^{\frac{1}{2}} U_{\beta }\varphi \, {\mathrm d}t \right| \le C \int _0^{\frac{1}{2}} |\varphi |, \quad \forall \varphi \in C^1_c (0,1/2 ), \end{aligned}$$

(5.12)

for some positive constant C independent of $\varphi .$

We start observing that, integrating by parts and applying the coarea formula (5.3),

$$\begin{aligned} 0= & {} \int _{\Omega } \Delta u \varphi (u)\, {\mathrm d}{\mathfrak {m}}=-\int _{\Omega }|\nabla u|^2\varphi '(u)\, {\mathrm d}{\mathfrak {m}}\\= & {} -\int _0^1 \int |\nabla u| \, {\mathrm d}\textrm{Per}(\{u<r\}) \varphi '(r)\,{\mathrm d}r,\quad \forall \varphi \in C^1_c (0,1 ), \end{aligned}$$

in particular $\int |\nabla u| \, {\mathrm d}\textrm{Per}(\{u<r\})=D$ for a.e. $r\in (0,1)$, for some constant D. Therefore using again the coarea formula

$$\begin{aligned} \left| \int _0^{\frac{1}{2}} U_{\beta } \varphi \, {\mathrm d}t\right|&\le \int u^{\beta \frac{1-N}{N-2}} |\nabla u|^{\beta +2}|\varphi (u)|\, {\mathrm d}{\mathfrak {m}}\\&\le \left\| |\nabla u|^{\beta }u^{\beta \frac{1-N}{N-2}} \right\| _{L^{\infty }(\{u\le 1/2\})} \int |\nabla u|^2|\varphi (u)| {\mathrm d}{\mathfrak {m}}\\&\overset{(4.3)}{=} \left\| |\nabla u|^{\beta }u^{\beta \frac{1-N}{N-2}} \right\| _{L^{\infty }(\{u\le 1/2\})} \int _0^{\frac{1}{2}} \int |\nabla u| \, {\mathrm d}\textrm{Per}(\{u<r\}) |\varphi (r)|\, {\mathrm d}r \\&= D\left\| |\nabla u|^{\beta }u^{\beta \frac{1-N}{N-2}} \right\| _{L^{\infty }(\{u\le 1/2\})} \int _0^{\frac{1}{2}}|\varphi |, \quad \forall \varphi \in C^1_c (0,1/2). \end{aligned}$$

Therefore to prove (5.12) it remains to show that $\left\| |\nabla u|^{\beta }u^{\beta \frac{1-N}{N-2}} \right\| _{L^{\infty }(\{u\le 1/2\})}<+\infty$. Let $R_0$ be as in Proposition 5.2 and observe that u, being positive and satisfying $\liminf _{x\rightarrow \partial \Omega }u(x)\ge 1$, is bounded away from zero in ${B_{R_0}(x_0)\cap \Omega }$. Moreover again thanks to $\liminf _{x\rightarrow \partial \Omega }u(x)\ge 1$ we have $\textsf {d} (\partial \Omega ,\{u\le 1/2\})>0.$ Therefore, since $u \in \textsf {LIP} _{\textsf {loc} }(\Omega )$, we have

$$\begin{aligned} \left\| |\nabla u|^{\beta }u^{\beta \frac{1-N}{N-2}} \right\| _{L^{\infty }(\{u\le 1/2\}\cap B_{R_0}(x_0))}<+\infty . \end{aligned}$$

Moreover combining Proposition 5.1 and the lower bound in (5.1),we obtain

$$\begin{aligned} \Vert |\nabla u|^{\beta }u^{\beta \frac{1-N}{N-2}}\Vert _{L^\infty (\Omega \cap B_{R_0}(x_0)^c)} \le \left\| \left( \frac{C(N)}{\textsf {d} (.,x_0)^{N-2}u} \right) ^\beta \right\| _{L^\infty (\Omega \cap B_{R_0}(x_0)^c)}<+\infty . \end{aligned}$$

Combining the two estimates we conclude. $\square$

We are now ready to prove the main monotonicity result.

Proof of Theorem 5.4

We start observing that, thanks to (5.8), setting $\mu :=(U_{\beta }(t)'t^2)'\ge 0$ we have

$$\begin{aligned} U_{\beta }^{'-}(t)t^2-U_{\beta }^{'-}(s)s^2&=\mu ([s,t))\ge \int _s^t\int g {\mathrm d}\textrm{Per}(\{u<r\}) \, {\mathrm d}r\\&\overset{(4.3)}{=}\int _{\{s<u<t\}} \Phi (u) {\mathrm d}{\mathfrak {m}}, \end{aligned}$$

for every $0<s<t\le 1$, with $\Phi (u)= C_{\beta ,N}u^2|\nabla |\nabla u^{\frac{1}{2-N}}|^{\frac{\beta }{2}}|^2$ when $\beta >\frac{N-2}{N-1}$, $\Phi (u)=0$ when $\beta =\frac{N-2}{N-1}$ and where g is a Borel representative of $\frac{{\mathrm d}( \Phi (u) {\mathfrak {m}}{_{|_{\Omega }}})}{{\mathrm d}(|\nabla u|{\mathfrak {m}}{_{|_{\Omega }}}))}$. Therefore to conclude it is enough to prove that there exists a sequence $s_n\rightarrow 0^+$ such that $U_{\beta }^{'-}(s_n){s_n^2}\rightarrow 0.$

To achieve this, we first prove that $U^{'-}_{\beta }(t)\ge 0$ for every $t \in (0,1)$. We assume by contradiction that there exists $T\in (0,1)$ such that $U^{'-}_{\beta }(T)<0$. From (5.10)

$$\begin{aligned} U^{'-}_{\beta }(s)\le U^{'-}_{\beta }(T)\frac{T^2}{s^2}, \quad \forall s<T, \end{aligned}$$

from which integrating with respect to s on the interval (t, T)

$$\begin{aligned} U_{\beta }(T)-U_{\beta }(t)\le U^{'-}_{\beta }(T)T^2 \left( \frac{1}{t}-\frac{1}{T}\right) . \end{aligned}$$

Sending $t\rightarrow 0^+$ and recalling that $U^{'-}_{\beta }(T)<0$, we obtain $U_\beta (t)\rightarrow +\infty$ as $t\rightarrow 0^+$, which, however, contradicts (5.11).

Since $U^{'-}_{\beta }(t)\ge 0$, we have that $U_{\beta }$ is non-decreasing and also nonnegative, hence it admits a limit as $t\rightarrow 0^+.$ In particular $U^{'-}_{\beta } \in L^1(0,\frac{1}{2}),$ therefore

$$\begin{aligned} a_n:=\int _{2^{-(n+1)}}^{2^{-n}} U_{\beta }^{'-}(t) {\mathrm d}t\rightarrow 0, \text { as } n\rightarrow +\infty . \end{aligned}$$

Moreover from Markov inequality,we have $|\{U_{\beta }^{'-}>a_n2^{n+2} \}\cap (2^{-(n+1)},2^{-n})|\le \frac{1}{2} 2^{-(n+1)}$, thus for every n we can find $s_n \in (2^{-(n+1)},2^{-n})$ such that $U_{\beta }^{'-}(s_n)\le a_n2^{n+2}$. Therefore

$$\begin{aligned} U_{\beta }^{'-}(s_n)s_n\le a_n2^{n+2}s_n<4a_n\rightarrow 0 \end{aligned}$$

and the proof is complete. $\square$

6 Functional Versions of the Rigidity and Almost Rigidity

6.1 From Outer Functional Cone to Outer Metric Cone

The following result is a variant of the “from volume cone to metric cone” theorem for $\textrm{RCD}$ spaces (see [40]). The two main differences with the work in [40] are that here we start from a function satisfying an equation (instead that from a condition on the measure), from which we deduce a conical structure on the complement of a bounded set (instead that on a ball). Related to this type of results, we mention also [4, 23] and [24] where functional versions of the splitting theorem in nonsmooth setting have been obtained. Finally we recall that the almost splitting theorems in the smooth setting proved in [29] were also based on a functional formulation similar to the one we are considering here.

Theorem 6.1

Let $(X,\textsf {d} ,{\mathfrak {m}})$ be an $\textrm{RCD}(0,N)$ space with $N \in [2,\infty )$ and $U \subset X$ be open with $\partial U$ bounded. Suppose there exists a positive and continuous function ${\textbf {u}}\in D({\varvec{\Delta }},U)$ such that $\Delta {\textbf {u}}=N$ ${\mathfrak {m}}$-a.e. in U, $|\nabla \sqrt{2{\textbf {u}}}|^2=1$ ${\mathfrak {m}}$-a.e. in U, ${\textbf {u}}_0:=\limsup _{x\rightarrow \partial U}{\textbf {u}}(x)<+\infty$ and $\{{\textbf {u}}>{\textbf {u}}_0\}\ne \emptyset$. Then

• (i) there exists unique an $\textrm{RCD}(N-2,N-1)$ space $(Z,\textsf {d} _Z,{\mathfrak {m}}_Z)$ with $\text {diam}(Z)\le \pi$ and a bijective measure preserving local isometry $S:\{{\textbf {u}}>{\textbf {u}}_0\}\rightarrow Y\setminus \overline{B_{r}}(O_Y)$, with $r:=\sqrt{2 {\textbf {u}}_0}$ and where $(Y,\textsf {d} _Y,{\mathfrak {m}}_Y)$ is the Euclidean N cone built over Z with vertex $O_Y,$

• (ii)

  • if $D_Z:=\text {diam}(Z)<\pi$ then local isometry of point i) is an isometry between $Y\setminus \overline{B_{r_Z}}(O_Y)$ and $\left\{ {\textbf {u}}>r_{Z}^2/2\right\}$, where $r_Z:=r(1-\sin {D_Z/2})^{-1}>r,$

  • if $\text {diam}(Z)=\pi$, then $(X,\textsf {d} ,{\mathfrak {m}})$ isomorphic to $(Y,\textsf {d} _Y,{\mathfrak {m}}_Y)$,

• (iii) the function ${\textbf {u}}$ has the following explicit form

  $$\begin{aligned} {\textbf {u}}(x)=\frac{1}{2}\textsf {d} _Y(S(x),O_Y)^2=\frac{1}{2}(\textsf {d} (x,\partial \{{\textbf {u}}> {\textbf {u}}_0\})+\sqrt{2{\textbf {u}}_0})^2, \quad \forall x \in \{{\textbf {u}}>{\textbf {u}}_0\},\nonumber \\ \end{aligned}$$

  (6.1)

  in particular the level set $\{u=\frac{t^2}{2}\}$, for every $t>{\textbf {u}}_0$, is Lipschitz path connected and isometric (with its induced intrinsic distance) to $(Z,t\textsf {d} _Z)$.

Remark 6.2

It might be worth to remark that the hypotheses of Theorem 6.1 are not stable with respect to taking products with ${{\,\mathrm{{\mathbb {R}}}\,}}$. This is because $\partial U$ does not remains bounded under this operation and (since the natural transformation would be ‘${\textbf {u}}\rightarrow {\textbf {u}}(x)+\frac{t^2}{2}$’) the new function would also not satisfy $\limsup _{x\rightarrow \partial U}{\textbf {u}}(x)<+\infty$. This is particularly relevant to the extra rigidity statement present in ii).

We observe that the uniqueness part of Theorem 6.1 is an immediate consequence of the rest of the statement. Indeed, from the last part of iii), we deduce that the metric space $(Z,\textsf {d} _Z)$ (and thus $(Y,\textsf {d} _Y)$) is uniquely determined up to isometries. Moreover, since S is measure preserving, the measure ${\mathfrak {m}}_Y$ is uniquely determined as well, hence from the definition of the measure in an N cone,we obtain that also ${\mathfrak {m}}_Z$ is uniquely determined.

As already said, the proof of the above Theorem is mainly an adaptation of the proof in [40]. However, some parts will require new arguments. The first main point is that in [40],the starting point is the gradient flow of the distance function $\textsf {d}$, which is used to deduce analytical information on $\textsf {d}$. Here instead we start from an analytical information, i.e. a PDE, and we want to build a flow. This will be done through the tool of Regular Lagrangian Flows. One of the main tools we need to develop in this regard is an a-priori estimate of local type, which seems to be missing in literature and does not follows immediately from the standard global a-priori estimates in [16].

The second main difference is that here the analysis takes place in the complementary of a bounded set, while in [40] all the work is done inside a fixed ball. Among other things, this difference will mainly affect the way in which we deduce that the cone is itself an $\textrm{RCD}(0,N)$ space. Indeed in [40] this follows from the fact that a whole ball with centre $x_0$ is isometric to a ball centred at the tip of the cone, therefore any blow-up of the space at $x_0$ will converge to the said cone. Then from the closedness of the $\textrm{RCD}(0,N)$ condition,the conclusion follows. However, in our case, the same argument cannot be applied, indeed our isometry is by nature far from the tip of the cone. This issue will be overcome noticing that our isometry is almost global, meaning that the space is isometric to the cone outside a bounded set. This allow us to deduce that any blow down of the space will converge to the cone, which gives the conclusion again by the closedness of the $\textrm{RCD}(0,N)$ class.

Since they are interesting on their own and independent of the rest of the proof, we isolate the two ingredients that we just described in the following two subsections. The remaining part of the argument will be outlined in Appendix A.

6.1.1 The Blow Down Argument

Proposition 6.3

Let $(\textrm{X},\textsf {d} _{\textrm{X}},{\mathfrak {m}}_{\textrm{X}})$ be a m.m.s. and let $V\subset \textrm{X}$ be closed and bounded. Suppose that there exists an Euclidean N cone, $(Y,\textsf {d} _Y,{\mathfrak {m}}_Y)$ over a m.m.s. Z, $N \in [1,\infty )$, with tip $O_Y$ and a bijective local isometry $T:V^c \rightarrow Y\setminus B_R(O_Y)$, for some $R>0$, which is measure preserving, i.e. $T_*{\mathfrak {m}}_{\textrm{X}}{_{|_{V^c}}}={\mathfrak {m}}_Y{_{|_{B_R(O_Y)}}}$. Then for every $x_0\in \textrm{X}$ and every sequence $r_n\rightarrow +\infty$ it holds that

$$\begin{aligned} (\textrm{X},r_n^{-1}\textsf {d} _{\textrm{X}},r_n^{-N}{\mathfrak {m}}_{\textrm{X}},x_0)\overset{{\text{ p }mGH}}{\longrightarrow }(Y,\textsf {d} _Y,{\mathfrak {m}}_Y,O_Y). \end{aligned}$$

In particular if $\textrm{X}$ is an ${\textsf {RCD}}(0,N)$ space, then Y is an ${\textsf {RCD}}(0,N)$ space as well.

Finally if $\textrm{X}$ is ${\textsf {RCD}}(0,N)$ and $\text {diam}(Z)=\pi$ then $\textrm{X}$ is isomorphic to Y as m.m.s..

Remark 6.4

Observe that in Proposition 6.3, the assumptions that V is bounded and that T is surjective onto the complementary of a bounded set are crucial. Otherwise we would easily build a counterexample taking the product $X \times {{\,\mathrm{{\mathbb {R}}}\,}}$ (cf. with Remark 6.2).

Proof of Proposition 6.3

Fix $x_0 \in \textrm{X}$ and observe that, up to increase R and enlarge V, it is not restrictive to assume that $x_0\in V$.

Set $D:=\text {diam}(V)$, $\delta _n:=\frac{1}{4r_n}(D+R)$ and define $\textrm{X}_n:=(\textrm{X},r_n^{-1}\textsf {d} _{\textrm{X}},r_n^{-N}{\mathfrak {m}}_{\textrm{X}},x_0)$. Without loss of generality we will assume that $r_n\ge 1.$

Since Y is an Euclidean N cone,it follows that $Y_n:=(Y,r_n^{-1}\textsf {d} _Y,r_n^{-N}{\mathfrak {m}}_Y)$ is isomorphic to $(Y,\textsf {d} _Y,{\mathfrak {m}}_Y)$ via the map $i_n:Y_n \rightarrow Y$, defined as $i_n(t,z):=(t/r_n,z)$ in polar coordinates, which satisfies $i_n(B_R(O_Y))=B_{r_n^{-1}R}O_Y$ for every $R>0.$ Observe that in particular ${i_n}_*{\mathfrak {m}}_Y=r_n^N{\mathfrak {m}}_Y.$

We extend T to the whole $\textrm{X}$ by setting $T(x)=O_Y$ for every $x \in V$ and we denote this new map again by T. It is straightforward to check that

$$\begin{aligned} |\textsf {d} _X(x_1,x_2)-\textsf {d} _Y(T(x_1),T(x_2))|\le 2(R+D), \quad \forall \, x_1,x_2 \in X. \end{aligned}$$

(6.2)

Define now the map $T_n : \textrm{X}_n \rightarrow Y$ as $T_n=i_n\circ T.$ It follows from (6.2) and the properties of $i_n$ that $T_n$ is a $\delta _n$-isometry. Moreover it can be readily checked that $B^Y_{R-D}(O_Y)\subset T(B^{\textrm{X}}_{R}(x_0))$, for any $R>D.$ In particular it follows that $B^Y_{R-\delta _n}(O_Y)\subset T_n(B^{\textrm{X}_n}_{R}(x_0))$, for every $R>D$. Finally we let $\varphi \in C_b(Y)$ be of bounded support, since T is measure preserving we have

$$\begin{aligned} r_n^{-N}\int \varphi \circ T_n {\mathrm d}{\mathfrak {m}}&=r_n^{-N}\int _V \varphi \circ T_n {\mathrm d}{\mathfrak {m}}+r_n^{-N}\int _{Y\setminus B_R(O_Y)} \varphi \circ i_n {\mathrm d}{\mathfrak {m}}_Y\\&=r_n^{-N}\int _V \varphi \circ {\mathrm d}{T_n}_*{\mathfrak {m}}+\int _{Y\setminus B_{r_n^{-1}R}(O_Y)} \varphi {\mathrm d}{\mathfrak {m}}_Y. \end{aligned}$$

Passing to the limit, observing that the first term on the right-hand side vanishes as $r_n\rightarrow +\infty$, we obtain $r_n^{-N}\int \varphi \circ T_n {\mathrm d}{\mathfrak {m}}\rightarrow \int \varphi {\mathrm d}{\mathfrak {m}}_Y$ as $r_n\rightarrow +\infty$. This concludes the first part.

The second part follows immediately from the closedness of the $\textrm{RCD}(0,N)$ condition under pmGH-convergence.

Suppose now that $\textrm{X}$ is an $\textrm{RCD}(0,N)$ space and $\text {diam}(Z)=\pi$. Then Y must contain a line. Therefore, since from the previous part Y is an $\textrm{RCD}(0,N)$ space, it follows from the splitting theorem [45, 46] that Y is isomorphic to $({{\,\mathrm{{\mathbb {R}}}\,}}\times Y',\textsf {d} _{Eucl}\times \textsf {d} ',{\mathcal {L}}^1\otimes {\mathfrak {m}}_Y')$ for some m.m.s. $(Y',\textsf {d} ',{\mathfrak {m}}_{Y'})$. In particular $O_Y=({{\bar{t}}}, {{\bar{y}}})$ for some ${{\bar{t}}} \in {{\,\mathrm{{\mathbb {R}}}\,}}$ and ${{\bar{y}}} \in Y'$ and ${\mathfrak {m}}_{Y}(B_r(O_Y))={\mathfrak {m}}_{Y}(B_r(s,{{\bar{y}}})),$ for any $r>0$ and any $s\in {{\,\mathrm{{\mathbb {R}}}\,}}.$

Therefore taking s big enough, we have that $O':=(s,\bar{y})\in Y$ satisfies $O'\in \{\textsf {d} _Y(.,O_Y)>R+1\}.$ Therefore, since $T|_{V^c}$ is a measure preserving local isometry, ${\mathfrak {m}}_Y(B_r(O'))={\mathfrak {m}}_X(B_r(T^{-1}(O')))$ holds for every $r \in (0,1).$ Hence

$$\begin{aligned} \lim _{r\rightarrow 0^+} \frac{{\mathfrak {m}}(B_r(T^{-1}(O')))}{r^N}= & {} \lim _{r\rightarrow 0^+}\frac{{\mathfrak {m}}_Y(B_r(O'))}{r^N}\\= & {} \lim _{r\rightarrow 0^+}\frac{{\mathfrak {m}}_Y(B_r(O_Y))}{r^N}={\mathfrak {m}}_Y(B_1(O_Y))=:\theta , \end{aligned}$$

since $O_Y$ is the vertex of Y. On the other hand, since $\textrm{X}_n:=(\textrm{X},r_n^{-1}\textsf {d} _{\textrm{X}},r_n^{-N}{\mathfrak {m}}_{\textrm{X}},x_0)\overset{{\text{ p }mGH}}{\longrightarrow }(Y,\textsf {d} _Y,{\mathfrak {m}}_Y,O_Y)$ we have

$$\begin{aligned} \lim _{r_n\rightarrow +\infty } \frac{{\mathfrak {m}}(B_{r_n}(T^{-1}(O')))}{{r_n}^N}= & {} \lim _{r_n\rightarrow +\infty }\frac{{\mathfrak {m}}_X(B_{r_n}(x_0))}{r_n^N}\\= & {} \lim _n {\mathfrak {m}}_{X_n}(B_1(x_0))={\mathfrak {m}}_Y(B_1(O_Y))=\theta . \end{aligned}$$

From Bishop-Gromov inequality,we deduce that $\frac{{\mathfrak {m}}(B_r({T^{-1}}(O')))}{r^N}=\theta$ for every $r>0$ and from [40, Thm. 1.1] we must have that $\textrm{X}$ is a cone, which must evidently coincide with Y. $\square$

6.1.2 Local a-Priori Estimate for Regular Lagrangian Flows in RCD Spaces

The following local version of the a-priori estimates in [16, Prop. 4.6] will be crucial for the argument of Appendix A to work (see Proposition A.6).

Proposition 6.5

Let $\{v_t,\mu _t\}_{t \in [0,T]}$ be as in Theorem 2.29. Assume additionally that $\{\mu _t\}_{t\in [0,T]}$ are all concentrated in a common bounded Borel set B. Then setting $\rho _t :=\frac{{\mathrm d}\rho _t}{{\mathrm d}{\mathfrak {m}}}$, $t \in [0,T]$, it holds that

$$\begin{aligned} \sup _{t \in (0,T)} \Vert \rho _t\Vert _{L^{\infty }}\le \Vert \rho _0\Vert _{L^{\infty }} e^{\int _0^T \Vert \textrm{div}(v_t)^-\Vert _{L^{\infty }(B_t) }\, {\mathrm d}t}\,, \end{aligned}$$

(6.3)

for any family $\{B_t\}_{t\in [0,T]}$ of Borel sets such that $\rho _t=0$ ${\mathfrak {m}}$-a.e. in $B_t^c$ and the map $(x,t)\mapsto {\chi }_{B_t}(x)$ is Borel.

For the proof of Proposition 6.5,we will need the following :

Lemma 6.6

(Commutator estimate [16, Lemma 5.8]) Let $\textrm{X}$ be an $\textrm{RCD}(K,\infty )$ m.m.s., then there exists a positive constant $C=C(K)>0$ such that the following holds. Let $v \in W^{1,2}_{C}(TX)$ with $\textrm{div}(v)\in L^\infty ({\mathfrak {m}})$, then

$$\begin{aligned}{} & {} \int \langle \nabla h_t (f),v\rangle g {\mathrm d}{\mathfrak {m}}+ \int f \textrm{div}(h_t(g)v_t) {\mathrm d}{\mathfrak {m}}\nonumber \\{} & {} \quad \le C \left( \Vert \nabla v\Vert _{L^2(T^{\otimes 2}X)}+\Vert \textrm{div}(v)\Vert _{L^\infty } \right) \Vert f\Vert _{L^2\cap L^4}\Vert g\Vert _{L^2\cap L^4}, \end{aligned}$$

(6.4)

for every $f,g \in L^2({\mathfrak {m}})\cap L^4({\mathfrak {m}})$ and every $t>0$. In particular for fixed g,the left-hand side of (6.4) defines a functional in $(L^2({\mathfrak {m}}) \cap L^4({\mathfrak {m}}))^*=L^2({\mathfrak {m}})+L^{4'}({\mathfrak {m}})$, denoted by ${\mathscr {C}}^t(g,v)$, which satisfies

$$\begin{aligned} \Vert {\mathscr {C}}^t(g,v)\Vert _{L^2({\mathfrak {m}})+L^{4'}({\mathfrak {m}})}\le C \left( \Vert \nabla _{sym} v\Vert _{L^2(T^{\otimes 2}X)}+\Vert \textrm{div}(v)\Vert _{L^\infty } \right) \Vert g\Vert _{L^2\cap L^4}. \end{aligned}$$

(6.5)

Moreover it holds that

$$\begin{aligned} \Vert {\mathscr {C}}^t(g,v)\Vert _{L^2({\mathfrak {m}})+L^{4'}({\mathfrak {m}})}\rightarrow 0, \quad \quad \text { as } t \rightarrow 0^+. \end{aligned}$$

(6.6)

We can now pass to the proof of the local a-priori estimate.

Proof of Proposition 6.3

We start with the preliminary observation that, combining the fact that $\mu _t={F_t}_*\mu _0$ with (2.23) and recalling that B is bounded, we have

$$\begin{aligned} \sup _{t \in [0,T]} \Vert \rho _t\Vert _{L^q({\mathfrak {m}})}<+\infty , \quad \forall \,q\in [1,\infty ]. \end{aligned}$$

(6.7)

To conclude it is sufficient to prove that for every $p>1$ the function $[0,T]\ni t \mapsto \int (\rho _t)^p {\mathrm d}{\mathfrak {m}}$ is absolutely continuous and

$$\begin{aligned} \frac{{\mathrm d}}{{\mathrm d}t} \int (\rho _t)^p {\mathrm d}{\mathfrak {m}}\le (p-1)\int (\rho _t)^p \textrm{div}(v_t)^- {\mathrm d}{\mathfrak {m}}, \quad \text { for a.e. } t\in (0,T). \end{aligned}$$

(6.8)

Indeed (6.3) would follow first applying Gronwall Lemma (noticing that $\rho _t=0$ ${\mathfrak {m}}$-a.e. outside $B_t$) and then letting $p \rightarrow +\infty .$

So we fix $p>1.$ Pick a sequence $s_n \downarrow 0$ and define $\rho _t^n:= h_{s_n}\rho _t.$ From the fact that $\rho _t$ is a solution of the continuity equation and the selfadjointedness of the heat flow, we obtain that for every $f \in \textsf {LIP} _{bs}(\textrm{X}),$ the function $t\mapsto \int f\rho _t^n {\mathrm d}{\mathfrak {m}}$ is absolutely continuous and

$$\begin{aligned} \frac{{\mathrm d}}{{\mathrm d}t}\int f \rho _t^n {\mathrm d}{\mathfrak {m}}= & {} \int \langle \nabla h_{s_n}f,v_t\rangle \rho _t \, {\mathrm d}{\mathfrak {m}}\nonumber \\= & {} \int f[{\mathscr {C}}^{s_n}(\rho _t,v_t)-\text {div}(\rho _t^n v_t)] {\mathrm d}{\mathfrak {m}}, \quad \text { for a.e. } t\in (0,T), \end{aligned}$$

(6.9)

where ${\mathscr {C}}^{s_n}(\rho _t,v_t)$ is defined as in Lemma 6.4. Set now $\eta _t^n:={\mathscr {C}}^{s_n}(\rho _t,v_t)-\textrm{div}(\rho _t^n v_t)$. From the Leibniz rule and the $L^\infty$-to Lipschitz regularization of the heat flow (2.11) we have that

$$\begin{aligned} \Vert \textrm{div}(\rho _t^n v_t)\Vert _{L^2}\le \Vert \rho _t\Vert _{L^2} \Vert \textrm{div}(v_t)\Vert _{L^\infty }+c(K)\frac{\Vert \rho _t\Vert _{L^\infty }}{\sqrt{s_n}}\Vert |v_t|\Vert _{L^2}. \end{aligned}$$

This bound together with (6.5), (6.7) and the hypotheses on $v_t$, guarantees that $\eta _t^n \in L^1((0,T),L^2({\mathfrak {m}})+L^{4'}({\mathfrak {m}})).$ Denote by V the Banach space $L^2({\mathfrak {m}})+L^{4'}({\mathfrak {m}})$ and observe that $L^2\cap L^4=V^*$. Then (6.9) can be restated as: for a weakly*-dense set of $\varphi \in V^*$ the function $[0,T] \ni t \mapsto \varphi (\rho _t^n)$ is absolutely continuous and $\frac{{\mathrm d}}{{\mathrm d}t}\varphi (\rho _t^n)=\varphi (\eta _t^n).$ It follows (see e.g. Remark 4.9 in [16]) that $\rho _t^n$ is absolutely continuous in $L^1((0,T),V)$ and strongly differentiable a.e. with $\frac{{\mathrm d}}{{\mathrm d}t}\rho _t^n=\eta _t^n.$

Pick a convex function $\beta : [0+\infty ) \rightarrow [0+\infty )$ such that $\beta (t)=t^p$ for every $t \le 2\sup _{t \in (0,T)} \Vert \rho _t\Vert _{L^\infty }<+\infty$ and such that $\beta '$ is globally bounded. In particular from the maximum principle for the heat flow we have that $\beta (\rho _t^n)=(\rho _t^n)^p$ for every t and n. Moreover, since $\rho _t$ are uniformly bounded in $L^q({\mathfrak {m}})$ for every $1\le q \le \infty$, from the contractivity of the heat flow we have also that $\rho _t^n$ are bounded in $L^q({\mathfrak {m}})$ for every $1\le q \le \infty$, uniformly in t and n. Finally, observing that $\beta '(t)/t$ is globally bounded, we deduce that $\beta '(\rho _t^n)$ are again bounded in $L^q({\mathfrak {m}})$ for every $1\le q <\infty$, uniformly in t and n.

Observe now that from the convexity of $\beta$, we have that

$$\begin{aligned} \int \beta (\rho _t^n)-\beta (\rho _s^n) {\mathrm d}{\mathfrak {m}}\le \int \beta '(\rho _t^n) (\rho _t^n-\rho _s^n) {\mathrm d}{\mathfrak {m}}, \quad \forall t,s \in [0,T]. \end{aligned}$$

(6.10)

This in turn gives

$$\begin{aligned} \int \beta (\rho _t^n)-\beta (\rho _s^n) {\mathrm d}{\mathfrak {m}}\le & {} \sup _{t \in [0,T]}\Vert \beta '(\rho _t^n)\Vert _{L^2\cap L^4({\mathfrak {m}})} \Vert \rho _t^n-\rho _s^n \Vert _{L^2+L^{4'}}\\\le & {} \sup _{t \in [0,T]}\Vert \beta '(\rho _t^n)\Vert _{L^2\cap L^4} \int _s^t \Vert \eta _r^n\Vert _{L^2+L^{4'}} {\mathrm d}r, \end{aligned}$$

for every $t,s \in [0,T],$ with $s\le t.$ Hence the function $\int \beta (\rho _t^n){\mathrm d}{\mathfrak {m}}$ is absolutely continuous in [0, T] and from (6.10),we deduce that

$$\begin{aligned} \frac{{\mathrm d}}{{\mathrm d}t}\int \beta (\rho _t^n){\mathrm d}{\mathfrak {m}}\le \int \beta '(\rho _t^n) \eta _t^n {\mathrm d}{\mathfrak {m}}, \quad \text { for a.e. } t\in (0,T). \end{aligned}$$

Then from the definition of $\eta _t^n$ and $\beta$ and integrating by parts,we obtain

$$\begin{aligned} \frac{{\mathrm d}}{{\mathrm d}t}\int \beta (\rho _t^n){\mathrm d}{\mathfrak {m}}&\le - \int [\beta '(\rho _t^n)\rho _t^n-\beta (\rho _t^n)]\textrm{div}(v_t) {\mathrm d}{\mathfrak {m}}+ \int {\mathscr {C}}^{s_n}(\rho _t,v_t) \beta '(\rho _t^n) {\mathrm d}{\mathfrak {m}}\\&\le (p-1)\int (\rho _t^n)^p \textrm{div}(v_t)^- {\mathrm d}{\mathfrak {m}}+ p\int {\mathscr {C}}^{s_n}(\rho _t,v_t) (\rho _t^n)^{p-1} {\mathrm d}{\mathfrak {m}}, \end{aligned}$$

for a.e. $t \in (0,T).$ Observe now that combining (6.5) with (6.6), an application of dominated convergence gives that $\int _0^T\Vert {\mathscr {C}}^{s_n}(\rho _t,v_t)\Vert _{L^2+L^{4'}} \rightarrow 0$ as $s_n \rightarrow 0$. Then, recalling that $\textrm{div}(v_t)^- \in L^\infty ((0,T),L^\infty ({\mathfrak {m}}))$ and (6.7), we can let $s_n \rightarrow 0$ in the above and obtain at once the absolute continuity of $\int (\rho _t)^p {\mathrm d}{\mathfrak {m}}$ together with (6.8). $\square$

6.2 From Almost Outer Functional Cone to Almost Outer Metric Cone

Our aim in this section is to prove the following.

Theorem 6.7

For every $\varepsilon \in (0,1/3)$, $R_0>0$, $\gamma >\frac{1}{2}\frac{N-2}{N-1}$, $N\in [2,\infty )$, $L>0$ there exists $0<\delta =\delta (\varepsilon ,\gamma ,N,R_0,L)$ such that the following holds. Let $(\textrm{X},\textsf {d} ,{\mathfrak {m}},x_0)$ be a pointed $\textrm{RCD}(-\delta ,N)$ m.m.s. with ${\mathfrak {m}}(B_1(x_0))\in (\varepsilon ,\varepsilon ^{-1})$. Let $U\subset \textrm{X}$ be open with $U^c\subset B_{R_0}(x_0)$ and $v\in D({\varvec{\Delta }},U)\cap C(U)$ be positive and such that $\limsup _{x\rightarrow \partial U}v(x)\le 1$, $\Delta v=N|\nabla \sqrt{2v}|^2$, $v \ge 1+\varepsilon$ in $B_{R_0}(x_0)^c\ne \emptyset$ and $\Vert |\nabla \sqrt{v}|\Vert _{L^\infty (U)}\le L$. Suppose furthermore that

$$\begin{aligned} \int _{U} \frac{1}{v^{N-2}} \left| \nabla |\nabla \sqrt{v}|^\gamma \right| ^2\, {\mathrm d}{\mathfrak {m}}<\delta , \end{aligned}$$

(6.11)

where it is intended that $v^{N-2}\equiv 1$ in the case $N=2$.

Then there exists a pointed $\mathrm{RCD(0,N)}$ space $(\textrm{X}',\textsf {d} ',{\mathfrak {m}}',x')$ such that

$$\begin{aligned} \textsf {d} _{pmGH}((\textrm{X},\textsf {d} ,{\mathfrak {m}},x_0),(\textrm{X}',\textsf {d} ',{\mathfrak {m}}',x'))<\varepsilon \end{aligned}$$

(6.12)

and $(\textrm{X}',\textsf {d} ',{\mathfrak {m}}',x')$ is a truncated cone outside a compact set $K\subset B_{2R_0}(x'),$ i.e. there exists an $\mathrm{RCD(0,N)}$ N cone Y, with vertex $O_Y$, over an $\textrm{RCD}(N-2,N-1)$ space Z, and a measure preserving local isometry $T: \textrm{X}'\setminus K\rightarrow Y\setminus {{\bar{B}}}_{r}(O_Y)$, for some $r>L^{-1}.$

Remark 6.8

Observe that inequality (6.12) is non-trivial because $\textrm{X}'$ is a cone outside a compact set $K\subset B_{2R_0}(x')$ with $R_0$ independent of $\varepsilon >0$. This means that for an arbitrary large radius $R\gg R_0$, up to choosing $\varepsilon >0$ sufficiently small, we have that the ball $B_{R}(x_0)$ is as close as we want in the GH-sense to a ball of radius $R>0$, which is a truncated cone up to removing a set of diameter less than $4R_0$ (which is fixed and independent of R).

Observe that (6.11) makes sense thanks to Corollary 4.5.

Notice also that the assumption $u \ge 1+\varepsilon$ in $B_{R_0}(x_0)^c$ is necessary as the function $v\equiv 1$ shows. We will also prove another version of the above result, which is Theorem 6.9. Before passing to its statement, we need some definitions and notations.

For any couple of compact metric spaces $(X_1,\textsf {d} _1),\, (X_2,\textsf {d} _2)$,we define their Gromov Hausdorff distance as

$$\begin{aligned} \begin{aligned} \textsf {d} _{GH}((X_1,\textsf {d} _2),(\textrm{X}_2,\textsf {d} _2)):=\inf \{&\varepsilon >0 : \exists f: X_1\rightarrow X_2 \ \text { such that} \ f(\textrm{X}_1) \text { is } \varepsilon \text {-dense in } \textrm{X}_2\\&\text { and } |\textsf {d} _1(x,y)-\textsf {d} _2(f(x),f(y))|\le \varepsilon , \, \forall x,y \in X_1\}. \end{aligned} \end{aligned}$$

For a sequence of compact metric spaces $(X_n,\textsf {d} _n)$ converging to $(\textrm{X},\textsf {d} )$ in the GH-sense, we say that a sequence of maps $f_n : \textrm{X}_n \rightarrow \textrm{X}$ realizes the convergence if there exists a sequence $\varepsilon _n \rightarrow 0$ such that $f(\textrm{X}_n)$ is $\varepsilon _n$-dense in $\textrm{X}$ and $|\textsf {d} _n(x,y)-\textsf {d} (f_n(x),f_n(y))|\le \varepsilon _n$, $\forall x,y \in X_n$.

Moreover we recall the notion of Sturm distance ${\mathbb {D}}$ for compact m.m.s. which was first introduced in [80] in the case of renormalized spaces (see also [54]):

$$\begin{aligned} {\mathbb {D}}((X_1,\textsf {d} _2,{\mathfrak {m}}_2),(\textrm{X}_2,\textsf {d} _2,{\mathfrak {m}}_1)):=\inf \left| \log \frac{{\mathfrak {m}}_1(\textrm{X}_1)}{{\mathfrak {m}}_2(\textrm{X}_2)}\right| +W_2\left( {\iota _1}_*{\tilde{{\mathfrak {m}}}}_1,{\iota _2}_*\tilde{\mathfrak {m}}_2\right) , \end{aligned}$$

where ${\tilde{{\mathfrak {m}}}}_i=\frac{{\mathfrak {m}}_i}{{\mathfrak {m}}(\textrm{X}_i)} ,$ $i=1,2$ and the infimum is taken among all the complete and separable metric spaces $(Y,\textsf {d} )$ and isometric embeddings $X_i \overset{\iota _i}{\hookrightarrow }Y$, $i=1,2.$ Observe that ${\mathbb {D}}$ is well defined since we are assuming that $\mathop {\textrm{supp}}\nolimits ({\mathfrak {m}}_i)=\textrm{X}_i.$

It is important to recall that an upper bound on ${\mathbb {D}}$ does not imply in general an upper bound on $\textsf {d} _{GH}$, indeed this holds only if we restrict ourselves to a family of uniformly doubling metric measure spaces. However, we will need to apply ${\mathbb {D}}$ to spaces which are not a-priori uniformly doubling, for this reason we will need to work both with $\textsf {d} _{GH}$ and with ${\mathbb {D}}.$

Finally, for any $(\textrm{X},\textsf {d} )$ complete metric space and $A\subset \textrm{X}$ we define the intrinsic metric on A to be the distance function $\textsf {d} ^{A}: A \rightarrow [0,+\infty ]$ defined by

$$\begin{aligned} \textsf {d} ^A(x,y):=\inf _\gamma \, L(\gamma ),\quad \forall x,y \in A, \end{aligned}$$

where the infimum is taken among all curves $\gamma \in AC([0,1],\textrm{X})$ with values in A and such that $\gamma (0)=x$, $\gamma (1)=y.$

If we also assume that A is relatively compact, then for every $x,y \in A$ such that $\textsf {d} (x,y)<+\infty$, there exists an absolutely continuous curve $\gamma$ with values in ${{\bar{A}}}$ such that $\textsf {d} ^A(x,y)=L(\gamma ).$

We are ready to state the second version of Theorem 6.7, which is more in the spirit of the ‘volume annulus implies metric annulus’ theorem of Cheeger and Colding, which is also based on a functional formulation similar to the present one (see [29, Theorem 4.85]).

Theorem 6.9

For every $\varepsilon \in (0,1/3)$, $R_0>0$, $\gamma > \frac{1}{2}\frac{N-2}{N-1}$, $N\in [2,\infty )$, $L>0$, $\eta \in (0,\varepsilon ^{-1})$ there exists $0<\delta =\delta (\varepsilon ,\gamma ,N,R_0,L,\eta )$ such that, given $\textrm{X}$, U and v as in Theorem 6.7, there exists an $\mathrm{RCD(0,N)}$ N cone $(Y,\textsf {d} _Y,{\mathfrak {m}}_Y)$ with vertex $O_Y$, over an $\textrm{RCD}(N-2,N-1)$ space Z, and a constant $\lambda \in (0,L)$ such that the following holds.

For every $1+\varepsilon +\eta<t_1\le t_2<\varepsilon ^{-1}$ satisfying $\{ \sqrt{v}\le t_2+2\eta \}\subset B_{R_0}(x_0)$, it holds

$$\begin{aligned} \textsf {d} _{GH}\left( (\{t_1\le \sqrt{v}\le t_2\}, \textsf {d} _{\textrm{X}}^{\eta }),\left( \{t_1\le \lambda \textsf {d} _{O_Y}\le t_2\}, \textsf {d} _{Y}^{\eta }\right) \right) <\varepsilon , \end{aligned}$$

(6.13)

where $\textsf {d} _{O_Y}:=\textsf {d} _Y(.,O_Y)$ and $\textsf {d} _{\textrm{X}}^{\eta }$ and $\textsf {d} _{Y}^{\eta }$ denote the intrinsic metrics on $\{t_1-\eta< \sqrt{v}< t_2+\eta \}$ and on $\{t_1-\eta<\lambda \textsf {d} _{O_Y}< t_2+\eta \}$ (see definition above). Moreover, provided that $t_1+\varepsilon <t_2$,

$$\begin{aligned} {\mathbb {D}}\left( \big (\{t_1\le \sqrt{v}\le t_2\}, \textsf {d} _{\textrm{X}}^{\varepsilon },{\mathfrak {m}}{_{|_{\{t_1\le \sqrt{v}\le t_2\}}}}\big ),\big (\{t_1\le \lambda \textsf {d} _{O_Y}\le t_2\}, \textsf {d} _{Y}^{\varepsilon },{\mathfrak {m}}_Y{_{|_{\{t_1\le \lambda \textsf {d} _{O_Y}\le t_2\}}}}\big ) \right) <\varepsilon .\nonumber \\ \end{aligned}$$

(6.14)

Moreover the cone Y can be taken so that the conclusion of Theorem 6.7 holds (with the same $\varepsilon$, $R_0$ and L) with Y and for some $\textrm{RCD}(0,N)$ space $\textrm{X}'$.

We point out that in general we cannot say anything better than $\lambda \le L$. This is immediately seen by taking $v=L^2|x|^2$ in ${{\,\mathrm{{\mathbb {R}}}\,}}^n$ and $U= {{\,\mathrm{{\mathbb {R}}}\,}}^n\setminus {{\bar{B}}}_{1/L}(0).$

It is important to notice that the information in (6.13) is not contained in (6.14), indeed, as said above, it is not clear to us if, fixed $\varepsilon ,\gamma ,N,R_0,L$ as in Theorem 6.9, the metric measure spaces $(\{\sqrt{t_1}\le \sqrt{v}\le \sqrt{t_2}\}, \textsf {d} _{\textrm{X}}^{\eta },{\mathfrak {m}}{_{|_{\{t_1\le \sqrt{v}\le t_2\}}}})$, for arbitrary $v,t_1,t_2$ as in the hypotheses, satisfy some uniform doubling condition.

For the proof of Theorem 6.9, we will need the following elementary lemma. The proof is a direct consequence of the definition of distance in a cone and will be omitted.

Lemma 6.10

Let $(Y,\textsf {d} )$ be an Euclidean cone of vertex $O_Y$ and for any $0<a<b$ let $\textsf {d} _{a,b}$ be the intrinsic metric on $\{a<d(.,O_Y)<b\}$. Then for every $0<\varepsilon<a<b$ it holds

$$\begin{aligned} \textsf {d} _{a-\varepsilon ,b+\varepsilon }\le \textsf {d} _{a,b}\le \frac{a}{a-\varepsilon }\textsf {d} _{a-\varepsilon ,b+\varepsilon }, \quad \text { in } \{a<d(.,O_Y)<b\}. \end{aligned}$$

Moreover for any two sequences $(a_n),(b_n)$ such that $a_n \rightarrow a$, $b_n \rightarrow b$ it holds

$$\begin{aligned} (\{a_n\le d(.,O_Y)\le b_n\},\textsf {d} _{a_n-\varepsilon ,b_n+\varepsilon })\overset{GH}{\rightarrow }\ (\{a\le d(.,O_Y)\le b\},\textsf {d} _{a-\varepsilon ,b+\varepsilon }) \end{aligned}$$

and the map $f_n(t,z):=\left( \frac{(t-a_n)(b-a)}{(b_n-a_n)}+a,z \right)$ (in polar coordinates) realizes such convergence.

Finally for proof of Theorem 6.9,we will need the following result (see below for the definition of mGH-convergence):

Proposition 6.11

([54, Prop. 3.30]) Let $(X_n,\textsf {d} _n,{\mathfrak {m}}_n)$ be compact m.m. spaces mGH-converging to a compact m.m.s. $(\textrm{X}_\infty ,\textsf {d} _\infty ,{\mathfrak {m}}_\infty )$. Then

$$\begin{aligned} {\mathbb {D}}((X_n,\textsf {d} _n,{\mathfrak {m}}_n),(\textrm{X}_\infty ,\textsf {d} _\infty ,{\mathfrak {m}}_\infty ))\rightarrow 0. \end{aligned}$$

Definition 6.12

(measure Gromov Hausdorff convergence) We say that the sequence $(\textrm{X}_n,\textsf {d} _n,{\mathfrak {m}}_n)$ of m.m.s. measure Gromov Hausdorff-converges (mGH-converges in short) to a compact m.m.s. $(\textrm{X}_\infty ,\textsf {d} _\infty ,{\mathfrak {m}}_\infty )$, if there are Borel maps $f_n:X_n\rightarrow X_\infty$ such that

• (1) $\sup _{x,y\in B_{R_n}(x_n)}|\textsf {d} _n(x,y)-\textsf {d} _\infty (f_n(x),f_n(y))|\le \varepsilon _n$,

• (2) $f_n(\textrm{X}_n)$ is $\varepsilon _n$-dense in $\textrm{X}_\infty$,

• (3) ${f_n}_*{\mathfrak {m}}_n\rightharpoonup {\mathfrak {m}}_\infty$ in duality with $C(\textrm{X}_\infty )$.

Notice that if $(\textrm{X}_n,\textsf {d} _n,{\mathfrak {m}}_n) \rightarrow (\textrm{X}_\infty ,\textsf {d} _\infty ,{\mathfrak {m}}_\infty )$ in the mGH-sense, then $\textsf {d} _{GH}((\textrm{X}_n,\textsf {d} _n,),(\textrm{X}_\infty ,\textsf {d} _\infty ))\rightarrow 0.$

Proof of Theorems 6.7 and 6.9

The proof of Theorem 6.7 is essentially the same as Theorem 6.9, except that it stops earlier. For this reason,we will prove both theorems together. The reader interested only in the proof of the first result can ignore the second half of the argument.

Proof of Theorem 6.7:

We argue by contradiction. Suppose that there exist numbers $\varepsilon \in (0,1/3),N\in [2,\infty )$, $R_0>0$, $\gamma \ge \frac{1}{2}\frac{N-2}{N-1}$, $L>0$, a sequence $\delta _n \rightarrow 0,$ a sequence $(X_n,\textsf {d} _n,{\mathfrak {m}}_n,x_n)$ of $\textrm{RCD}(-\delta _n,N)$ m.m.s., a sequence of open sets $U_n\subset X_n$, with $U_n^c \subset B_{R_0}(x_n)$, functions $v_n\in D({\varvec{\Delta }},U_n)$ satisfying $\Delta v_n=2N|\nabla \sqrt{v_n}|^2$ such that:

(a):

$\limsup _{x\rightarrow \partial U_n}v_n(x)\le 1$, $v_n \ge 1+\varepsilon$ in $B_{R_0}(x_0)^c\ne \emptyset$, $\,\Vert |\nabla \sqrt{v}_n|\Vert _{L^\infty (U_n)}\le L,$

(b):

${\mathfrak {m}}_n(B_1(x_n))\in (\varepsilon ,\varepsilon ^{-1})$,

(c):

(6.11) holds (with $v_n$, ${\mathfrak {m}}_n$, $\gamma$ and $\delta =\delta _n$),

(d):

for every n the conclusion of Theorem 6.7 does not hold.

We first observe that, since $v_n \ge 1+\varepsilon$ in $B_{R_0}(x_0)^c$, removing the set $\{v_n\le 1\}$ from $U_n$ does not effect neither the hypotheses of the theorems nor their conclusions, therefore it is not restrictive to assume that $v_n>1$ in $U_n$.

Moreover by compactness, up to passing to a nonrelabelled subsequence, we can assume that the p.m.m. spaces $(X_n,\textsf {d} _n,{\mathfrak {m}}_n,x_n)$ pmGH-converge to a to an $\mathrm{RCD(0,N)}$ pointed m.m.s. $(X_\infty ,\textsf {d} _\infty ,{\mathfrak {m}}_\infty ,x_\infty ).$

Passing to the extrinsic approach, we consider a proper metric space $(Y,\textsf {d} _Y)$ that realizes such convergence, in particular we identify the metric spaces $X_n$ and $\textrm{X}_\infty$ with the corresponding subsets of Y such that $\textsf {d} _Y(x_n,x_\infty )\rightarrow 0$, ${\mathfrak {m}}_n \rightharpoonup {\mathfrak {m}}_\infty$ in duality with $C_{bs}(Y)$ and $\textsf {d} ^Y_H(B^{\textrm{X}_n}_R(x_n),B^{\textrm{X}_\infty }_R(x_\infty ))\rightarrow 0$ for every $R>0.$ In particular for every $x \in \textrm{X}_\infty$, there exists a sequence $y_n \in \textrm{X}_n$ such that $\textsf {d} _{Y}(x,y_n)\rightarrow 0$ and conversely for every $R>0$ and every sequence $y_n \in B^{\textrm{X}_n}_{R}(x_n)$ there exists a subsequence converging to a point $x \in \textrm{X}_\infty$. These two facts will be used repeatedly in the proof without further notice.

Define the compact sets $K_n=U_n^c\subset \textrm{X}_n$ and observe that, since $K_n\subset B_{R_0}(x_n)$, they are all contained in a common ball in Y centred at $x_\infty .$ Hence from the metric version of Blaschke’s theorem (see [20, Theorem 7.3.8]), there exists a compact set $K_\infty \subset \textrm{X}_\infty$ such that, up to a subsequence, $\textsf {d} ^Y_H(K_\infty ,K_n)\rightarrow 0.$

Define the open (in the topology of $\textrm{X}_\infty$) set $U_\infty :=X_\infty \setminus K_\infty$ and for every $r>0$, define the open sets $U_n^{<r}=\{x \in X_n \ : \ \textsf {d} _n(x,K_n)<r\}$ and $U^{<r}_\infty =\{x \in X_\infty \ : \ \textsf {d} _\infty (x, K_\infty )<r\}$. Analogously we define the sets $U_n^{>r},U_n^{\le r},U_n^{\ge r}$ and the corresponding ones for $n=\infty .$

From the assumptions $\limsup _{x\rightarrow \partial U_n}v_n(x)\le 1$ and $\Vert |\nabla \sqrt{v}_n|\Vert _{L^\infty (U_n)}\le L$, applying Proposition 2.10,it follows that

$$\begin{aligned} v_n \le (1+RL)^2, \text { in } B_R(x_n)\cap U_n, \quad \{v_n\le 1+\varepsilon /4\} \text { in } U_n^{\le 4\rho }\cap U_n, \end{aligned}$$

(6.15)

for every $R>0$ and for some small constant $0<\rho =\rho (\varepsilon ,L)<\varepsilon$, independent of n.

For every n and every $k \in {\mathbb {N}}$ with $k\ge R_0+100$, thanks to Proposition 2.13, there exists a cut-off function $\eta _k^n \in \textsf {Test} (X)$, $0\le \eta _k^n\le 1$, such that $\mathop {\textrm{supp}}\nolimits \eta _n^k\subset U_n^{>\rho /2}\cap B_{k+2}(x_n)$, $\eta _n^k=1$ on $U_n^{\ge \rho }\cap B_{k+1}(x_n)$ and $\textrm{Lip}\eta _n^k+|\Delta \eta _n^k|\le C$, for some constant C depending only on $\varepsilon ,N,L$. Observe also that we can choose $\eta _n^k$ so that $\eta _n^k=\eta _n^{k+1}$ in $B_{k+1}(x_n)$, for every k.

Define the functions $v_n^k:=v_n\eta _n^k, \, \tilde{v}_n^k:=\sqrt{v_n}\eta _n^k$ and observe that from (6.15) and the assumption $\Vert |\nabla \sqrt{v}_n|\Vert _{L^\infty (U_n)}\le L$ they are equi-Lipschitz, equibounded in n and all supported on $B_{k+2}(x_n).$ Hence by Ascoli-Arzelà (see Prop. 2.21), up to a subsequence, as $n \rightarrow +\infty$, they converge uniformly to functions $v_\infty ^k,\tilde{v}_\infty ^k\in C(X_\infty )$ with support in ${{\bar{B}}}_{k+2}(x_\infty )$.

From Proposition 2.23, it follows that $v_n^k,\tilde{v}_n^k$ converge also strongly in $L^2$, respectively, to $v_\infty ^k,{\tilde{v}}_\infty ^k$. It is clear from the construction and the uniform convergence that $v_\infty ^k=v_\infty ^{k+1}$ on $B_{k}(x_\infty )$. Therefore the assignment $v_\infty :=v_\infty ^k$ in $B_{k}(x_\infty )$ for every k, well defines a function $v_\infty \in C(\textrm{X}_\infty ).$ Analogously we can define ${\tilde{v}}_\infty \in C(\textrm{X}_\infty )$ and we observe that $\tilde{v}_\infty =\sqrt{v_\infty }$ in $U_\infty ^{\ge 2\rho }$.

We make the following two claims:

(A):

$v_\infty \le 1+\varepsilon /4$ in $U_\infty ^{\le 3\rho }$,

(B):

$\{v_\infty >1+\varepsilon /4 \}\ne \emptyset$, $v_\infty \in D({\varvec{\Delta }})$ and (up to multiplying $v_\infty$ by a positive constant $C_0$) it holds that $\Delta v_\infty =N$, $|\nabla v_\infty |^2=2v_\infty$ ${\mathfrak {m}}$-a.e. in $\{v_\infty >1+\varepsilon /4 \}$.

We start with claim (A). It is clearly enough prove that

$$\begin{aligned} v_\infty ^k\le 1+\varepsilon /4, \quad \text { in } {U_\infty ^{\le 3\rho }} \end{aligned}$$

for any k. Pick any $y \in U_\infty ^{\le 3\rho }$, then there exists a sequence $y_n \in X_n$ such that $y_n\rightarrow y$ in Y and by uniform convergence $v_n^k(y_n)\rightarrow v_\infty ^k(y)$. Moreover it must hold that $\textsf {d} _Y(y_n,K_n)<4\rho$ if n is big enough. If $y_n \notin U_n$, then $v_n^k(y_n)=0$ by construction. If instead $y_n \in U_n$ from the second in (6.15) and the fact that $\eta _k^n\le 1$ we deduce that $v_n^k(y_n)\le v_n(y_n)\le 1+\varepsilon /4.$ Combining these two observations, we get claim (A).

We pass to the proof of claim (B). It is easy to check, since $v_n\ge 1$ in $U_n$ (recall the observation made at the beginning of the proof), that $\sup _n \Vert \Delta v_{n}^k\Vert _{L^\infty (X_n)}<+\infty$ and $\sup _n \Vert \Delta \tilde{v}_{n}^k\Vert _{L^\infty (X_n)}<+\infty .$ Moreover by Bishop-Gromov inequality,we have $\sup _n{\mathfrak {m}}_n(B_R(x_n))<+\infty$, for every $R\ge 1$. Therefore $\sup _n \Vert \Delta v_{n}^k\Vert _{L^2(X_n)},$ $\sup _n\Vert \Delta {\tilde{v}}_{n}^k\Vert _{L^2(X_n)}<+\infty$ and applying Theorem 2.24 we deduce that $v_\infty ^k,\tilde{v}_\infty ^k \in D(\Delta )$, that $\Delta v_n^k$ converges to $\Delta v_\infty ^k$ weakly in $L^2$ and that $|\nabla \tilde{v}_n^k|\rightarrow |\nabla {\tilde{v}}_\infty ^k|$ strongly in $L^2$. The locality of Laplacian follows that $v_\infty \in D({\varvec{\Delta }}).$ Additionally, since $L':=\sup _n \Vert |\nabla \tilde{v}_n^k|\Vert _{L^\infty }<+\infty$, applying a) of Proposition 2.23 (with $\varphi (t)=(t\wedge L')^{\alpha }$), we also deduce that $| \nabla {\tilde{v}}_n^k|^{\alpha }\rightarrow |\nabla \tilde{v}_\infty ^k|^{\alpha }$ strongly in $L^2,$ for every $\alpha >0.$ We make the intermediate claim that

$$\begin{aligned} \Delta v_\infty = N |\nabla \sqrt{2v_\infty }|^2, \quad {\mathfrak {m}}\text {-a.e. in } U_{\infty }^{>2\rho }\cap B_{k}(x_\infty ). \end{aligned}$$

(6.16)

In particular from Corollary 4.5,this implies that $|\nabla \sqrt{v_\infty }|^{\gamma } \in W ^{1,2}_{\textsf {loc} }(U_{\infty }^{>2\rho }\cap B_{k}(x_\infty ))$. To prove (6.16) pick any $\varphi \in \textsf {LIP} _c(U_{\infty }^{>2\rho }\cap B^{\textrm{X}_\infty }_{k}(x_\infty ))$. Consider also a function $\eta \in \textsf {LIP} (Y)$ such that $\eta \equiv 1$ in $\mathop {\textrm{supp}}\nolimits \, \varphi$, $\textsf {d} _Y(\mathop {\textrm{supp}}\nolimits \eta , K_\infty )>2\rho$ and $\mathop {\textrm{supp}}\nolimits \eta \subset B^Y_k(x_\infty )$. Moreover, since $\textsf {d} _H^Y(K_n,K_\infty )\rightarrow 0,$ for n big enough, we have $\{y \ : \ \textsf {d} _Y(y,K_\infty )>2\rho \}\subset \{y \ : \ \textsf {d} _Y(y,K_n)>\rho \}$ and analogously, since $x_n\rightarrow x_\infty$ in Y, for n big enough $B^Y_{k}(x_\infty )\subset B^Y_{k+1}(x_n).$ Therefore $\mathop {\textrm{supp}}\nolimits \eta \cap X_n \subset U_n^{>\rho }\cap B^{\textrm{X}_n}_{k+1}(x_n)$ for n big enough. We now extend $\varphi$ to a function $\varphi '\in \textsf {LIP} (Y)$ and define ${\bar{\varphi }}=\eta \varphi ' \in \textsf {LIP} _{bs}(Y).$ Since by the locality of the Laplacian and gradient we have $\Delta v_n^k=\Delta v_n=N|\nabla \sqrt{2 v_n}|^2=2N|\nabla {\tilde{v}}_n^k|^2$ ${\mathfrak {m}}_n$-a.e. in $\mathop {\textrm{supp}}\nolimits {\bar{\varphi }}$, we can compute

$$\begin{aligned} \int \varphi \Delta v_\infty ^k {\mathrm d}{\mathfrak {m}}_\infty&=\int {\bar{\varphi }} \Delta v_\infty ^k {\mathrm d}{\mathfrak {m}}_\infty =\lim _n \int {\bar{\varphi }} \Delta v_n^k {\mathrm d}{\mathfrak {m}}_n=\lim _n \int {\bar{\varphi 2N}}|\nabla {\tilde{v}}_n^k|^2{\mathrm d}{\mathfrak {m}}_n\\&\quad =\int {\bar{\varphi }} 2 N |\nabla {\tilde{v}}_\infty ^k|^2{\mathrm d}{\mathfrak {m}}_\infty =\int {\bar{\varphi }} 2 N |\nabla \sqrt{v}_\infty |^2{\mathrm d}{\mathfrak {m}}_\infty . \end{aligned}$$

This and the locality of the Laplacian prove (6.16).

For every n and every k as above,we consider a cut-off function $\xi _{n}^k\in \textsf {LIP} (\textrm{X}_n)$ analogous to $\eta _n^k$ but with smaller support, more precisely we require that $0\le \xi _k^n\le 1$, $\mathop {\textrm{supp}}\nolimits \xi _n^k\subset U_n^{>\rho }\cap B_{k+2}(x_n)$, $\xi _n^k=1$ on $U_n^{\ge 2\rho }\cap B_{k+1}(x_n)$ and $\textrm{Lip}\xi _n^k\le C'$, for some constant $C'$ depending only on $\varepsilon ,N,L$. Up to a subsequence, from Ascoli-Arzelà, we have that $\xi _n^k \rightarrow \xi _\infty ^k$ uniformly, for some $\xi _\infty ^k\in \textsf {LIP} (\textrm{X}_\infty )$ satisfying $\xi _\infty ^k=1$ in $U_\infty ^{>2\rho }\cap B^{\textrm{X}_\infty }_{k}(x_\infty ).$ In particular the same convergence holds also strongly in $L^2$.

We set $w_{n,k}:=\xi _{n}^k|\nabla \sqrt{v_n}|^{\gamma }\in W ^{1,2}(\textrm{X}_n)$ and $w_{\infty ,k}:=\xi _{\infty }^k|\nabla \tilde{v}_\infty ^k|^{\gamma } \in W ^{1,2}(\textrm{X}_\infty )$ and observe that by construction and the locality of the gradient $w_{n,k}=\xi _{n}^k|\nabla {\tilde{v}}_n^k|^\gamma$ ${\mathfrak {m}}_n$-a.e.. In particular, since we proved that $|\nabla {\tilde{v}}_n^k|^\gamma \rightarrow |\nabla {\tilde{v}}_\infty ^k|^{\gamma }$ strongly in $L^2$ and recalling $\sup _n \Vert |\nabla {\tilde{v}}_n^k|\Vert _{L^\infty }<+\infty$, we have from Proposition 2.23 that $w_{n,k}\rightarrow w_{\infty ,k}$ strongly in $L^2$.

Combining (6.11) with the first in (6.15) and $\textrm{Lip}\xi _{n,k}\le C'$ we deduce that $\sup _n \Vert |\nabla w_{n,k}|\Vert _{L^2({\mathfrak {m}}_n)}<+\infty$. We now apply Lemma 2.25 with the open set $A=\{\textsf {d} _Y({K_\infty },.)>3\rho \}\cap B^Y_{k}(x_\infty )$ that, combined with the observation that $A\cap X_n \subset U_n^{\ge 2\rho }\cap B_{k+1}(x_n)$ for n big enough, gives

$$\begin{aligned}&\int _{U_{\infty }^{>3\rho }\cap B_{k}(x_\infty )} |\nabla |\nabla {\tilde{v}}_\infty ^k|^{\gamma } |^2\, {\mathrm d}{\mathfrak {m}}_\infty \le \liminf _n \int _{U_n^{\ge 2\rho }\cap B_{k+1}(x_n)} |\nabla |\nabla \sqrt{ v_n}|^{\gamma } |^2\, {\mathrm d}{\mathfrak {m}}_n\\&\quad \overset{(5.15)}{\le } (1+RL)^{2N-4}\liminf _n \int _{U_n} v_n^{2-N}|\nabla |\nabla \sqrt{ v_n}|^{\gamma } |^2\, {\mathrm d}{\mathfrak {m}}_n\overset{(5.11)}{=}0. \end{aligned}$$

Therefore, from the locality of the gradient and the arbitrariness of k,we obtain that $|\nabla |\nabla \sqrt{ v_\infty }|^{\gamma } |=0$ ${\mathfrak {m}}_\infty$-a.e. in $U_{\infty }^{>3\rho }.$ Consider now the open set $\{v_\infty>1+\varepsilon /4\}\subset U_\infty ^{>3\rho }$. Observe that from the assumption $v_n \ge 1+\varepsilon$ in $B_{R_0}(x_n)^c\ne \emptyset$ and (6.15),we deduce that for every n there exists $y_n \in \textrm{X}_n\cap B_{2R_0}(x_n)\cap U_n^{>4\rho }$ such that $v_n^k(y_n)=v_n(y_n)\ge 1+\varepsilon$ for every k, therefore by compactness and uniform convergence we deduce that $\{v_\infty >1+\varepsilon /4\}\ne \emptyset$. From (A) it holds $\partial \{v_\infty >1+\varepsilon /4\}=\{v_\infty =1+\varepsilon /4\}$, in particular since $v_\infty ^{(2-N)/2}$ ( $\ln (v_\infty ^{-1/2})$ if $N=2$) is harmonic in $U_\infty ^{>2\rho }$ (recall (6.16)), from the maximum principle,it follows that the connected components of $\{v_\infty >1+\varepsilon /4\}$ are unbounded. Let $U'$ be one of these components. It follows that $|\nabla \sqrt{v_\infty }|\equiv C$ ${\mathfrak {m}}$-a.e. in $U'$ for some constant C, that must be positive. Indeed if $C=0$, we would have that $v_\infty$ is constant in $U'$, but since $\partial U'\subset \{v_\infty =1+\varepsilon /4\}$, $v_\infty$ should be constantly equal to $1+\varepsilon /4$, which contradicts $U'\subset \{v_\infty >1+\varepsilon /4\}.$ Finally,the assumption $v_n \ge 1+\varepsilon$ in $B_{R_0}(x_n)^c$ ensures that $\textrm{X}_\infty \setminus U'\subset B_{2R_0}(x_\infty )$. It follows that the function $(2C^2)^{-1}v_\infty {_{|_{U'}}}$ satisfies the hypotheses of Theorem 6.1 with $U=U'$. In particular $\textrm{X}_\infty$ has Euclidean volume growth and from Corollary 3.9,it has one end, from which we deduce that $\{v_\infty >1+\varepsilon /4\}$ is connected. Therefore repeating the above argument for $U'=\{v_\infty >1+\varepsilon /4\}$ proves claim (B) with $C_0:=(2C^2)^{-1}$.

Combining (A) and (B), from Theorem 6.1,we deduce the existence of an $\textrm{RCD}(0,N)$ N cone $(Y',\textsf {d} _{Y'},{\mathfrak {m}}_{Y'})$ with vertex $O_{Y'}$ and a bijective measure preserving local isometry $T: \{1+\varepsilon /4<v_\infty \}\rightarrow \{ \ r<\textsf {d} _{Y'}(O_{Y'},.) \}$, for some $r>0$ which also satisfies (recall (6.1))

$$\begin{aligned} \sqrt{v_\infty }(x)=\lambda \textsf {d} _{Y'}(O_{Y'},T(x)), \quad \text { for every }x \in \{ v_\infty > 1+\varepsilon /4\}, \end{aligned}$$

(6.17)

where $\lambda :=(2C_0)^{-1/2}$ ($C_0$ being the constant in (B)). We claim that $\lambda \le \sup _n \Vert |\nabla \sqrt{v_n}|\Vert _{L^\infty }$, which in particular gives that $r\ge \lambda ^{-1}(1+\varepsilon /4)\ge L^{-1}.$ Indeed from (6.17), the fact that $Y'$ is geodesic and the fact that T is a local isometry we deduce that for every $x \in \{ v_\infty > 1+\varepsilon /4\}$ and every $r'>0$ small enough, there exists $y \in B^{\textrm{X}_\infty }_{r'}(x)$ such that $|\sqrt{v_\infty (x)}-\sqrt{v_\infty (y)}|=\lambda \textsf {d} _\infty (x,y),$. The claim now follows from uniform convergence and the Sobolev-to-Lipschitz property. Since, as observed above, $\textrm{X}_\infty \setminus \{1+\varepsilon /4<v_\infty \}^c\subset B_{2R_0}(x_\infty )$, the conclusion of Theorem 6.7 holds for every sufficiently large n, which contradicts item d) above. This concludes the proof of Theorem 6.7.

Proof of Theorem 6.9: We argue by contradiction exactly as above, except that we substitute the assumption d) with the following:

$d')$:

there exist a number $\eta >0$ and two sequences $(t_1^n),(t_2^n)\subset (1+\varepsilon +\eta ,\varepsilon ^{-1})$ with $t_1^n\le t_2^n$, such that $\{\sqrt{v}_n\le t_2^n+2\eta \}\subset B_{R_0}(x_n)$ and the conclusion of Theorem 6.9 is false with $t_1^n,t_2^n$, for every n.

Since assumption d) was not used until the very end of the proof of Theorem 6.7 above, we can, and will, repeat all the first part of the proof up to this point together with all the constructions and objects introduced along the argument.

Up to passing to a subsequence,we can assume that $t_i^n \rightarrow t_i^\infty \in [1+\varepsilon +\eta ,\varepsilon ^{-1}]$, $i=1,2.$

It will be useful later to remark that

$$\begin{aligned} \sqrt{v_n}={\tilde{v}}_n^k\, \quad \text { in } \{1+\varepsilon /2\le \sqrt{v_n}\le t_2^n+2\eta \}, \text { for every } k, \end{aligned}$$

(6.18)

which follows from the second in (6.15) and the assumption $\{\sqrt{v_n}\le t_2^n+2\eta \}\subset B_{R_0}(x_n).$

We claim that

$$\begin{aligned} a_n:=\sup _{s \in (t_1^\infty -\eta ,t_2^\infty +\eta )} \textsf {d} _H^Y(\{\sqrt{v_n}=s\},\{\sqrt{v_\infty }=s\})\rightarrow 0, \quad \text { as }n \rightarrow +\infty . \end{aligned}$$

(6.19)

(We point out that this does not follows from the uniform convergence, indeed we need first to prove some regularity of the level sets of $\sqrt{v_\infty }$). The key observation is that for every $\varepsilon '>0$,there exists $\delta '>0$ such that for every $t \in [1+\varepsilon ,2\varepsilon ^{-1}]$ it holds

$$\begin{aligned} B^{X_\infty }_{\varepsilon '}(x)\cap \{\sqrt{v_\infty }=t'\}\ne \emptyset , \quad \forall x \in \{\sqrt{v_\infty }=t\}, \forall t' \in [t-\delta ',t+\delta ']. \end{aligned}$$

(6.20)

This is an immediate consequence of (6.17), the fact that T is a local isometry and the fact that, since $Y'$ is a cone, for every $y' \in Y'$,there exists a ray emanating from $O_{Y'}$ and passing through $y'$.

Suppose now that (6.19) does not hold. Then, up to a passing to nonrelabelled subsequence, there exists a sequence $(s_{n})\subset (t_1^\infty -\eta ,t_2^\infty -\eta )$ and $\varepsilon '>0$ such that $s_{n}\rightarrow {{\bar{s}}} \in [t_1^\infty -\eta ,t_2^\infty +\eta ] \subset [1+\varepsilon ,2\varepsilon ^{-1}]$ and

$$\begin{aligned} \textsf {d} _H^Y(\{\sqrt{v_n}=s_{n}\},\{\sqrt{v_\infty }=s_{n}\})>\varepsilon ', \quad \forall n. \end{aligned}$$

Therefore, up to passing to a further subsequence, there exists either a sequence $y_{n}\in \{\sqrt{v}_{n}=s_{n}\}$ such that $\textsf {d} _Y(y_{n},\{\sqrt{v_\infty }=s_{n}\})>\varepsilon '$, for all n, or a sequence $y_{n}\in \{\sqrt{v_\infty }=s_{n}\}$ such that $\textsf {d} _Y(y_{n},\{v_{n}=s_{n}\})>\varepsilon '$, for all n. In the first case, since by assumption $\{\sqrt{v}_n=s_n\}\subset \{t_1^n-2\eta \le \sqrt{v_n}\le t_2^n+2\eta \}\subset B_{R_0}(x_n)$ for n big enough, up to passing to a further subsequence we have that $y_{n} \rightarrow y_\infty \in \textrm{X}_\infty$ and by uniform convergence (recall (6.18)) that $\sqrt{v_\infty }(y_\infty )={{\bar{s}}}$. In particular $\textsf {d} (y_\infty ,\{\{\sqrt{v_\infty }=s_{n}\}\})>\varepsilon '/2$ for every n big enough, which contradicts (6.20). In the second case, again up to a subsequence and from the continuity of $\sqrt{v_\infty }$, we have that $y_{n} \rightarrow y_\infty \in \{\sqrt{v_\infty }={{\bar{s}}}\}$. Moreover from (6.20),it follows the existence of a $\delta '>0$ such that there exist $y_{\infty }^+,y_\infty ^-\in B^{\textrm{X}_\infty }_{\varepsilon '/4}(y_\infty )$ such that $\sqrt{v_\infty }(y_{\infty }^\pm )={{\bar{t}}}\pm \delta '$. Finally, from uniform convergence (recall again (6.18)), for every n big enough there exist $y_{n}^+, y_{n}^-\in \textrm{X}^{n}$ such that $\textsf {d} _Y(y_{n}^\pm ,y_{\infty }^\pm )<\varepsilon '/4$ and $|v_{n}(y_{n}^\pm )-({{\bar{s}}}\pm \delta ')|<\delta '/2.$ In particular by the continuity of $v_{n}$, for every k big enough, there exists $z_{n}$ which lies on a geodesic connecting $y_{n}^+$ and $y_{n}^-$ such that $z_{n}\in \{v_{n}=s_{n}\}$. From the triangle inequality it follows that $\textsf {d} (z_n,y_n)<\varepsilon '$ if n is big enough, which is a contradiction since $y_n \in \{\sqrt{v_\infty } =s_{n}\}$.

From (6.19) it follows that $\textsf {d} _H^Y(\{t_1^n\le \sqrt{v_n}\le t_2^n\},\{t_1^n\le \sqrt{v_\infty }\le t_2^n\})\rightarrow 0$ as $n \rightarrow +\infty$. Moreover it is clear that $\textsf {d} _H^Y(\{t_1^n\le \sqrt{v_\infty }\le t_2^n\},\{t_1^\infty \le \sqrt{v_\infty }\le t_2^\infty \})\rightarrow 0$ as $n \rightarrow +\infty$ (recall (6.17)), therefore

$$\begin{aligned} b_n:=\textsf {d} _H^Y(\{t_1^n\le \sqrt{v_n}\le t_2^n\},\{t_1^\infty \le \sqrt{v_\infty }\le t_2^\infty \})\rightarrow 0, \quad \text { as } n \rightarrow +\infty .\nonumber \\ \end{aligned}$$

(6.21)

In particular, since both sets are compact, we can build a Borel map $f_n : \{t_1^n\le \sqrt{v_n}\le t_2^n\}\rightarrow \{t_1^\infty \le \sqrt{v_\infty }\le t_2^\infty \}$ that has $b_n$-dense image and such that $\textsf {d} _Y(x,f_n(x))\le 2b_n$ for all $x \in \{t_1^n\le \sqrt{v_n}\le t_2^n\}.$

We claim that

$$\begin{aligned} {f_n}_*\left( {\mathfrak {m}}_n {_{|_{\{t_1^n\le \sqrt{v}_n\le t_2^n\}}}}\right) \rightharpoonup {\mathfrak {m}}_\infty {_{|_{\{t_1^\infty \le \sqrt{v}_\infty \le t_2^\infty \}}}}, \quad \text {in duality with }C(\{t_1^\infty \le \sqrt{v}_\infty \le t_2^\infty \}).\nonumber \\ \end{aligned}$$

(6.22)

From the fact that $\textsf {d} _Y(.,f_n(.))\le 2b_n$ and using dominated convergence, it is enough to show that

$$\begin{aligned} {\mathfrak {m}}_n {_{|_{\{t_1^n\le \sqrt{v}_n\le t_2^n\}}}}\rightharpoonup {\mathfrak {m}}_\infty {_{|_{\{t_1^\infty \le \sqrt{v}_\infty \le t_2^\infty \}}}}, \quad \text {in duality with }C_{bs}(Y). \end{aligned}$$

To prove the above we first define for every $\delta >0$ the closed set $C_{\delta }:=\{y \in Y \ : \ \textsf {d} _Y(y,\{\sqrt{v_\infty }=t_1^\infty \}\cup \{\sqrt{v_\infty }=t_2^\infty \})\le \delta \}$ and observe that for every $\varepsilon '>0$ there exists $\delta '$ such that

$$\begin{aligned} {\mathfrak {m}}_{\infty }(C_{\delta '})<\varepsilon '. \end{aligned}$$

(6.23)

This can be seen using the fact that T is a measure preserving local isometry, the Bishop-Gromov inequality and formula (6.17).

We also define for any $\delta >0$ the sets $A_\delta :=\{y \in Y \ : \ \textsf {d} _Y(y,X_\infty \setminus \{t_1^\infty \le \sqrt{v_\infty }\le t_2^\infty \})\ge \delta \}$ and $B_\delta :=\{y\in Y\ : \ \textsf {d} _Y(y, \{t_1^\infty \le \sqrt{v_\infty }\le t_2^\infty \}\le \delta )\}$. We claim that $B_{\delta _1}\setminus A_{\delta _2} \subset C_{2\delta _1+2\delta _2},$ for every $\delta _1,\delta _2>0.$ To see this let $y \in B_{\delta _1}\setminus A_{\delta _2}$, which implies $\textsf {d} (y,\{t_1^\infty \le \sqrt{v_\infty }\le t_2^\infty \})\le \delta _1,\textsf {d} (y,X_\infty \setminus \{t_1^\infty \le \sqrt{v_\infty }\le t_2^\infty \})< \delta _2$. Taking two points $y_1,y_2\in \textrm{X}_\infty$ which realize these two distances we must have that $\textsf {d} _\infty (y_1,y_2)=\textsf {d} _Y(y_1,y_2)\le \delta _1+\delta _2$. Moreover any geodesics in $\textrm{X}_\infty$ from $y_1$ to $y_2$, by the continuity of $v_\infty$, must intersects $\{\sqrt{v_\infty }=t_1^\infty \}\cup \{\sqrt{v_\infty }=t_2^\infty \},$ from which the claim follows.

We finally fix $\varphi \in C_{bs}(Y)$ and $\varepsilon '>0$ arbitrary. Let $\delta '=\delta '(\varepsilon ')$ be the one given by (6.23) and pick $\eta \in C_b(Y)$ such that $0\le \eta \le 1$, $\eta =1$ in $A_{\delta '/4}$ and $\mathop {\textrm{supp}}\nolimits (\eta )\subset A_{\delta '/8}.$ Observe that by uniform convergence (recall also (6.18)), for n big enough we have that $A_{\delta '/8}\cap \mathop {\textrm{supp}}\nolimits (\varphi )\cap X_n\subset \{t_1^n\le \sqrt{v_n}\le t_2^n\}\subset B_{\delta '/4},$ therefore

$$\begin{aligned}&\limsup _{n} \left| \int \varphi \,{\mathrm d}{\mathfrak {m}}_n{_{|_{\{t_1^n\le \sqrt{v_n}\le t_2^n\}}}}-\int \varphi \,{\mathrm d}{\mathfrak {m}}_\infty {_{|_{\{t_1^\infty \le \sqrt{v_\infty }\le t_2^\infty \}}}} \right| \\&\quad \le \limsup _n \left| \int \varphi \eta \,{\mathrm d}{\mathfrak {m}}_n-\int \varphi \eta \,{\mathrm d}{\mathfrak {m}}_\infty \right| +\\&\qquad + \limsup _n \Vert \varphi \Vert _\infty {\mathfrak {m}}_n(B_{\delta '/4}\setminus A_{\delta '/4})+\Vert \varphi \Vert _\infty {\mathfrak {m}}_\infty (\{t_1^\infty \le \sqrt{v_\infty }\le t_2^\infty \}\setminus A_{\delta '/4})\\&\quad \le \limsup _n {\mathfrak {m}}_n(C_{\delta '})+{\mathfrak {m}}_\infty (C_{\delta '})\overset{(5.23)}{\le }2\varepsilon '. \end{aligned}$$

From the arbitrariness of $\varepsilon '$ and $\varphi \in C_{bs}(Y)$,the convergence in (6.22) follows.

We now pass to the study of the behaviour of $f_n$ with respect to the intrinsic metrics. More precisely for every $\tau >0$ we set $A^\tau _n:=\{t_1^n-\tau< \sqrt{v_n}<t_2^n+\tau \}$, $A^\tau _\infty :=\{t_1^\infty -\tau< \sqrt{v_\infty }<t_2^\infty +\tau \}$, $A^\tau _{\infty ,n}:=\{t_1^n-\tau< \sqrt{v_\infty }<t_2^n+\tau \}$ and denote by $\textsf {d} ^\tau _n, \textsf {d} ^\tau _\infty ,\textsf {d} ^\tau _{\infty ,n}$ the intrinsic metrics on $A_n^\tau ,\, A_\infty ^\tau ,A^\tau _{\infty ,n}$, respectively (see the beginning of this section). It is clear that the metrics $\textsf {d} ^\tau _n, \textsf {d} ^\tau _\infty ,\textsf {d} ^\tau _{\infty ,n}$ induce on the sets $\{t_1^n\le \sqrt{v_n}\le t_2^n\},\{t_1^\infty \le \sqrt{v_\infty }\le t_2^\infty \},\{t_1^n\le \sqrt{v_\infty }\le t_2^n\}$ the same topology induced by the metrics $\textsf {d} _n,\textsf {d} _\infty$.

Notice also that, from (6.17) and since T is a local isometry on $\{\sqrt{v}_\infty >1+\varepsilon /4\}$,

$$\begin{aligned}{} & {} \left( \{s\le \sqrt{v_\infty }\le t\},\textsf {d} _\infty ^{s,t,\tau }\right) \text { is isometric to } (\{s\le \lambda \textsf {d} _{O_{Y'}}\le t\},\textsf {d} _{Y'}^{s,t,\tau }), \,\,\, \nonumber \\{} & {} \quad \forall \tau \in (0,\eta ),\, \forall \,t\ge s > 1+\varepsilon +\eta , \end{aligned}$$

(6.24)

where $\textsf {d} _\infty ^{s,t,\tau }$ and $\textsf {d} _{Y'}^{s,t,\tau }$ are the intrinsic metrics,respectively, on $\{s-\tau< \sqrt{v_\infty }<t+\tau \}$ and on $\{s-\tau< \lambda \textsf {d} _{O_{Y'}}<t+\tau \}$, the isometry being T itself, which also measure preserving. In particular there exists a constant $D>0$ such that for every $\tau \in (0,\eta )$ it holds $\text {diam}(\{t_1^\infty \le \sqrt{v_\infty }\le t_2^\infty \},\textsf {d} _\infty ^{\tau })\le D.$

Observe that from (6.18) we deduce that the functions $\sqrt{v_n}, \sqrt{v_\infty }$ are equi-Lipschitz on $\{t_1^n-\eta \le \sqrt{v_n}\le t_2^n+\eta \}$, $\{t_1^\infty -\eta \le \sqrt{v_\infty }\le t_1^\infty +\eta \}$ and we fix $M\ge 2$ a bound on their Lipschitz constant.

Putting $\varepsilon _n :=2\max (b_n,a_n)$ (where $a_n,b_n$ are the ones in (6.19) and (6.21)) it is not restrictive to assume both that $\sqrt{\varepsilon _n}<\eta /(2M)$ and that $|t_1^n-t_1^\infty |,|t_2^n-t_2^\infty |<\varepsilon _n$, for every n.

Pick any $x_0,x_1 \in \{t_1^n\le \sqrt{v_n}\le t_2^n\}$ and set $y_i=f_n(x_i)\in \{t_1^\infty \le \sqrt{v_\infty }\le t_2^\infty \}$, $i=0,1$, where $f_n$ was defined above and recall that $\textsf {d} _Y(x_i,f_n(x_i))\le \varepsilon _n$. Consider an absolutely continuous curve $\gamma : [0,1]\rightarrow \overline{A_\infty ^{\varepsilon -2M\sqrt{\varepsilon _n}}}$ such that $\gamma (i)=y_i$, $i=0,1$ and $L(\gamma )= \textsf {d} _{\infty }^{\eta -2M\sqrt{\varepsilon _n}}(y_0,y_1)\le D.$ Letting $N_n:=\lfloor 2D/\sqrt{\varepsilon _n}\rfloor$, there exist $0=t_0<t_1<...<t_{N_n}=1$ such that $\textsf {d} _\infty (\gamma (t_i),\gamma (t_{i+1})\le L(\gamma {_{|_{[t_i,t_{i+1}]}}}))\le L(\gamma )/N_n,$ for every $i=0,...,N_n-1$. Thanks to (6.19) and the M-Lipschitzianity of $\sqrt{v_n}$ there exist points $x_i\in A_n^{\eta -M\sqrt{\varepsilon _n}}$ $i=1,...,N_n-1$, such that $\textsf {d} _Y(x_i,\gamma (t_i))<\varepsilon _n$, $i=1,...,N_n-1$, and in particular $|\textsf {d} _n(x_i,x_{i+1})-\textsf {d} _\infty (\gamma (t_i),\gamma (t_{i+1})|\le 2\varepsilon _n,$ for every $i=0,...,N_n.$ Therefore $\textsf {d} _n(x_i,x_{i+1})< L(\gamma )/N_n+2\varepsilon _n\le \sqrt{\varepsilon _n}$, and thus any geodesic (in $\textrm{X}_n$) $\gamma _i$ from $x_i$ to $x_{i+1}$ has image contained in $A_n^{\eta }$. We define ${\bar{\gamma }}:[0,1]\rightarrow A_n^{\eta }$ as the concatenation of all the geodesics $\gamma _i$ (appropriately reparametrized), in particular

$$\begin{aligned} \textsf {d} _n^{\eta }(x_0,x_1)\le & {} L({\bar{\gamma }})\le N_n \left( \frac{L(\gamma )}{N_n}+2\varepsilon _n\right) \nonumber \\\le & {} \textsf {d} _\infty ^{\eta -2M\sqrt{\varepsilon _n}}(f_n(x_0),f_n(x_1))+4D\sqrt{\varepsilon _n}. \end{aligned}$$

(6.25)

Conversely pick an absolutely continuous curve ${\bar{\gamma }}: [0,1]\rightarrow \overline{A_n^{\eta }}$ such that ${\bar{\gamma }}(i)=x_i$, $i=0,1$ and $L({\bar{\gamma }})= d_n^{\eta }(x,y)\le 2D,$ which exists thanks to (6.25). Arguing exactly as above we can construct an absolutely continuous curve $\gamma : [0,1]\rightarrow A_\infty ^{\eta +2M\sqrt{\varepsilon _n}}$ such that $\gamma (i)=y_i$, $i=0,1$ and

$$\begin{aligned} \textsf {d} _\infty ^{\eta +2M\sqrt{\varepsilon _n}}(f_n(x_0),f_n(x_1))\le L( \gamma )\le \textsf {d} _n^{\eta }(x_0,x_1)+4D\sqrt{\varepsilon _n}. \end{aligned}$$

Recalling (6.24),we are in position to apply Lemma 6.10 and deduce that $\textsf {d} _\infty ^{\eta \pm 2M\sqrt{\varepsilon _n}}(f_n(x_0),f_n(x_1))\rightarrow \textsf {d} _\infty ^{\eta }(f_n(x_0),f_n(x_1))$ as $n\rightarrow +\infty$, uniformly in $x_0,x_1 \in \{t_1^n\le \sqrt{v_n}\le t_2^n\}$. Moreover again from Lemma 6.10,we have that the image of $f_n$ is $cb_n$-dense in $\{ t_1^\infty \le \sqrt{v}_\infty \le t_2^\infty \}$ w.r.t. the metric $\textsf {d} _\infty ^{\eta }$, for some constant c independent of n.

Combing this with the above inequalities and (6.22),we obtain

$$\begin{aligned} \left( \{t_1^n\le \sqrt{v_n}\le t_2^n\},\textsf {d} _n^{\eta },\mu _n \right) \overset{mGH}{\rightarrow }\ \left( \{ t_1^\infty \le \sqrt{v_\infty }\le t_2^\infty \},\textsf {d} _{\infty }^{\eta },\mu _\infty \right) , \end{aligned}$$

(6.26)

with $\mu _n:={\mathfrak {m}}_n{_{|_{\{t_1^n\le \sqrt{v}_n\le t_2^n\}}}}$, $\mu _\infty :={\mathfrak {m}}_\infty {_{|_{\{t_1^\infty \le \sqrt{v}_\infty \le t_2^\infty \}}}}$ and where, if $t_1^\infty =t_2^\infty$, the convergence is intended only in the GH-sense. Finally from Lemma 6.10 and recalling (6.24) we have that $\left( \{t_1^n\le \sqrt{v_\infty }\le t_2^n\},\textsf {d} _{\infty ,n}^{\eta } \right) \overset{GH}{\rightarrow }\ \left( \{ t_1^\infty \le \sqrt{v_\infty }\le t_2^\infty \},\textsf {d} _{\infty }^{\eta } \right) ,$ and that such convergence can be realized by a map $g_n: \{t_1^n\le \sqrt{v_\infty }\le t_2^n\} \rightarrow \{ t_1^\infty \le \sqrt{v_\infty }\le t_2^\infty \}$ that (if $t_1^n\ne t_2^n$) also satisfies ${g_n}_*\left( {\mathfrak {m}}_\infty {_{|_{\{t_1^n\le \sqrt{v_\infty }\le t_2^n\}}}}\right) =\frac{(t_2^\infty )^N-(t_1^\infty )^N)}{((t_2^n)^N-(t_1^n)^N)}{\mathfrak {m}}_\infty {_{|_{\{ t_1^\infty \le \sqrt{v_\infty }\le t_2^\infty \}}}}$. In particular

$$\begin{aligned} \left( \{t_1^n\le \sqrt{v_\infty }\le t_2^n\},\textsf {d} _{\infty ,n}^{\eta },\mu _{\infty ,n} \right) \overset{mGH}{\rightarrow }\ \left( \{ t_1^\infty \le \sqrt{v_\infty }\le t_2^\infty \},\textsf {d} _{\infty }^{\eta },\mu _\infty \right) , \end{aligned}$$

(6.27)

with $\mu _{\infty ,n}:={\mathfrak {m}}_\infty {_{|_{\{t_1^n\le \sqrt{v_\infty }\le t_2^n\}}}}$ and where in the case $t_1^\infty =t_2^\infty$ the convergence is intended only in the GH-sense.

If $t_1^\infty =t_2^\infty$, combining (6.26) with (6.27), and recalling (6.24), we obtain that (6.13) holds for n big enough. Therefore, since we are assuming $d')$ (see above), up to a subsequence, we must have that either (6.14) is false every n big enough or the last claim about Y in Theorem 6.9 is false. The latter cannot happen, indeed in the first half of the proof we proved precisely that Theorem 6.7 holds with the same Y and with the same $\varepsilon ,L, R_0.$ Hence we must be n the first case and in particular $t_1^\infty +\varepsilon <t_2^\infty$ and we can apply Proposition 6.11 together with (6.26) and (6.27) and obtain that (6.14) holds for n big enough, which is a contradiction. This concludes the proof of Theorem6.9. $\square$

7 Rigidity and Almost Rigidity from the Monotonicity Formula

7.1 Rigidity

The following rigidity result follows almost immediately combining the explicit lower bound on the derivative of $U_{\beta }'$ in (5.6) and Theorem 6.1 about “from outer functional cone to outer metric cone”.

Theorem 7.1

Let $\textrm{X},\, \Omega ,\, u , U_\beta$, with $\beta >\frac{N-2}{N-1}$, be as in Theorem 5.4 and suppose that $U^{'-}_{\beta }(t_0)=0$ for some $t_0\in (0,1]$.Then the hypotheses of Theorem 6.1, and in particular also its conclusions, are satisfied choosing ${\textbf {u}}=Cu^\frac{2}{2-N}$, ${\textbf {u}}_0=Ct_0^\frac{2}{2-N}$ and $U=\{u<t_0\}$, for some constant $C>0$.

Proof

Suppose $U_{\beta }^{'-}(t_0)=0$ for some $t_0\in (0,1]$ and observe that, thanks to (5.6), since $C_{\beta ,N}>0$, we must have that $|\nabla |\nabla u^{\frac{1}{2-N}}|^{\beta /2}|=0$ ${\mathfrak {m}}$-a.e. in $\{u<t_0\}$. We claim that $\{u<t_0\}$ is connected. Indeed, if $t_0<1$, from the continuity of u follows that $\partial \{u<t_0\}\subset \{u=t_0\}$, hence from the strong maximum principle we deduce that all the connected components of $\{u<t_0\}$ are unbounded. Moreover $\partial \{u<t_0\}$ is bounded and thus from Corollary 3.9,it follows that $\{u<t_0\}$ is connected. If $t_0=1,$ we conclude observing that $\{u<1\}$ is the union of the sets $\{u<t\}$ with $t<1.$ Therefore we have that $|\nabla u^{\frac{1}{2-N}}|^2\equiv C$ ${\mathfrak {m}}$-a.e. in $\{u<t_0\}$, for some constant C. We now claim that $C>0$. Indeed if $C=0$ we would have that $\nabla u=-(2-N)u^{\frac{N-1}{N-2}}\, \nabla u^{\frac{1}{2-N}} =0$ ${\mathfrak {m}}$-a.e. in $\{u<t_0\}$ and therefore u would be constant in $\{u<t_0\}$ (recall (2.8)). However, u goes to 0 at $+\infty$ and $\{u<t_0\}$ is unbounded, therefore $u\equiv 0$ in $\{u<t_0\}$, but this violates the positivity of u. Setting $v=u^{\frac{1}{2-N}}$, by the chain rule for the Laplacian, the harmonicity of u and by locality we have

$$\begin{aligned} \Delta \frac{v^2}{2}=\frac{1}{2} \Delta (u^{\frac{2}{2-N}})=\frac{N}{(2-N)^2}\frac{|\nabla u|^2}{u^{2\frac{N-1}{N-2}}}=CN, \quad {\mathfrak {m}}\text {-a.e. in } \{u<t_0\}. \end{aligned}$$

Moreover $|\nabla v^2/2|^2=v^2|\nabla v|^2=2C\frac{v^2}{2}.$ Therefore the function ${\textbf {u}}=C^{-1}v^2/2$ satisfies the hypotheses of Theorem 6.1 with $U=\{u<t_0\}$ and ${\textbf {u}}_0=C^{-1}t_0^{2/(2-N)}/2$. This concludes the proof. $\square$

7.2 Almost Rigidity

The goal of this subsection is to prove the following.

Theorem 7.2

For all numbers $\varepsilon \in (0,1/3)$, $R_0>0$, $\beta > \frac{N-2}{N-1}$, $N\in (2,\infty )$ and for every function $f:(1,+\infty )\rightarrow {{\,\mathrm{{\mathbb {R}}}\,}}^+$ in $L^1(1,+\infty )$ there exists $0<\delta =\delta (\varepsilon ,\beta ,N,f)$ such that the following holds. Let $(\textrm{X},\textsf {d} ,{\mathfrak {m}},x_0)$ be a pointed $\textrm{RCD}(0,N)$ m.m.s. such that ${\mathfrak {m}}(B_1(x_0))\le \varepsilon ^{-1}$ and $\frac{s}{{\mathfrak {m}}(B_s(x_0))}\le f(s)$ for $s\ge \varepsilon ^{-1}$. Let u be a solution to (P) with $\Vert u\Vert _{L^\infty (\Omega )}\le \varepsilon ^{-1}$ and such that there exists $t\in (\varepsilon ,1]$ satisfying $\text {diam}(\{u>t-\varepsilon t\}^c)<R_0$, $\textsf {d} (x_0,\{u\le t\})>\varepsilon$, $\Vert |\nabla u|\Vert _{L^\infty (\{u<t\})}\le \varepsilon ^{-1}$ and

$$\begin{aligned} {U^-_\beta }'(t)<\delta . \end{aligned}$$

(7.1)

Then there exists a pointed $\mathrm{RCD(0,N)}$ space $(\textrm{X}',\textsf {d} ',{\mathfrak {m}}',x')$ such that

$$\begin{aligned} \textsf {d} _{pmGH}((\textrm{X},\textsf {d} ,{\mathfrak {m}},x_0),(\textrm{X}',\textsf {d} ',{\mathfrak {m}}',x'))<\varepsilon \end{aligned}$$

(7.2)

and $(\textrm{X}',\textsf {d} ',{\mathfrak {m}}',x')$ is a truncated cone outside a compact set $K\subset B_{2R_0}(x'),$ i.e. there exists an $\mathrm{RCD(0,N)}$ Euclidean N cone Y, with tip $O_Y$, over an $\textrm{RCD}(N-2,N-1)$ space Z and a measure preserving local isometry $T: \textrm{X}'\setminus K\rightarrow Y\setminus {{\bar{B}}}_{r}(O_Y)$, for some $r>0.$

Remark 7.3

Observe that inequality (7.2) is non-trivial because $\textrm{X}'$ is a cone outside a compact set $K\subset B_{2R_0}(x')$ with $R_0$ independent of $\varepsilon >0$ (cf. with Remark 6.8).

We will also prove the following alternative version of the above statement (see Sect. 6.2 for the definition of ${\mathbb {D}}, \textsf {d} _{GH}$ and of intrinsic metric).

Theorem 7.4

For all numbers $\varepsilon \in (0,1/3)$, $R_0>0$ $\beta > \frac{N-2}{N-1}$, $N\in (2,\infty )$, $\eta >0$ and for every function $f:(1,+\infty )\rightarrow {{\,\mathrm{{\mathbb {R}}}\,}}^+$ in $L^1(1,+\infty )$ there exists $0<\delta =\delta (\varepsilon ,\beta ,N,f,\eta )$ such that, given $\textrm{X}$, v and t as in Theorem 6.7, there exists an $\mathrm{RCD(0,N)}$ Euclidean N cone $(Y,\textsf {d} _Y,{\mathfrak {m}}_Y)$, with tip $O_Y$, over an $\textrm{RCD}(N-2,N-1)$ space Z and a constant $\lambda >0$ such that the following holds. For every $(1+\varepsilon +\eta )t^{\frac{1}{2-N}}<t_1<t_2<\varepsilon ^{-1}t^{\frac{1}{2-N}}$ it holds

$$\begin{aligned} \textsf {d} _{GH}\left( (\{t_1\le u^{\frac{1}{2-N}}\le t_2\}, \textsf {d} _{\textrm{X}}^{\eta }),\left( \{t_1\le \lambda \textsf {d} _{O_Y}\le t_2, \textsf {d} _{Y}^{\eta }\right) \right) <\varepsilon , \end{aligned}$$

where $\textsf {d} _{O_Y}:=\textsf {d} _Y(.,O_Y)$ and $\textsf {d} _{\textrm{X}}^{\eta }$ and $\textsf {d} _{Y}^{\eta }$ denote the intrinsic metrics on $\{t_1-\eta t^{\frac{1}{2-N}}< u^{\frac{1}{2-N}}<t_2+\eta t^{\frac{1}{2-N}}\}$ and on $\{t_1-\eta<\lambda \textsf {d} _{O_Y}< t_2+\eta \}$. Moreover, provided that $t_1+\varepsilon t<t_2$,

$$\begin{aligned} {\mathbb {D}}\left( \big (\{t_1\le u^{\frac{1}{2-N}}\le t_2\}, \textsf {d} _{\textrm{X}}^{\eta },{\mathfrak {m}}{_{|_{\{t_1\le u^{\frac{1}{2-N}}\le t_2\}}}}\big ),\big (\{t_1\le \lambda \textsf {d} _{O_Y}\le t_2\}, \textsf {d} _{Y}^{\eta },{\mathfrak {m}}_Y{_{|_{\{t_1\le \lambda \textsf {d} _{O_Y}\le t_2\}}}}\big ) \right) <\varepsilon . \end{aligned}$$

Before passing to the proof,we explain why the bound on $\Vert |\nabla u|\Vert _{L^\infty (\{u<t\})}$ is natural and often satisfied. The immediate observation is that from the gradient bound for harmonic functions (2.22) we deduce

$$\begin{aligned} |\nabla u|\le \frac{C(N)}{\varepsilon }, \quad {\mathfrak {m}}\text {-a.e. in } \{u<1\}\cap \{x \ : \ \textsf {d} (x,\partial \Omega )>\varepsilon \}, \quad \forall \, \varepsilon >0. \end{aligned}$$

(7.3)

In particular for fixed $\varepsilon >0$, thanks to the assumption $\liminf _{x\rightarrow \partial \Omega }u(x)\ge 1$, for t sufficiently small (but depending on u) the gradient bound $\Vert |\nabla u|\Vert _{L^\infty (\{u<t\})}\le C(N)\varepsilon ^{-1}$ is always satisfied. An estimate on the value of t can be given in the case $B_{\varepsilon }(x_0)\subset \Omega .$ Indeed applying the lower bound for u given by (5.1), it is immediately seen that $\{u<t\}\subset \{x \ : \ \textsf {d} (x,\partial \Omega )>\varepsilon \}$ for any $0<t < \frac{1}{2}\left( \frac{\varepsilon }{\text {diam}(\Omega ^c)+\varepsilon }\right) ^{N-2}.$

Something more explicit can be said if we consider u to be an electrostatic potential. Indeed combining (7.3) with the continuity estimate (8.1) one can easily prove the following:

Proposition 7.5

For all numbers $\varepsilon \in (0,1/3)$, $N\in (2,\infty )$ there exists $0<C=C(\varepsilon ,N)$ such that the following holds. Let $(\textrm{X},\textsf {d} ,{\mathfrak {m}})$ be a noncompact RCD(0, N) m.m.s. and let $E\subset \textrm{X}$ be open and bounded with uniformly $\textsf {Cap}$-fat boundary with parameters $(\varepsilon ,\varepsilon )$ (see Definition B.3). Let u be the capacitary potential relative to E (see Theorem 8.4). Then

$$\begin{aligned} |\nabla u|\le C(\varepsilon ,N), \quad {\mathfrak {m}}\text {-a.e. in } \{u<t\},\text { for every } t \in (0,1-\varepsilon ). \end{aligned}$$

We pass to the proof of Theorem 7.2 and Theorem 7.4, which are almost corollaries of Theorem 6.7 and Theorem 6.9.

Proof of Theorem 7.2 and Theorem 7.4

Observe first that by Bishop-Gromov inequality

$$\begin{aligned} {\mathfrak {m}}(B_1(x_0))\ge {\mathfrak {m}}(B_{\varepsilon ^{-1}})\varepsilon ^N\ge \varepsilon ^{N-1}/f(\varepsilon ^{-1}). \end{aligned}$$

From the second in (5.1) we have that there exist a positive constant $C_1=C_1(\varepsilon ,R_0,N)$, such that

$$\begin{aligned} u(x) \le C_1\int _{\textsf {d} (x,x_0)}^\infty f(s)\,{\mathrm d}s, \quad \forall { x\in B_{\varepsilon ^{-1}\vee R_0}(x_0)^c}. \end{aligned}$$

(7.4)

In particular, since $t>\varepsilon$, $\text {diam}(\{u<t\}^c)\le \text {diam}(\{u<\varepsilon \}^c)\le C_2$, for some constant $C_2$ depending only on $\varepsilon ,R_0,N,f$. Therefore, again since $t>\varepsilon$, up to rescaling u as $ut^{-1}$ we can assume that $t=1$ (observe also that, called ${\tilde{U}}_{\beta }$ the function relative to $t^{-1}u$, it holds that $\tilde{U}_{\beta }(s)=U_\beta (ts)t^{\beta \frac{N-1}{N-2}-\beta -1}$, $s\in (0,1)$).

Define $v:=u^{\frac{2}{2-N}}$ and set $\Omega '=\{u<1\}=\{v>1\}$. Note also that $\text {diam}(\{v\le 1+c_N\varepsilon \}^c)<R_0$ for some constant $c_N\le 1$. From (P) we have that $\Delta v=N|\nabla \sqrt{2v}|^2$, ${\mathfrak {m}}$-a.e. in $\Omega '$ and that $\limsup _{x\rightarrow \partial \Omega '}v(x)\le 1$.

To apply Theorems 6.7 and 6.9 it still remains to check the bounds on $|\nabla \sqrt{v}|.$

Observe that from the assumption $\textsf {d} (x_0,\{u\le 1\})>\varepsilon$ and the first in (5.1) we deduce that

$$\begin{aligned} u_n(x) \ge c\textsf {d} (x_n,x)^{2-N}, \text { for every } x \in \Omega ', \end{aligned}$$

(7.5)

for some positive constant $c=c(\varepsilon )$. Combining (7.5), the assumption $\Vert |\nabla u|\Vert _{L^\infty (\{u<1\})}\le \varepsilon ^{-1}$ and the gradient estimate (2.22), it easily follows that

$$\begin{aligned} |\nabla \sqrt{v}|=(N-2)^{-1}|\nabla u|u^{\frac{1-N}{N-2}}\le C_3, \end{aligned}$$

for some positive constant $C_3=C_3(\varepsilon )$. Finally from (5.6) and (7.1) we have

$$\begin{aligned} \int _{\{u<1\}} \frac{1}{v^{N-2}} \left| \nabla |\nabla \sqrt{v}|^{\beta /2} \right| ^2\, {\mathrm d}{\mathfrak {m}}\le C_{\beta ,N}^{-1}{U_\beta ^-}'(1)<C_{\beta ,N}^{-1}\delta . \end{aligned}$$

We are therefore in position to apply Theorem 6.7 and conclude the proof of Theorem 7.2.

Theorem 7.4 follows from Theorem 6.9 and observing that $\{u=s^{\frac{1}{2-N}}\}=\{\sqrt{v}=s\}$ and that, thanks to (7.4), for every $t>0$

$$\begin{aligned} \{\sqrt{v}\le t\}\subset B_{R_1}(x_0), \end{aligned}$$

for some $R_1=R_1(t,\varepsilon ,N,f,R_0)$. $\square$

8 The Electrostatic Potential

It was already discussed in Sect. 3 that a solution (P) is not granted, in particular already for Riemannian manifolds, the existence of solutions implies nonparabolicity. We also showed (see Corollary 3.5) that the Green function solves (P). In this short section,we provide another example of solution to (P) given by the electrostatic potential. We recall that this type of solution was crucial in the recent work [6].

Definition 8.1

(Electrostatic potential) Given $(\textrm{X},\textsf {d} ,{\mathfrak {m}})$ an (unbounded) infinitesimally Hilbertian m.m.s. and $E \subset \textrm{X}$ open and bounded, an electrostatic potential for E is a function $u \in D({\varvec{\Delta }}, \textrm{X}\setminus \bar{E})\cap C(\textrm{X}\setminus E)$ solution to

$$\begin{aligned} {\left\{ \begin{array}{ll} {\varvec{\Delta }}{_{|_{\textrm{X}\setminus {{\bar{E}}}}}} u=0,\\ u=1, &{} \text { in } \partial E,\\ u(x) \rightarrow 0 &{} \text {as }\textsf {d} (x,\partial E)\rightarrow +\infty . \end{array}\right. } \end{aligned}$$

Remark 8.2

In an $\textrm{RCD}(K,N)$ space if an electrostatic potential for E exists, then it is also unique, and this follows immediately from the maximum principle (see Proposition 2.15).

Remark 8.3

If u is a solution to (P), then the function $(1-\varepsilon )^{-1}u{_{|_{\{u\le 1-\varepsilon \}}}}$ is immediately seen to be the electrostatic potential for the open set $\textrm{X}\setminus \{u\le 1-\varepsilon \}.$

We pass to our main existence result for the electrostatic potential, which holds for sets with sufficiently regular boundary, namely with $\textsf {Cap}$-fat regular boundary. We refer to Appendix B for the definition of $\textsf {Cap}$-fat regular boundary and for examples of sets satisfying this condition.

Theorem 8.4

Let $(\textrm{X},\textsf {d} ,{\mathfrak {m}})$ be a nonparabolic $\textrm{RCD}(0,N)$ m.m.s. and let $E \subset \textrm{X}$ be open and bounded with $\textsf {Cap}$-fat boundary. Then the electrostatic potential u for E exists. Moreover the following continuity estimate holds: for every $x \in \partial E$ it holds

$$\begin{aligned} 1-u(y)\le \textsf {d} (y,x)^{\alpha _x}, \quad \forall y\in B_{r_x/2}(x)\cap E^c, \end{aligned}$$

(8.1)

for some positive constant $\alpha _x=\alpha (r_x,c_x,N)>0$, where $r_x,c_x$ are the $\textsf {Cap}$-fatness parameters of x.

Finally the function

$$\begin{aligned} {\tilde{u}}:={\left\{ \begin{array}{ll} u, &{} \textrm{X}\setminus E,\\ 1, &{} {{\bar{E}}}, \end{array}\right. } \end{aligned}$$

belongs to $S^2(\textrm{X})$ (recall Definition 2.1) and

$$\begin{aligned} \int _{\textrm{X}} |D {\tilde{u}}|^2\, {\mathrm d}{\mathfrak {m}}\le \lim _{r\rightarrow +\infty } \textsf {Cap} (E,B_r(x)), \quad \forall x \in E. \end{aligned}$$

(8.2)

Let us make some comments before passing to proof of this result. We first observe that the limit in (8.2) does not depend on $x \in E$ and that it is actually an $\inf$ and thus finite. It is not true in general that ${\tilde{u}} \in W ^{1,2}(\textrm{X})$, indeed it might not be square integrable, as can be seen taking E to be a ball in ${{\,\mathrm{{\mathbb {R}}}\,}}^3.$ Nevertheless, if the measure satisfies $0< \liminf _{r\rightarrow +\infty } r^{-\lambda }{\mathfrak {m}}(B_r(x))\le \limsup _{r\rightarrow +\infty } r^{-\lambda }{\mathfrak {m}}(B_r(x))<+\infty$ for some $\lambda >2$, then $u \in L^p({\mathfrak {m}})$ for every $p >\frac{\lambda }{\lambda -2}$ (see for example the upper bound in (5.1)).

Proof of Theorem 8.4

We will actually build ${\tilde{u}}$ and then define u to be the restriction of ${\tilde{u}}$ to $E^c.$ The argument is by compactness. Fix $x_0 \in E$ and set $B_n:=B_n(x_0)$ with $n\in {\mathbb {N}}$ and $n>\text {diam}(E)+100$. Theorem B.1 guarantees the existence of a function $u_n\in W ^{1,2}_{0}(B_n)\cap C(B)$ harmonic in $B_n\setminus {{\overline{E}}}$, such that $0\le u_n\le 1$, $u_n=1$ on ${{\bar{E}}}$ and $\int _{\textrm{X}}|\nabla u_n|^2\, {\mathrm d}{\mathfrak {m}}=\textsf {Cap} (E,B_n)$. Moreover from the comparison principle in Proposition B.9,we must have that $u_n\le u_{n+1}$ in $B_n.$

It follows from Lemma 2.17 that, up to a (non-relabelled) subsequence, $u_n$ converges in $\textrm{X}\setminus {{\bar{E}}}$ uniformly on compact sets to a function ${\tilde{u}} \in C(\textrm{X}\setminus \bar{E})$ harmonic in $\textrm{X}\setminus {\bar{E}}.$ In particular $u_n \rightarrow {\tilde{u}}$ ${\mathfrak {m}}$-a.e.. Moreover, since $\textsf {Cap} (E,B_r(x_0))$ is decreasing in r, we have that $\sup _n \Vert |\nabla u_n|\Vert _{L^2(\textrm{X})}<+\infty .$ Therefore from the lower semicontinuity of weak upper gradients (2.3), we deduce that ${\tilde{u}} \in \textrm{S}^2(\textrm{X})$ and

$$\begin{aligned} \int _{\textrm{X}}|\nabla {\tilde{u}}|^2\, {\mathrm d}{\mathfrak {m}}\le \liminf _n \int _{\textrm{X}}|\nabla u_n|^2\, {\mathrm d}{\mathfrak {m}}\le \liminf _n \textsf {Cap} (E,B_n)=\lim _{r\rightarrow +\infty }\textsf {Cap} (E,B_r(x_0)). \end{aligned}$$

The continuity estimate follows directly from the fact that $u_n\le {\tilde{u}}\le 1$ for every n and from the continuity estimate in Theorem B.1, observing that in the case $K=0$ we can drop the dependence on the diameter of E.

It remains to show that u goes to 0 at infinity. We prove it by comparison with the quasi Green function $G^1_{x_0}$ (recall (3.2)). We have that $G^1_{x_0}$ is Lipschitz and superharmonic in $\textrm{X}$. Moreover $G_{x_0}^1$ is positive, hence $\lambda G_{x_0}^1\ge {\chi }_{E}$ for a large enough constant $\lambda >0$. In particular the comparison principle in Theorem B.1 implies that $u_n \le \lambda G_{x_0}^1$ for every n, which in turn gives $u \le \lambda G_{x_0}^1.$ Finally from the estimate for the Green function in (3.3),we have

$$\begin{aligned} G_{x_0}^1\le G_{x_0}(x) \le \int _{\textsf {d} (x,x_0)}^\infty \frac{s}{{\mathfrak {m}}(B_s(x))}\, d s, \end{aligned}$$

in particular $G_{x_0}^1(x)\rightarrow 0$ at infinity. This concludes the proof. $\square$

Appendices

Appendix: From Outer Functional Cone to Outer Metric Cone - Additional Details

This section is devoted to the proof of Theorem 6.1 given the two results presented in Section 6.1 (that is Proposition 6.3 and Proposition 6.5).

Up to Sect. A.4 (included),the proof is in large part the same as [40], hence some steps will be only outlined; however, there will be differences and new arguments that will be explained in detail and emphasized along the exposition.

After Sect. A.4 (in particular in Sects. A.5, A.6 and A.7),the argument diverges from the one in [40], indeed (following a suggestion of an anonymous referee) we replace the second-order analysis of [40] with a more direct argument which uses the differentiation formula in [60] and is inspired by [24, Sect. 3.1]. This strategy is in turn analogous to the one previously employed by Cheeger and Colding [29] to derived quantitative almost splitting results via Hessian estimates. We remark that these type of arguments (as for the one used in [24]) were not possible at the time of the writing of [45] and [40] (which contains, respectively,the first versions of the splitting theorem and the volume cone to metric cone theorem in $\textrm{RCD}$ spaces), since the differentiation formula in [60] was not yet available.

We also mention the recent [33] where a similar argument exploiting [60] is used to prove a version of the (almost) volume annulus to metric annulus in $\textrm{RCD}$ setting.

1.1 The Gradient Flow of ${\textbf {u}}$ and Its Effect on the Measure

From the chain rule for the Laplacian (4.2), the positivity of ${\textbf {u}}$ and recalling that $\Delta {\textbf {u}}=N$ and $|\nabla {\textbf {u}}|^2=2{\textbf {u}}$ ${\mathfrak {m}}$-a.e. in U, it follows that

$$\begin{aligned} v:={\left\{ \begin{array}{ll} {\textbf {u}}^{\frac{2-N}{2}}, &{} \text {if } N>2,\\ \ln (\frac{1}{\sqrt{\textbf {u}}} ), &{} \text {if } N=2, \end{array}\right. }\quad \text {is harmonic in } U. \end{aligned}$$

In particular the maximum principle and the fact that ${\textbf {u}}_0=\limsup _{x\rightarrow \partial U}{\textbf {u}}(x)$ ensure that

$$\begin{aligned} \text {every connected component of } \{{\textbf {u}}>{\textbf {u}}_0\} \text { is unbounded.} \end{aligned}$$

(A.1)

Since by assumption $\{{\textbf {u}}>{\textbf {u}}_0\}$ is nonempty we must have that $\textrm{X}$ is unbounded.

For technical reasons,we will work locally, in particular we fix a set V open and relatively compact in U and consider $\eta \in \textsf {Test} (\textrm{X})$ such that $\eta =1$ in ${{\overline{V}}}$, $0\le \eta \le 1$ and $\mathop {\textrm{supp}}\nolimits \eta \subset U$, which exists thanks to Proposition 2.13. We then define

$$\begin{aligned} u:=\eta {\textbf {u}}. \end{aligned}$$

Since ${\textbf {u}}\in \textsf {Test} _\textsf {loc} ( U)$ from (2.19) we deduce that $u\in \textsf {Test} (X)$.

We point out that we would like to take right away V to be of the form $\{t_0<{\textbf {u}}<T_0\}$; however, to ensure that this set is relatively compact in U, we need first to know that ${\textbf {u}}$ blows up at infinity. This will be proved in Lemma A.2.

We now consider the regular Lagrangian flow $F : [0,T] \times X \rightarrow X$ associated to the autonomous vector field $v=-\nabla u.$ Observe that since $\Delta u\in L^\infty ({\mathfrak {m}})$ and $u \in \textsf {Test} (\textrm{X})$ the assumptions of Theorem 2.29 are satisfied. In particular the flow F exists unique. Moreover, again thanks to $\Delta u\in L^\infty ({\mathfrak {m}})$ and Remark 2.30 we can extend the map F to $(-\infty ,+\infty )\times X$.

Proposition A.1

• 1. For ${\mathfrak {m}}$-a.e. $x\in X$ it holds that $F_t(F_s(x))=F_{s+t}(x)$ for every $s,t \in {{\,\mathrm{{\mathbb {R}}}\,}}.$

• 2. For ${\mathfrak {m}}$-a.e. $x\in U$ it holds that $(-\infty ,\infty )\ni t \mapsto F_t(x)$ is continuous. Moreover denoted by $(a_x,b_x)$ the maximal interval such that $F_t(x)\in U$ for all $t \in (a_x,b_x)$ (which in particular satisfies $a_x<0, b_x>0$ and possibly $a_x=-\infty$ or $b_x=+\infty$), it holds

  $$\begin{aligned} {\textbf {u}}(F_t(x))=e^{-2t}{\textbf {u}}(x), \quad \forall t\in (a_x,b_x) \end{aligned}$$

  (A.2)

  and

  $$\begin{aligned} \textsf {d} (F_s(x),F_t(x))\le |e^{-t}-e^{-s}|\sqrt{2{\textbf {u}}(x)}, \quad \forall t,s\in (a_x,b_x). \end{aligned}$$

  (A.3)

Proof

The first is just (2.25).

(A.2) follows observing that from (2.24), since $|\nabla u|^2=2u$ ${\mathfrak {m}}$-a.e. in U, for ${\mathfrak {m}}$-a.e. $x\in U$ it holds that

$$\begin{aligned} \frac{{\mathrm d}}{{\mathrm d}t}u(F_t(x))=-2u(F_t(x)), \quad \text {for a.e. } t\in (a_x,b_x). \end{aligned}$$

(A.3) instead can be derived from the fact that, recalling Remark 2.27, for ${\mathfrak {m}}$-a.e. $x\in U$ it holds $\overset{.}{|F_t(x)|}=\sqrt{2u(F_t(x))}$ for a.e. $t \in (a_x,b_x)$. $\square$

Recall that, as remarked at the beginning of the section, $\textrm{X}$ is unbounded, hence the following result makes sense.

Lemma A.2

$$\begin{aligned} {\textbf {u}}(x)\rightarrow +\infty \quad \text { as } \textsf {d} (x, U^c)\rightarrow +\infty . \end{aligned}$$

(A.4)

Proof

Suppose (A.4) is false. Then we can find a ball $B_{2R}({{\bar{x}}})\subset U$ such that $R>100\sqrt{{\textbf {u}}({{\bar{x}}})}+1.$ We choose $\eta \in \textsf {Test} (\textrm{X})$ such that $\eta =1$ in ${\overline{B}}_{R}({{\bar{x}}})$, $0\le \eta \le 1$ and $\mathop {\textrm{supp}}\nolimits \eta \subset U$, which exists from Proposition 2.13. We define $u:={\textbf {u}}\eta \in \textsf {Test} (\textrm{X})$ and consider the regular Lagrangian flow $F_t$ relative to $-\nabla u.$ Then from (A.3) (with the choice $V=B_R(\bar{x})$), the continuity of ${\textbf {u}}$ and the choice of R we can find $x'\in B_1({{\bar{x}}})$ such that the curve $F_t(x')$ is contained in $B_R({{\bar{x}}})$ for all $t>0.$ This together with (A.2) contradicts the positivity of ${\textbf {u}}$. $\square$

From now until the very last part of the proof, we fix $t_0,T_0 \in {{\,\mathrm{{\mathbb {R}}}\,}}^+$ such that ${\textbf {u}}_0<t_0+1<T<T_0-1$ and $T_0-T>T-t_0$, where T is to be chosen later.

Thanks to both (A.4) and ${\textbf {u}}_0=\limsup _{x\rightarrow \partial U} {\textbf {u}}(x)$ we have that $\{t_0<{\textbf {u}}<T_0\}$ is compactly contained in U. Hence we can pick a cut-off function $\eta \in \textsf {Test} (\textrm{X})$ such that $\eta =1$ in $\{t_0\le {\textbf {u}}\le T_0\}$, $0\le \eta \le 1$, $\mathop {\textrm{supp}}\nolimits \eta \subset U$ and define $u:=\eta {\textbf {u}}\in \textsf {Test} (\textrm{X}).$ As above we consider $F_t$ the flow relative to $-\nabla u$, which is defined for all positive and negative times.

Define for every $a,b\in [{\textbf {u}}_0,\infty )$, the open set

$$\begin{aligned} A_{a,b}:=\{a<{\textbf {u}}<b\}. \end{aligned}$$

From the definition of u, the hypotheses on ${\textbf {u}}$ and the locality of the gradient and the Laplacian, we have

$$\begin{aligned} \begin{aligned}&\Delta u=N, \quad {\mathfrak {m}}\text {-a.e. in } A_{t_0,T_0}.\\&|\nabla u|^2=2u, \quad {\mathfrak {m}}\text {-a.e. in } A_{t_0,T_0}. \end{aligned} \end{aligned}$$

(A.5)

The following can be proven arguing as in [40, Sect. 3.6.1]; however, we give a shorter proof, which use the improved Bochner inequality (2.17).

Proposition A.3

$$\begin{aligned} \textrm{Hess}(u)=\textsf {id} \quad {\mathfrak {m}}\text {-a.e. in } A_{t_0,T_0}. \end{aligned}$$

Proof

Localizing (2.17) to $A_{t_0,T_0}$ and recalling (A.5) we obtain

$$\begin{aligned} N\ge |\textrm{Hess}(u)|_{HS}^2+\frac{(N-\textsf {tr} \textrm{Hess}(u))^2}{N-\dim (\textrm{X})} \quad {\mathfrak {m}}\text {-a.e. in } A_{t_0,T_0}. \end{aligned}$$

By Cauchy-Swartz and recalling (2.15), (2.16) we observe that $|\textrm{Hess}(u)|_{HS}^2=\sum _{1\le i,j\le \dim (\textrm{X})} \textrm{Hess}(u)(e_i,e_j)^2\ge \sum _{i=1}^{\dim (\textrm{X})}\textrm{Hess}(u)(e_i,e_i)^2 \ge \frac{\textsf {tr} \textrm{Hess}(u)^2}{N}$. Plugging this in the above inequality and applying again Cauchy-Swartz we obtain

$$\begin{aligned} N\ge \frac{\textsf {tr} \textrm{Hess}(u)^2}{N}+\frac{(N-\textsf {tr} \textrm{Hess}(u))^2}{N-\dim (\textrm{X})}\ge N \quad {\mathfrak {m}}\text {-a.e. in } A_{t_0,T_0}. \end{aligned}$$

Hence all the inequality we used were actually equalities, in particular $\textrm{Hess}(u)(e_i,e_j)=0$ ${\mathfrak {m}}\text {-a.e. in } A_{t_0,T_0}$, for every $i\ne j$ and $\textrm{Hess}(u)(e_i,e_i)=1$ ${\mathfrak {m}}\text {-a.e. in } A_{t_0,T_0}$ for every $i=1,...,\dim (\textrm{X})$, which concludes the proof. $\square$

Proposition A.4

• 1. For ${\mathfrak {m}}$-a.e. $x\in X$ it holds that $F_t(F_s(x))=F_{s+t}(x)$ for every $s,t \in {{\,\mathrm{{\mathbb {R}}}\,}}.$

• 2. For ${\mathfrak {m}}$-a.e. $x\in A_{t_0,T_0}$ it holds that $F_t(x)\in A_{t_0,T_0}$ and

  $$\begin{aligned} u(F_t(x))=e^{-2t}u(x), \end{aligned}$$

  (A.6)

  for every $t\in (\frac{1}{2}\log {\frac{u(x)}{T_0}},\frac{1}{2}\log {\frac{u(x)}{t_0}}),$ moreover

  $$\begin{aligned} \textsf {d} (F_s(x),F_t(x))=|e^{-t}-e^{-s}|\sqrt{2u(x)}, \end{aligned}$$

  (A.7)

  for every $s,t\in (\frac{1}{2}\log {\frac{u(x)}{T_0}},\frac{1}{2}\log {\frac{u(x)}{t_0}}),$ in particular the curve $(\frac{1}{2}\log {\frac{u(x)}{T_0}},\frac{1}{2}\log {\frac{u(x)}{t_0}}) \ni t \mapsto F_t(x)$ is supported on a geodesic.

Proof

Everything except for the equality in (A.7) follow Proposition A.1 together with the observation that in this case $a_x= \frac{1}{2}\log {\frac{u(x)}{T_0}}$ and $b_x= \frac{1}{2}\log {\frac{u(x)}{t_0}}$.

To show equality in (A.7), it is enough to show it for $t=0$ and $s>0$. Hence we fix $s \in (0,\frac{1}{2}\log {\frac{u(x)}{t_0}})$. Thanks to (A.3), we only need to show that

$$\begin{aligned} \textsf {d} (F_s(x),x)\ge (1-e^{-s})\sqrt{2u(x)}. \end{aligned}$$

(A.8)

We make the intermediate claim that

$$\begin{aligned} \textsf {d} (x, U^c)\ge \sqrt{2{\textbf {u}}(x)}-\sqrt{2{\textbf {u}}_0}, \quad \forall x \in U. \end{aligned}$$

(A.9)

To prove it we first observe that, since ${\textbf {u}}$ is positive, we have that $\sqrt{{\textbf {u}}}\in W ^{1,2}_{\textsf {loc} }( U)$ and $|\nabla \sqrt{2{\textbf {u}}}|=1$ ${\mathfrak {m}}$-a.e. in U.

The properness of the space $\textrm{X}$ ensures that there exists $\bar{x}\in \partial U$ such that $\textsf {d} (x,{{\bar{x}}})=\textsf {d} (x,\partial U)$. Moreover, since $\textrm{X}$ is geodesic, there exists a sequence $x_n\subset U$ such that $x_n\rightarrow x$ and $\textsf {d} (x,x_n)\le \textsf {d} (x,\partial U)$. Hence recalling that $|\nabla \sqrt{2{\textbf {u}}}|=1$ ${\mathfrak {m}}$-a.e., we are in position to apply (2.7) and deduce that

$$\begin{aligned} \textsf {d} (x,{{\bar{x}}})=\lim _n \textsf {d} (x,x_n)\ge \liminf _n \sqrt{2{\textbf {u}}(x)}-\sqrt{2{\textbf {u}}(x_n)}\ge \sqrt{2{\textbf {u}}(x)}-\sqrt{2{\textbf {u}}_0}, \end{aligned}$$

which proves (A.9).

From (A.3) and by how we chose s, we have that $\textsf {d} (F_s(x),x)\le (1-e^{-s})\sqrt{2{\textbf {u}}(x)}\le \sqrt{2{\textbf {u}}(x)}-\sqrt{2t_0}\le \sqrt{2{\textbf {u}}(x)}-\sqrt{2{\textbf {u}}_0}.$ Hence from (A.9) we deduce $\textsf {d} (F_s(x),x)\le \textsf {d} (x,\partial U)$ and applying again (2.7) combined with (A.6) we obtain (A.8). $\square$

Lemma A.5

For every $t \in (0,\frac{1}{2}\log \frac{T_0}{t_0})$ it holds that

$$\begin{aligned} {\mathfrak {m}}(A_{e^{2t}t_0,T_0})=e^{Nt}{\mathfrak {m}}(A_{t_0,e^{-2t}T_0}). \end{aligned}$$

(A.10)

Proof

The argument is essentially the same as in [40, Prop. 3.7] but reversed, indeed here we start from $\Delta u=N$ and $|\nabla u|^2=2u$ and deduce information on the measure. $\square$

The following result has not a direct counterpart in [40]; however, it morally substitutes the bound (3.1) in [40, Prop. 3.2] (cf. with (A.12) below). We remark that the proof of the following Proposition relies on the local estimate of Proposition 6.5.

Proposition A.6

For every $t \in (0,\frac{1}{2}\log \frac{T_0}{t_0})$ it holds that

$$\begin{aligned} ({F_t}_*{\mathfrak {m}}){_{|_{A_{t_0,e^{-2t}T_0}}}}={F_t}_*\left( {\mathfrak {m}}{_{|_{A_{e^{2t}t_0,T_0}}}}\right) =e^{Nt}{\mathfrak {m}}{_{|_{A_{t_0,e^{-2t}T_0}}}}. \end{aligned}$$

(A.11)

Proof

Consider the probability measure $\mu _0=\frac{{\mathfrak {m}}{_{|_{A_{e^{2t}t_0,T_0}}}}}{{\mathfrak {m}}(A_{e^{2t}t_0,T_0})}.$ By Theorem 2.29$\{{F_s}_*\mu _0\}_{s\in [0,t]}$ are all Borel probability measures, absolutely continuous with respect to ${\mathfrak {m}}$ and solve the continuity equation with initial datum $\mu _0$. Moreover (A.6) implies that ${F_s}_*\mu _0$ is concentrated on $A_{t_0,T_0}$ for every $s \in [0,t]$. Therefore setting ${F_t}_*\mu _0=\rho _t{\mathfrak {m}}$ we are in position to apply (6.3), that combined with (A.5) gives

$$\begin{aligned} \Vert \rho _t\Vert _{\infty } \le {\mathfrak {m}}(A_{e^{2t}t_0,T_0})^{-1}e^{Nt}. \end{aligned}$$

(A.12)

However, applying now (A.10) and observing that and that ${F_t}_*\mu _0$ is concentrated in $A_{t_0,e^{-2t}T_0}$, again thanks to (A.6), we can compute

$$\begin{aligned} 0\le \int _{A_{t_0,e^{-2t}T_0}} {\mathfrak {m}}(A_{e^{2t}t_0,T_0})^{-1}e^{Nt}-\rho _t {\mathrm d}{\mathfrak {m}}= {\mathfrak {m}}(A_{e^{2t}t_0,T_0})^{-1}e^{Nt}{\mathfrak {m}}(A_{t_0,e^{-2t}T_0})-1=0, \end{aligned}$$

that gives the second in (A.11).

The first in (A.11) follows directly from (A.6). $\square$

Having at our disposal (A.11) and (A.7), we can argue exactly as in [40, Cor. 3.8] to obtain the following

Proposition A.7

(Continuous disintegration) We have

$$\begin{aligned} u_*{\mathfrak {m}}{_{|_{A_{t_0,T_0}}}}=cr^{\frac{N}{2}-1} {\mathcal {L}}^1{_{|_{(t_0,T_0)}}}, \end{aligned}$$

where $c:=\frac{N}{2}\frac{{\mathfrak {m}}(A_{t_1,t_2})}{t_1^{N/2}-t_2^{N/2}}$, for any $t_1,t_2 \in {{\,\mathrm{{\mathbb {R}}}\,}}^+$ with $t_0\le t_2<t_2\le T_0$. Moreover there exists a weakly continuous family of Borel measures $(t_0,T_0)\ni r\mapsto {\mathfrak {m}}_r \in {\mathcal {P}}(X)$ such that

$$\begin{aligned} \int \varphi {\mathrm d}{\mathfrak {m}}= c \int _{t_0}^{T_0} \int \varphi {\mathrm d}{\mathfrak {m}}_r r^{N/2-1} {\mathrm d}r, \quad \forall \varphi \in C_c(A_{t_0,T_0}). \end{aligned}$$

(A.13)

Finally, for every $t \in (0,\log \frac{T_0}{t_0})$ the measures ${\mathfrak {m}}_r$ satisfies

$$\begin{aligned} {F_t}_*{\mathfrak {m}}_r={\mathfrak {m}}_{e^{-2t}r}, \quad \text { for a.e. } r\in (e^{2t}t_0,T_0). \end{aligned}$$

The following result has not a counterpart in [40], since it deals with large scales, while the analysis in [40] is local.

Corollary A.8

$\textrm{X}$ has Euclidean volume growth, in particular $\{{\textbf {u}}>{\textbf {u}}_0\}$ is unbounded, connected with $\{{\textbf {u}}>{\textbf {u}}_0\}^c$ bounded.

Proof

Combining (A.6) and (A.7), it can be shown that $A_{t_0,T_0}\subset B_{4\sqrt{T_0}+C}(x_0)$ for every $T_0>2t_0$ for some fixed constant $C>0$ (recall that what we proved so far holds for an arbitrary $T_0>T$). Therefore $A_{t_0,\frac{(R-C)^2}{16}}\subset B_R(x_0)$ for every R big enough and the conclusion follows using (A.13).

Since $\textrm{X}$ has Euclidean volume growth, it is not compact and not a cylinder in the sense of i) of Proposition 3.8. Now observe that $\partial \{{\textbf {u}}>{\textbf {u}}_0\}$ is bounded (as a consequence of (A.4)), hence compact, and that each connected component of $\{{\textbf {u}}>{\textbf {u}}_0\}$ is unbounded (by (A.1)). Hence the conclusion follows from Proposition 3.8 and the fact that $\{{\textbf {u}}>{\textbf {u}}_0\}$ is not empty. $\square$

Lemma A.9

[40, Lemma 3.11] Let $f\in L^p({\mathfrak {m}})$ with $p<+\infty$, then the map $t\mapsto f\circ F_t$ is continuous in $L^p({\mathfrak {m}}).$ Moreover if $f\in W ^{1,2}(X)$ the map $t \mapsto f\circ F_t$ is $C^1$ in $L^2$ and its derivative is given by

$$\begin{aligned} \frac{{\mathrm d}}{{\mathrm d}t} f\circ F_t=-\langle \nabla f , \nabla u \rangle \circ F_t. \end{aligned}$$

1.2 Effect on the Dirichlet Energy

Thanks to (A.5) and (A.11), we can repeat almost verbatim the analysis done in [40, Sect. 3.2]. Indeed all the proofs contained there rely only on the analogous properties of the function $\textsf {b}$ and $\textsf {Fl} _t$, i.e. “$|D \textsf {b} |^2=2\textsf {b} ,\Delta \textsf {b} =N$ and ${\textsf {Fl} _t}_*{\mathfrak {m}}=e^{Nt}{\mathfrak {m}}$”.

This said, we will only state, adapted to our case and without proof, the final result in [40, Sect. 3.2] (i.e. Corollary 3.17), since it is the only statement that is needed for the rest of the argument.

Theorem A.10

Let $t \in (0,\log \frac{T_0}{t_0})$, and $f \in L^2({\mathfrak {m}})$ with support in $A_{t_0,e^{-2t}T_0}$. Then $f \in W ^{1,2}(X)$ if and only if $f\circ F_t \in W ^{1,2}(X)$ and in this case

$$\begin{aligned} |\nabla (f \circ F_t)|= e^{-t}|\nabla f |\circ F_t, {\mathfrak {m}}-a.e. \end{aligned}$$

1.3 Precise Representative of the Flow

The following proposition is the analogous of [40, Thm. 3.18]. We point out that the proof in [40] contains an oversight in the proof that $\textsf {Fl} _t$ has a locally Lipschitz representative. Indeed it is claimed that this follows from the fact that $\textsf {Fl} _t$ is Lipschitz in $\textsf {Fl} _t^{-1}(B_r(x_0))$ for every small enough ball $B_r(x_0)$. However, since $\textsf {Fl} _t$ is not yet proven to be continuous, we do not know enough information on the sets $\textsf {Fl} _t^{-1}(B_r(x_0))$ to ‘patch’ them and obtain the claimed local Lipschitzianity.

For this reason,we will give a complete proof which also fixes the original argument.

Proposition A.11

Let ${{{\mathcal {U}}}}_{t_0,T_0}\subset {{\,\mathrm{{\mathbb {R}}}\,}}\times \textrm{X}$ be the open set given by

$$\begin{aligned} {{{\mathcal {U}}}}_{t_0,T_0}:=\left\{ (x,t) \ : x \in A_{t_0,T_0} \text { and } t \in \left( \frac{1}{2}\log \frac{u(x)}{T_0} ,\frac{1}{2}\log \frac{u(x)}{t_0} \right) \right\} . \end{aligned}$$

Then the map

$$\begin{aligned} F : {{{\mathcal {U}}}}_{t_0,T_0}\rightarrow A_{t_0,T_0}, \end{aligned}$$

has a continuous representative w.r.t the measure ${\mathcal {L}}^1\otimes {\mathfrak {m}}$. Moreover for such representative (which we denote again by F) the map $F_t : A_{e^{2t}t_0,T_0}\rightarrow A_{t_0,e^{-2t}T_0}$ is locally $e^{-t}$-Lipschitz having $F_{-t}$ as its inverse, which is locally $e^{t}$-Lipschitz. Also for every $x \in A_{t_0,T_0}$ and every $s,t \in (\frac{1}{2}\log \frac{u(x)}{T_0} ,\frac{1}{2}\log \frac{u(x)}{t_0} )$

$$\begin{aligned} \textsf {d} (F_t(x),F_s(x))=|e^{-s}-e^{-t}|\sqrt{2u(x)} \end{aligned}$$

(A.14)

and

$$\begin{aligned} u(F_t(x))=e^{-2t}u(x). \end{aligned}$$

(A.15)

Finally for every $t \in (0,\log \frac{T_0}{t_0})$ and every curve $\gamma$ with values in $A_{e^{2t}t_0,T_0}$, putting $\tilde{\gamma }:= F_t\circ \gamma$ we have

$$\begin{aligned} |\overset{.}{{\tilde{\gamma }}}_s|=e^{-t}|\overset{.}{\gamma }_s|, \quad \text { for a.e. } s \in [0,1], \end{aligned}$$

(A.16)

meaning that one is absolutely continuous if and only if the other is absolutely continuous, in which case (A.16) holds.

Proof

Fix $t \in [0,\log \frac{T_0}{t_0})$. We start claiming that

$$\begin{aligned} \begin{aligned}&F_t{_{|_{A_{e^{2t}t_0,T_0}}}}\,\, \text { has a continuous representative that we denote by } {{\bar{F}}}_t \text { and}\\&{{\bar{F}}}_t(A_{e^{2t}t_0,T_0})=A_{t_0,e^{-2t}T_0}. \end{aligned} \end{aligned}$$

(A.17)

For the first part,it is sufficient to show that for every $a,b\in {{\,\mathrm{{\mathbb {R}}}\,}}$ such that $e^{2t}t_0<a<b<T_0$ the map $F_t{_{|_{A_{a,b}}}}$ has a continuous representative. Hence we fix such $a,b \in {{\,\mathrm{{\mathbb {R}}}\,}}$ and define the open sets $A':=A_{e^{-2t}a,e^{-2t}b}$ and $A:=A_{t_0,e^{-2t}T_0}$. Observe that the continuity of u implies $\textsf {d} (A',A^c)=:\delta >0.$ Consider now the countable family of 1-Lipschitz functions ${\mathcal {D}} \subset \textsf {LIP} (X)$ defined as

$$\begin{aligned} {\mathcal {D}}:=\{ f_{n,k} \ | \ n,k\in {\mathbb {N}}\}=\{ \max (\min (\textsf {d} (.,x_n),k-\textsf {d} (.,x_n)) ,0) \ | \ n,k\in {\mathbb {N}}\}, \end{aligned}$$

where $\{x_n\}_{n\in {\mathbb {N}}}$ is a dense subset of A. We pick a cut-off function $\eta \in \textsf {LIP} _c(A)$ such that $0\le \eta \le 1$ and $\eta \equiv 1$ in $A'$ and define the set $\eta {\mathcal {D}} \subset \textsf {LIP} _c(A)$ as $\eta {\mathcal {D}}:=\{ \eta f \ | \ f \in {\mathcal {D}}\}.$ For any $f \in {\mathcal {D}}$ it holds

$$\begin{aligned} \text {Lip} (f\eta ) \le \textrm{Lip}\eta \sup _A |f|+\textrm{Lip}f\le \textrm{Lip}\eta \, \text {diam}(A)+1=:L, \end{aligned}$$

hence the functions in $\eta {\mathcal {D}}$ are L-Lipschitz. We now make the key observation that

$$\begin{aligned} \textsf {d} (x,y)= & {} \sup _{f \in {\mathcal {D}}} |f(x)-f(y)|=\sup _{f \in {\mathcal {D}}} |\eta f(x)-\eta f(y)|\nonumber \\= & {} \sup _{f \in \eta {\mathcal {D}}} |f(x)- f(y)|, \end{aligned}$$

(A.18)

for every $x,y \in A'.$ Thanks to Corollary (A.10), we know that $f\circ F_t \in W ^{1,2}(X)$ for every $f \in \eta {\mathcal {D}}$ and $|D(f\circ F_t)|=e^{-t}|Df|\circ F_t\le e^{-t}L$ ${\mathfrak {m}}$-a.e. Then from the Sobolev-to-Lipschitz property of X, we deduce that $f \circ F_t$ has an L-Lipschitz representative. Thus there exists an ${\mathfrak {m}}$-negligible set $N\subset X$ such that for every $f \in \eta {\mathcal {D}}$ the restriction of $f\circ F_t$ to $X\setminus N$ is L-Lipschitz. Moreover from (A.6) it follows the existence of an ${\mathfrak {m}}$-negligible set $N'$ such that $F_t(A_{a,b}\setminus N')\subset A'.$ Therefore from (A.18) it follows that for every $x,y \in A_{a,b}\setminus (N\cup N')$

$$\begin{aligned} \textsf {d} (F_t(x),F_t(y))=\sup _{f \in \eta {\mathcal {D}}} |f(F_t(x))- f(F_t(y))|\le e^{-t}L \textsf {d} (x,y). \end{aligned}$$

This proves the first part of (A.17). We now show $\subset$ of the second part. From (A.6) it follows the existence of a negligible set N such that for every set U relatively compact in $A_{e^{2t}t_0,T_0}$ we have that $\bar{F}_t(U\setminus N)$ is relatively compact in $A_{t_0,e^{-2t}T_0}.$ Moreover, since negligible sets have empty interior we deduce that $\overline{U\setminus N}$ contains U. Therefore $\bar{F}_t(U)\subset {{\bar{F}}}_t(\overline{U\setminus N})\subset \overline{{{\bar{F}}}_t(U\setminus N)}$ which is contained in $A_{e^{2t}t_0,T_0}$ thanks to the first observation. We now show $\supset$. Again thanks to (A.6) the set $N:=A_{t_0,e^{-2t}T_0}\setminus {{\bar{F}}}_t(A_{e^{2t}t_0,T_0})$ is negligible. Pick any set U relatively compact in $A_{t_0,e^{-2t}T_0}$ and define $V :=\bar{F}_t^{-1}(U\setminus N)$ which is relatively compact in $A_{e^{2t}t_0,T_0}.$ Therefore, since ${{\bar{F}}}_t$ is continuous, the set $F_t({{\bar{V}}})$ is compact in $A_{t_0,e^{-2t}T_0}$ and, since negligible sets have empty interior, contains U. This concludes the proof of (A.17).

For any $t \in (\log \frac{t_0}{T_0},0)$,we can now argue exactly as above, to deduce that

$$\begin{aligned} \begin{aligned}&F_t{_{|_{A_{t_0,e^{2t}T_0}}}}\,\, \text { has a continuous representative that we denote by } {{\bar{F}}}_t \text { and}\\&{{\bar{F}}}_t(A_{t_0,e^{2t}T_0})=A_{e^{-2t}t_0,T_0}. \end{aligned} \end{aligned}$$

(A.19)

In particular, since $F_t(F_{-t})=\mathsf id$ ${\mathfrak {m}}$-a.e., we deduce that ${{\bar{F}}}_{-t}$ is the continuous inverse of ${{\bar{F}}}_t.$

Having proved (A.17) and (A.19) we can complete the proof arguing as in [40] with the obvious modifications. $\square$

From now on we denote by F a representative of $F: {{\,\mathrm{{\mathbb {R}}}\,}}\times \textrm{X}\rightarrow \textrm{X}$ which is continuous (in space and time) on $\mathcal{U}_{t_0,T_0}$.

1.4 Properties of Level Set $\{{\textbf {u}}=T\}$

In this short subsection,we prove that the level set $\{{\textbf {u}}=T\}$ is Lipschitz path connected when T is big enough. We remark that the argument will rely on Proposition 3.8 and is different from the one used in [40] to prove the same property for the “sphere”.

For the following result recall from Sect. A.1 that in our construction,we first choose T and then we choose $t_0$ and $T_0$ accordingly.

Lemma A.12

There exists $T>{\textbf {u}}_0$ and a constant $c>0$ (depending only on $T,t_0$ and ${\textbf {u}}$) such that, if $T_0$ is big enough, for every couple of points $x,y \in \{{\textbf {u}}=T\}$, there exists $\gamma \in \textsf {LIP} ([0,1],\textrm{X})$ joining x and y and such that $\gamma \subset A_{t_0+1,T_0-1 }$ and $\textrm{Lip}(\gamma )\le 5\textsf {d} (x,y).$

Proof

From (A.4), we know that $\textsf {d} (\{{\textbf {u}}= T\},\{{\textbf {u}}>t_0+1\}^c)\rightarrow +\infty$ as $T\rightarrow +\infty$. Moreover $\{{\textbf {u}}>t_0+1\}^c$ is bounded and, as proven in Corollary A.8, $\textrm{X}$ has Euclidean volume growth and in particular it is not a cylinder (since $N\ge 2$). Therefore from Proposition 3.8, provided T is big enough, we have that for every couple of points $x,y \in \{{\textbf {u}}=T\}$ there exists a Lipschitz path $\gamma$ from x to y and such that $\gamma \subset \{{\textbf {u}}>t_0+1\}$ and $\textrm{Lip}(\gamma )\le 5\textsf {d} (x,y)$. Therefore, again from (A.4) and since $\{{\textbf {u}}=T\}$ is bounded, if $T_0$ is big enough then it holds that $\gamma \subset A_{t_0+1,T_0-1}$. $\square$

Consider now the closed set $S_T:=\{{\textbf {u}}=T\}$ and the projection map $\textsf {Pr} : A_{t_0,T_0}\rightarrow S_T$ defined as

$$\begin{aligned} \textsf {Pr} (x):=F_{\frac{1}{2}\log {\frac{u(x)}{T}}}(x). \end{aligned}$$

Note that from Proposition A.11 and (A.14), we have that $\textsf {Pr}$ is well defined and locally Lipschitz.

Proposition A.13

The set $S_T$ is Lipschitz path connected, meaning that for every couple of points $x,y \in S_T$ there exists a Lipschitz curve $\gamma$ taking values in $S_T$, joining x and y. Moreover we can choose $\gamma$ so that $\textrm{Lip}(\gamma )\le c\,\textsf {d} (x,y),$ for some uniform constant $c>0.$

Proof

From Proposition A.12, we now that there exists a Lipschitz path connecting x and y taking values in $A_{t_0+1,T_0-1},$ which is relatively compact in $A_{t_0,T_0},$ then we simply consider the curve

$$\begin{aligned} {\tilde{\gamma }} :=\textsf {Pr} \circ \gamma , \end{aligned}$$

which remains a Lipschitz curve, since $\textsf {Pr}$ is locally Lipschitz. The claimed bound on $\textrm{Lip}({\tilde{\gamma }})$ follows from the bound on $\textrm{Lip}(\gamma )$ given in Proposition A.12 and again by the local Lipschitzianity of $\textsf {Pr} .$ $\square$

1.5 The Cosine Formula Holds

The goal of this section is to prove the following.

Proposition A.14

(Local cosine formula) For every $\varepsilon >0$ there exists $\rho =\rho (\varepsilon )>0$ such that for every $x \in A_{t_0+\varepsilon ,T_0-\varepsilon }$ it holds

$$\begin{aligned} \textsf {d} (x,y)^2=2u(x)+2u(y)-4\sqrt{u(x)u(y)}\left( 1- \frac{\textsf {d} (\textsf {Pr} (x),\textsf {Pr} (y)^2)}{4T} \right) , \quad \forall y \in B_\rho (x).\nonumber \\ \end{aligned}$$

(A.20)

To prove this statement, we need to construct some objects and prove some preliminary technical facts.

Fix any $x \in A_{t_0+\varepsilon ,T_0-\varepsilon }$ and $\delta >0$ small and to be chosen. For every $r>0$ and $y \in B_\delta (x)\cup B_\delta (\textsf {Pr} (x))$ define the measures $\mu ^r:=({\mathfrak {m}}(B_r(\textsf {Pr} (x))))^{-1}{\mathfrak {m}}{_{|_{B_r(\textsf {Pr} (x))}}}$ and $\nu ^r:=({\mathfrak {m}}(B_r(y)))^{-1}{\mathfrak {m}}{_{|_{B_r(y)}}}$. Let ${\bar{t}}\in {{\,\mathrm{{\mathbb {R}}}\,}}$ be such that $F_{{{\bar{t}}}}(\textsf {Pr} (x))=x$ (i.e. ${{\bar{t}}}=\frac{1}{2} \log \frac{T}{u(x)}$). For every $t \in [0,{{\bar{t}}}]$ (or in $[{\bar{t}},0]$ if ${{\bar{t}}}<0$) we also set $\mu _t^r:={F_t}_*\mu ^r$. Finally for everyone of these t’s we also consider the unique $W_2$-geodesic $\{\eta _s^{t,r}\}_{s \in [0,1]}$ going from $\nu ^r$ to $\mu _t^r$. For the main computations, we will need the following technical fact:

$$\begin{aligned}{} & {} \text { there exists } \rho =\rho (\varepsilon )>0 \text { such that, provided } r,\delta <\rho ,\, \nonumber \\{} & {} \quad \mathop {\textrm{supp}}\nolimits (\eta _s^{t,r})\subset A_{t_0,T_0} \text { for all } s \in [0,1]. \end{aligned}$$

(A.21)

Since the measures $\eta _s^{t,r}$ are concentrated on the support of geodesics going from $B_r(y)$ to $F_t(B_r(x))$, to check the above it is sufficient to prove the following.

Proposition A.15

For every $x \in A_{t_0+\varepsilon ,T_0-\varepsilon }$ there exists $\rho =\rho (\varepsilon )>0$ such that the following holds. Let ${{\bar{t}}}\in {{\,\mathrm{{\mathbb {R}}}\,}}$ be such that $F_{{{\bar{t}}}}(\textsf {Pr} (x))=x$ (i.e. ${{\bar{t}}}=\frac{1}{2} \log \frac{T}{u(x)}$). Then every geodesic $\gamma$ going from a point $y \in B_\rho (\textsf {Pr} (x))\cup B_\rho (x)$ to a point $x' \in F_t(B_\rho (\textsf {Pr} (x)))$, with $t \in [0,{{\bar{t}}}]$ (or $[{{\bar{t}}},0]$), $\gamma$ is contained in $A_{t_0,T_0}$.

This in turn will follow from the next simpler result, which roughly says that if two points are sufficiently close, then all the geodesics connecting one of them to the flow line of the other, are contained in $A_{t_0,T_0}.$

Lemma A.16

For every $\varepsilon >0$,there exists $\rho =\rho (\varepsilon )>0$ such that the following holds. For every $x,y \in A_{t_0+\varepsilon ,T_0-\varepsilon }$ such that $\textsf {d} (x,y)<\rho$ and every geodesic $\gamma$ going from y to a point of the type $z=F_t(x)\in A_{t_0+\varepsilon ,T_0-\varepsilon }$, $\gamma$ is contained in $A_{t_0,T_0}$.

Proof

We first notice that arguing as in the proof of (A.9), we have

$$\begin{aligned} \textsf {d} (x,\{{\textbf {u}}=t_0 \})\ge & {} \sqrt{2u(x)}-\sqrt{2t_0},\\ \textsf {d} (x,\{{\textbf {u}}=T_0 \})\ge & {} \sqrt{2T_0}-\sqrt{2u(x)}, \quad \forall x \in A_{t_0,T_0}. \end{aligned}$$

Fix $x,y \in A_{t_0+\varepsilon ,T_0-\varepsilon }$ with $\textsf {d} (x,y)<\rho$ and fix some point $z=F_t(x)\in A_{t_0+\varepsilon ,T_0-\varepsilon }.$ We will only consider the case $t>0$, since the other one is analogous. Notice that in this case $\textsf {d} (x,z)=\sqrt{2u(x)}-\sqrt{2u(z)}.$ Moreover, since u is Lipschitz, $|\sqrt{2u(y)}-\sqrt{2u(x)}|\le C \sqrt{\rho }$ for some C depending only on u.

Fix a geodesic $\gamma$ going from y to z. Suppose by contradiction that $\gamma$ exits $A_{t_0,T_0}$. This means that $\gamma _t \in \{{\textbf {u}}=t_0 \}\cup \{{\textbf {u}}=T_0 \}$ for some $t \in (0,1)$. Since $\gamma$ is a geodesic we also have

$$\begin{aligned} \textsf {d} (z,y)\ge \max \{ \textsf {d} (\gamma _t,z), \textsf {d} (\gamma _t,y)\}. \end{aligned}$$

Moreover from (A.14) and the triangle inequality we deduce that $\textsf {d} (z,y)\le \sqrt{2u(x)}-\sqrt{2u(z)}+\rho .$

We now have two possibilities: either $\gamma _t \in \{{\textbf {u}}=t_0 \}$ or $\gamma _t \in \{{\textbf {u}}=T_0 \}$. In the first one, combining the above observations we have

$$\begin{aligned} \sqrt{2u(x)}-\sqrt{2u(z)}+\rho\ge & {} \textsf {d} (z,y)\ge \textsf {d} (\gamma _t,y)\ge \sqrt{2u(y)}-\sqrt{2t_0}\\\ge & {} \sqrt{2u(x)}-C\sqrt{\rho }-\sqrt{2t_0}, \end{aligned}$$

which gives

$$\begin{aligned} \sqrt{2t_0}+\rho +C\sqrt{\rho }\ge \sqrt{2u(z)}\ge \sqrt{2(t_0+\varepsilon )}, \end{aligned}$$

that is clearly a contradiction provided $\rho$ is chosen small enough. In the case that $\gamma _t \in \{{\textbf {u}}=T_0 \}$ we analogously have

$$\begin{aligned} \sqrt{2(T-\varepsilon )}-\sqrt{2u(z)}+\rho\ge & {} \sqrt{2u(x)}-\sqrt{2u(z)}+\rho \ge \textsf {d} (z,y)\\\ge & {} \textsf {d} (\gamma _t,z)\ge \sqrt{2T_0}-\sqrt{2u(z)}, \end{aligned}$$

which is again a contradiction if $\rho$ is small enough. $\square$

We can now prove Proposition A.15, which as observed above, proves also (A.21).

Proof of Proposition A.15

Since $F_{{{\bar{t}}}}$ is locally Lipschitz in $A_{t_0,T_0}$ (recall Proposition A.11), there exists $\delta =\delta (\varepsilon )>0$ such that

$$\begin{aligned} F_{{{\bar{t}}}}(B_\delta (\textsf {Pr} (x))\subset A_{t_0+\varepsilon /2,T_0-\varepsilon /2} \end{aligned}$$

Moreover, since u is monotone along flow lines, we also have $F_t(B_\delta (\textsf {Pr} (x))\subset A_{t_0+\varepsilon /2,T_0-\varepsilon /2}$ for every $t \in [0,{{\bar{t}}}]\cup [{{\bar{t}}},0]$. Let now $\rho =\rho (\varepsilon /2)$ be the one given by Lemma A.16 corresponding to $\varepsilon /2$ and note that we can assume that $\rho <\delta .$ Then the conclusion for $y \in B_\rho (\textsf {Pr} (x))$ follows immediately from Lemma A.16.

For the cases in which y is close to x,we observe that, again by the local Lipschitzianity of $F_{{{\bar{t}}}}$, there exists $\bar{\rho }(\varepsilon ) \in (0,\rho /2)$ so that $F_{{{\bar{t}}}}(B_{\bar{\rho }}(\textsf {Pr} (x))) \subset B_{\rho /2}(x)$. In particular for every $y \in B_{{\bar{\rho }}}(x)$ and $x' \in F_{{{\bar{t}}}}(B_{\bar{\rho }}(\textsf {Pr} (x)))$,we have $\textsf {d} (y,x')<\rho /2+{\bar{\rho }}<\rho$. Therefore we conclude again by Lemma A.16 that every geodesic from y to $F_t(B_{{\bar{\rho }}}(\textsf {Pr} (x)))$, with $t \in [0,{{\bar{t}}}]\cup [{{\bar{t}}},0]$, is contained in $A_{t_0,T_0}.$ $\square$

Proof of Proposition A.14

It is sufficient to prove that for every $\varepsilon >0$ there exists $\rho (\varepsilon )>0$ such that, for every $x \in A_{t_0+\varepsilon ,T_0-\varepsilon }$ and every $y \in B_\rho (x)\cup B_{\rho }(\textsf {Pr} (x))$ it holds

$$\begin{aligned} \frac{\textsf {d} (x,y)^2}{2}=u(x)+u(y)+\sqrt{\frac{u(x)}{T}}\left( \frac{\textsf {d} (\textsf {Pr} (x),y)^2}{2}-T-u(y)\right) . \end{aligned}$$

(A.22)

Then (A.20) follows applying (A.22) first with $x=x' \in A_{t_0+\varepsilon ,T_0-\varepsilon }$ and $y=y' \in B_{\rho '}(x)$ (for some $\rho '<\rho (\varepsilon )$ to be chosen) and then with $x=y'$ and $y=\textsf {Pr} (x')$. The second application of (A.22) is possible because $\textsf {d} (y,\textsf {Pr} (x))=\textsf {d} (\textsf {Pr} (x'),\textsf {Pr} (y'))\le C_{\varepsilon } \textsf {d} (x',y')\le C_{\varepsilon } \rho '< \rho (\varepsilon )$ (recall that $\textsf {Pr}$ is locally Lipschitz) if we choose $\rho '(\varepsilon )$ small enough.

Fix $x \in A_{t_0+\varepsilon ,T_0-\varepsilon }$ and let ${{\bar{t}}}\in {{\,\mathrm{{\mathbb {R}}}\,}}$ be such that $F_{{{\bar{t}}}}(\textsf {Pr} (x))=x$. We will assume that ${{\bar{t}}}>0$, since the other case is exactly the same. Let $r,\delta <\rho (\varepsilon )$, where $\rho (\varepsilon )$ is given by (A.21). Then we also fix $y \in B_\rho (x)\cup B_{\rho }(\textsf {Pr} (x))$. Finally we define the measures $\mu ^r,\nu ^r,\mu _t^r,\eta _s^{t,r}$ as above, for $t \in [0,{{\bar{t}}}]$ and $s \in [0,1]$.

From Theorem 2.29, we have that the curve of measures $[0,1]\ni t \mapsto \mu _t^r$ satisfies the continuity equation with vector field $-\nabla u$. Then combining [50, Theorem 3.5] and [50, Proposition 3.10] we have that the function ${{\,\mathrm{{\mathbb {R}}}\,}}\ni t \mapsto W_2^2(\mu _t^r,\nu ^r)$ is absolutely continuous and

$$\begin{aligned} \frac{{\mathrm d}}{{\mathrm d}t} \frac{1}{2} W_2^2(\mu _t^r,\nu ^r)=- \int \langle \nabla \varphi _t, \nabla u\rangle \, {\mathrm d}\mu _t^r,\quad \text {a.e. } t, \end{aligned}$$

(A.23)

where $\varphi _t$ are suitable (Lipschitz) Kantorovich potentials from $\mu _t^r$ to $\nu .$ Moreover from the second differentiation formula along Wasserstein geodesics [60, Thm. 5.13] we have that, for every t, $[0,1]\ni s \mapsto \int u \,{\mathrm d}\eta _s^{t,r}$ is $C^2[0,1]$ and

$$\begin{aligned} \begin{aligned}&\frac{{\mathrm d}}{{\mathrm d}s}\int u \,{\mathrm d}\eta _s^{t,r}=\frac{1}{s} \int \langle \nabla u, \nabla \psi _s\rangle \,{\mathrm d}\eta _s^{t,r}, \quad \forall s \in [0,1],\\&\frac{{\mathrm d}^2}{{\mathrm d}s^2}\int u \,{\mathrm d}\eta _s^{t,r}= \frac{1}{s^2}\int \langle \textrm{Hess}(u)(\nabla \psi _s,\psi _s) {\mathrm d}\eta _s^{t,r},\quad \forall s \in [0,1],\\ \end{aligned} \end{aligned}$$

(A.24)

where $\psi _s$ are any choice of Kantorovich potentials from $\eta _s^{t,r}$ to $\nu ^r.$ We now fix $t\in [0,{{\bar{t}}}]$ such that the formula in (A.23) holds. Since $\eta _1^{t,r}=\mu _t^r$, we can combine it with the first in (A.24) at $s=1$ to get

$$\begin{aligned} \frac{{\mathrm d}}{{\mathrm d}s}{_{|_{s=1}}} \int u \,{\mathrm d}\eta _s^{t,r}=-\frac{{\mathrm d}}{{\mathrm d}t}{_{|_{t=1}}} \frac{1}{2} W_2^2(\mu _t^r,\nu ^r). \end{aligned}$$

(A.25)

Our goal is now to derive an explicit expression for $\frac{{\mathrm d}}{{\mathrm d}s}{_{|_{s=1}}} \int u \,{\mathrm d}\eta _s^{t,r}$. To do so we use the second in (A.24) and the fact that $\textrm{Hess}(u)=\textsf {id}$ on $\mathop {\textrm{supp}}\nolimits (\eta _s^{t,r})$ (which is ensured by (A.21))

$$\begin{aligned} \frac{{\mathrm d}^2}{{\mathrm d}s^2}\int u \,{\mathrm d}\eta _s^{t,r}=\frac{1}{s^2}\int \langle \textrm{Hess}(u)(\nabla \psi _s,\psi _s) {\mathrm d}\eta _s^{t,r}= \frac{1}{s^2} \int |\nabla \psi _s|^2 \, {\mathrm d}\eta _s^{t,r}\quad \forall s \in [0,1]. \end{aligned}$$

We now apply the metric version of Brenier theorem ( [8], see also [15]) to obtain that $\int |\nabla \psi _s|^2 \, {\mathrm d}\eta _s^{t,r}=W_2^2(\eta _s^{t,r},\nu ^r)$ for every $s \in [0,1]$, which combined with the fact that $\eta _s^{t,r}$ is a $W_2$-geodesic from $\mu _t^{r}$ to $\nu ^r$ gives

$$\begin{aligned} \frac{{\mathrm d}^2}{{\mathrm d}s^2}\int u \,{\mathrm d}\eta _s^{t,r}=W_2^2(\mu _t^{r},\nu ^r), \quad \forall s \in [0,1]. \end{aligned}$$

Since $s\mapsto \int u \,{\mathrm d}\eta _s^{t,r}$ is $C^2[0,1]$, we must have $\int u \,{\mathrm d}\eta _s^{t,r}=a+bs+W_2^2(\mu _t^{r},\nu ^r) \frac{s^2}{2}$ for every $s\in [0,1]$. Substituting $s\in \{0,1\}$ we deduce that

$$\begin{aligned} \int u{\mathrm d}\eta _s^t= & {} \int u {\mathrm d}\nu ^r+\left( \int u {\mathrm d}\mu _t^r-\int u {\mathrm d}\nu ^r-\frac{W_2(\mu _t^r,\nu ^r)^2}{2}\right) s\\{} & {} +W_2^2(\mu _t^r,\nu ^r)\frac{s^2}{2}, \quad \forall s \in [0,1], \end{aligned}$$

using which we can finally compute

$$\begin{aligned} \frac{{\mathrm d}}{{\mathrm d}s}{_{|_{s=1}}} \int u {\mathrm d}\eta _s^t=\int u {\mathrm d}\mu _t^r-\int u {\mathrm d}\nu ^r+\frac{W_2(\mu _r^t,\nu ^r)^2}{2}. \end{aligned}$$

Combining this with (A.25) we obtain that, setting $f(t):=W_2(\mu _t^r,\nu _r)^2/2$, $t \in [0,{{\bar{t}}}]$, f is absolutely continuous and

$$\begin{aligned} f'(t)=\int u {\mathrm d}\nu _r-\int u {\mathrm d}\mu _t^r-f(t), \quad \text {a.e. }t \in (0,{{\bar{t}}}). \end{aligned}$$

Note that the right-hand side is actually a continuous function in t, hence $f\in C^1[0,1].$ Since we know the value of f(0) we can now find f explicitly:

$$\begin{aligned} \frac{W_2(\mu _t^r,\nu _r)^2}{2}= & {} f(t)=e^{-t} \int _0^t e^{s}\left( \int u {\mathrm d}\nu _r-\int u {\mathrm d}\mu _s^r\right) {\mathrm d}s\\{} & {} +\frac{W_2(\mu _0^r,\nu _r)^2}{2} e^{-t}, \quad \forall \, t \in [0,{{\bar{t}}}]. \end{aligned}$$

We now let $r\rightarrow 0^+$ to obtain

$$\begin{aligned} \frac{\textsf {d} (F_t(\textsf {Pr} (x)),y)^2}{2}= & {} e^{-t} \int _0^t e^{s}(u(y)-u(F_s(\textsf {Pr} (x)))){\mathrm d}s\\{} & {} +\frac{\textsf {d} (\textsf {Pr} (x),y)^2}{2}e^{-t},\quad \forall \, t \in [0,{{\bar{t}}}]. \end{aligned}$$

Note that to pass the limit inside the integral sign, we can use that $\left| \int u {\mathrm d}\mu _s^r\right| \le \int |u|\circ F_s {\mathrm d}\mu ^r\le \Vert u\Vert _\infty$ and the dominated convergence theorem. Plugging in $u(F_s(\textsf {Pr} (x)))=e^{-2s}u(\textsf {Pr} (x))=e^{-2s}T$ (see (A.15)) and choosing $t={{\bar{t}}}$, we obtain

$$\begin{aligned} \frac{\textsf {d} (x,y)^2}{2}&=\frac{\textsf {d} (F_{{{\bar{t}}}}(\textsf {Pr} (x)),y)^2}{2}\\&=u(y)e^{-{{\bar{t}}}}(e^{{{\bar{t}}}}-1)+Te^{-{{\bar{t}}}}(e^{-{{\bar{t}}}}-1)+\frac{\textsf {d} (\textsf {Pr} (x),y)^2}{2}e^{-{{\bar{t}}}}=\\&=u(x)+u(y)+e^{-\bar{t}}\left( \frac{\textsf {d} (\textsf {Pr} (x),y)^2}{2}-T-u(y)\right) , \end{aligned}$$

that is precisely (A.22), since ${{\bar{t}}}=\frac{1}{2} \log \frac{T}{u(x)}$. $\square$

1.6 Intrinsic Metric on the Level Set

Definition A.17

We put $\textrm{X}':=S_T.$ For $x',y' \in S_T$ we define $\textsf {d} '(x',y')$ as

$$\begin{aligned} \textsf {d} '(x',y')^2:=\inf \int _0^1 |{\dot{\gamma }}_t|^2 {\mathrm d}t, \end{aligned}$$

where the infimum is taken among all Lipschitz path $\gamma : [0,1]\rightarrow X'\subset X$ joining $x'$ and $y'$ and the metric speed is computed w.r.t. the distance $\textsf {d}$.

Lemma A.18

$$\begin{aligned} \textsf {d} (x,y)\le \textsf {d} '(x,y)\le c \textsf {d} (x,y), \quad \text { for every } x,y \in \textrm{X}', \end{aligned}$$

(A.26)

where $c>0$ is the constant given in Proposition A.13

Proof

The first in (A.26) is immediate from the definition of $\textsf {d} ',$ while the second follows directly from Proposition A.13. $\square$

Corollary A.19

The topology induced by $\textsf {d} '$ on $X'$ is the same as the one induced by the inclusion $X'\subset X.$

We now define the measure ${\mathfrak {m}}'$ on $X'$ as

$$\begin{aligned} {\mathfrak {m}}':={\mathfrak {m}}_{T}, \end{aligned}$$

where ${\mathfrak {m}}_T$ is given in Proposition A.7. Observe that, thanks to Corollary A.19, ${\mathfrak {m}}'$ is a Borel probability measure on $X'.$

A straightforward computation, exploiting (A.13), gives also that

$$\begin{aligned} {\Pr }_* {\mathfrak {m}}|_{A_{a,b}}=c_{a,b}\, {\mathfrak {m}}', \end{aligned}$$

for every $a,b \in {{\,\mathrm{{\mathbb {R}}}\,}}$ such that $t_0\le a<b\le T_0$, where $c_{a,b}=c\int _a^b r^{N/2-1}{\mathrm d}r,$ with c as in Proposition A.7.

The only step that remains to conclude is to link $\textsf {d} '$ to the formula for the distance with separation of variables given by (A.20). To do so we define a new “distance” on $S_T$ by:

$$\begin{aligned} \textsf {D} (x,y):=\arccos \left( 1-\frac{\textsf {d} (x,y)^2}{4T}\right) , \quad \forall \, x,y \in S_T \text { such that } \textsf {d} (x,y)^2<4T, \end{aligned}$$

where we take the range of $\arccos (.)$ to be in $(-\pi /2,\pi /2).$ We point out that $\textsf {D}$ might not be a distance on the whole $S_T$; however, we will show that it is so at least locally. Note also that $c^{-1}\textsf {d} (x,y)\le \textsf {D} (x,y)\le c\textsf {d} (x,y)$ for some constant $c=c(T)>1.$

Our main goal is to prove the following.

Proposition A.20

($\textsf {D}$ locally coincides with $\textsf {d} '$) For every $p \in S_T$ there exists $r>0$ such that

$$\begin{aligned} \frac{\textsf {d} '(x,y)}{\sqrt{2T}}=\textsf {D} (x,y), \quad \forall x,y \in B_r(p)\cap S_T. \end{aligned}$$

To prove the above proposition we need first to show that $\textsf {D}$ is locally an intrinsic distance.

Proposition A.21

($\textsf {D}$ is a locally geodesic distance) For every $p \in S_T$ there exists $\delta >0$ such that

• i) $\textsf {D} (.,.)$ is a distance when restricted to $B_\delta (p)\cap S_T$,

• ii) there exists a constant $\lambda =\lambda (T)<1$ such that for every $x,y \in B_{\lambda \delta }(p)\cap S_T$ there exists a Lipschitz curve $\gamma :[0,1]\rightarrow \subset B_{\delta }(p)\cap S_T$ that is a geodesic for $\textsf {D}$, i.e. such that $\gamma _0=x,\gamma _1=y$ and $\textsf {D} (\gamma _t,\gamma _s)=|t-s|\textsf {D} (\gamma _0,\gamma _1)$ for every $t,s \in [0,1].$

Proof

We preliminarily note that for every $a,b \in [0,\pi /2]$, there exists $\lambda \in [0,1]$ such that

$$\begin{aligned} \sqrt{\lambda ^2+1-2\lambda \cos (a)}+ \sqrt{\lambda ^2+1-2\lambda \cos (b)}=\sqrt{2-2\cos (a+b)}. \end{aligned}$$

(A.27)

An immediate geometric proof of this fact is the following: take three points P, Q, S on the unit circle of centre O so that the length of the arcs PQ,QS and PS is,respectively,a, b and $a+b$, denote by M the intersection between the segments PS and QO, then $\lambda$ is precisely the distance from M to the centre O. Alternatively (A.27) follows also by continuity and the subadditivity of $\sqrt{2-2\cos (.)}$ in $[0,\pi ].$

Fix now $\varepsilon >0$ small so that $T\in (t_0+\varepsilon ,T_0-\varepsilon )$ and fix $p \in S_T.$ Let also $\rho =\rho (\varepsilon )>0$ be the one given by Proposition A.14. By continuity of u and $F_t$ (recall (A.14) and (A.15)) there exist $\delta =\delta (\varepsilon ,p)>0$, ${{\bar{t}}}={{\bar{t}}}(\varepsilon ,p)>0$ both small and such that $F_t(B_\delta (p))\subset B_{\rho /2}(p)$ for all $t \in [0,{{\bar{t}}}].$ In particular

$$\begin{aligned} \text {for every } x,y \in F_t(B_\delta ) \text { and every } t \in [0,{{\bar{t}}}], \text { (A.20) holds.} \end{aligned}$$

(A.28)

Proof of i): We only need to prove that the triangular inequality holds. We argue by contradiction assuming that there exist $x,y,z \in B_\delta (p)\cap S_T$ such that

$$\begin{aligned} \textsf {D} (x,z)>\textsf {D} (x,y)+\textsf {D} (y,z). \end{aligned}$$

We also let $\lambda$ be the one given by (A.27) and corresponding to $a=\textsf {D} (x,y), b=\textsf {D} (y,z)$. Finally let $t\ge 0$ be such that $e^{-2t}T={\lambda ^2 T}$. Note that $\lambda \rightarrow 1$ as $a+b\rightarrow 0$, hence up to decreasing $\delta$ we can assume that $t<{{\bar{t}}}.$ In particular thanks to (A.28) we can apply (A.20) to $x,F_t(y)$ and to $F_t(y),z$, that coupled with (A.27) gives

$$\begin{aligned} \textsf {d} (x,F_t(y))+\textsf {d} (F_t(y),z)= & {} \sqrt{4T-4T\cos \left( \textsf {D} (x,y)+\textsf {D} (y,z)\right) }\\< & {} \sqrt{4T-4T\cos \left( \textsf {D} (x,z)\right) }, \end{aligned}$$

where we have used the strict monotonicity of $\sqrt{1-\cos (.)}$. However, using again (A.20) and the definition of $\textsf {D}$ we see that $\sqrt{4T-4T\cos \left( \textsf {D} (x,z)\right) }=\textsf {d} (x,z)$, which is clearly a contradiction. This concludes the proof of i).

Proof of ii): It is sufficient to show that for every couple of points in $B_{\delta /2}(p)$,there exists a $\textsf {D}$-midpoint, then the conclusion follows from standard arguments (see,e.g. [20, Thm. 2.4.16]) and the fact that $S_T$ is closed (with respect to $\textsf {d}$) and that $\textsf {D}$ is comparable to $\textsf {d}$.

Let $x,y \in S_T\cap B_\delta (p)$. Take z such that $\textsf {d} (z,x)=\textsf {d} (z,y)=\frac{1}{2}\textsf {d} (x,z)$, which exists because $(\textrm{X},\textsf {d} )$ is geodesic. We claim that $\textsf {Pr} (z)$ is a $\textsf {D}$-midpoint for x and y. Observe that from (A.28), (A.20) holds for the couple of points x, y; x, z and z, x. From (A.20) we immediately see that

$$\begin{aligned} \textsf {D} (\textsf {Pr} (z),x)=\textsf {D} (\textsf {Pr} (z),y). \end{aligned}$$

Hence it is sufficient to show that $\textsf {D} (\textsf {Pr} (z),x)\le \frac{\textsf {D} (x,y)}{2}$. To this aim set ${\tilde{l}}:=\textsf {D} (\textsf {Pr} (z),x)$ and $l:=\textsf {d} (x,z)$ and observe that by (A.20)

$$\begin{aligned} \cos (\tilde{l})=\frac{2u(z)+2T-l^2}{4\sqrt{u(z)T}}=\frac{\frac{u(z)}{T}+1-(l/\sqrt{2T})^2}{2\sqrt{\frac{u(z)}{T}}} \end{aligned}$$

Moreover, since $\sqrt{2u}$ is 1-Lipschitz in $B_\delta (p)$ (provided $\delta$ is small enough), we see that $\sqrt{\frac{u(z)}{T}}\in (1-l/\sqrt{2T},1+l/\sqrt{2T})$ and that $l/\sqrt{2T}<1$ if $\delta$ is small enough. Next we observe that for any $a\in (0,1)$ the minimum of the function $\frac{t^2+1-a^2}{2t}$ for $t \in (1-a,1+a)$ is $\sqrt{1-a^2}$, which is achieved at $t=\sqrt{1-a^2}.$ Therefore

$$\begin{aligned} \cos ({\tilde{l}})\ge \sqrt{1-(l/\sqrt{2T})^2} \end{aligned}$$

and plugging in the identity $l^2=\textsf {d} (x,y)^2/4=T-T\cos (\textsf {D} (x,y))$ (obtained from (A.20)) we reach

$$\begin{aligned} \cos ({\tilde{l}})\ge \sqrt{\frac{T+T\cos (\textsf {D} (x,y))}{2T}}=\sqrt{\frac{1+\cos (\textsf {D} (x,y))}{2}}=\cos \left( \frac{\textsf {D} (x,y)}{2}\right) , \end{aligned}$$

which shows that ${\tilde{l}} \le \frac{\textsf {D} (x,y)}{2}$ and concludes the proof of ii). $\square$

We are now ready to prove that $\textsf {D}$ and $\textsf {d} '$ coincides inside small balls.

Proof of Proposition A.20

Fix $p \in S_T$ and let $\delta =\delta (p)>0$, $\lambda =\lambda (T)$ be the ones given by Proposition A.21. We fix $r>0$ small and to be chosen and fix $x,y \in B_{r}(p)\cap S_T.$ Since $\textsf {d} '$ is a geodesic distance there exists a constant speed geodesic from x to y (for $\textsf {d} '$) $\{\gamma _t\}_{t \in [0,1]}\subset S_T$. Moreover from (A.26) we have that $\gamma \subset B_\delta (p)$, provided r is chosen small enough. Note also that, since $\textsf {D}$ is comparable with $\textsf {d}$, $\gamma$ is a Lipschitz curve with respect to $\textsf {D}$ (which is a metric in $B_\delta (p)\cap S_T$). Take $t\in (0,1)$ such that $|{\dot{\gamma }}_t|$ exists, then since $|{\dot{\gamma }}_t|$ coincides with the metric speed computed using $\textsf {d}$, we have

$$\begin{aligned} \textsf {d} '(x,y)= & {} |{\dot{\gamma }}_t|=\lim _{h \rightarrow 0} \frac{\textsf {d} (\gamma _{t+h},\gamma _t)}{h}=\lim _{h \rightarrow 0}\frac{\sqrt{4T}}{h}\sqrt{1-\cos (\textsf {D} (\gamma _{t+h},\gamma _t))}\\= & {} \sqrt{2T}\lim _{h \rightarrow 0}\frac{\textsf {D} (\gamma _{t+h},\gamma _t)}{h}. \end{aligned}$$

In particular the length of the curve $\gamma$ computed with the metric $\textsf {D}$ is $\sqrt{2T}^{-1}\textsf {d} '(x,y)$ and since the length is always greater or equal than the distance between the two endpoints we just proved that

$$\begin{aligned} \textsf {D} (x,y)\le \frac{\textsf {d} '(x,y)}{\sqrt{2T}}. \end{aligned}$$

On the other hand, from ii) in Proposition A.21, if we choose $r<\lambda \delta$, there exists a curve $\gamma : [0,1]\rightarrow S_T\cap B_{\delta }$ that is a geodesic from x to y with respect to $\textsf {D}$. As above we have that $\gamma$ is a Lipschitz curve with respect to $\textsf {d} '$ and that, for any $t\in (0,1)$ such that the metric speed $|{\dot{\gamma }}_t|$ (computed with $\textsf {D}$) exists, we have

$$\begin{aligned} |{\dot{\gamma }}_t|=\frac{1}{\sqrt{2T}} \lim _{h \rightarrow 0} \frac{\textsf {d} (\gamma _{t+h},\gamma _t)}{h}. \end{aligned}$$

In particular the length of the curve $\gamma$ with respect to the metric $\textsf {d} '$ is $\sqrt{2T}\textsf {D} (x,y)$, which shows that

$$\begin{aligned} \textsf {d} '(x,y)\le \sqrt{2T}\textsf {D} (x,y). \end{aligned}$$

$\square$

1.7 Building the Cone and Conclusion

Let us define the $(Z,\textsf {d} _Z,{\mathfrak {m}}_Z)$ as $\left( X',\frac{\textsf {d} '}{\sqrt{2T}},{\mathfrak {m}}'\right)$. We then define the metric measure space $(Y,\textsf {d} _Y,{\mathfrak {m}}_Y)$ as the N-euclidean cone over the metric measure space $(Z,\textsf {d} _Z,{\mathfrak {m}}_Z)$.

For every $0<a<b<\infty$ we set $A^Y_{a,b}=\{ y \in Y \ : \ \textsf {d} _{Y}(y,O_Y)\in (\sqrt{2a},\sqrt{2b})\}=\{(r,z) \in Y: r \in (\sqrt{2a},\sqrt{2b})\}\subset Y.$

We then define the map $T : A^Y_{t_0,T_0} \rightarrow A_{t_0,T_0}$ as

$$\begin{aligned} T((r,z)):=F_{\frac{1}{2} \log \frac{2T}{r^2}}(z), \end{aligned}$$

which is well defined thanks to (A.15), and the map $S : A_{t_0,T_0} \rightarrow A^Y_{t_0,T_0}$ defined as

$$\begin{aligned} S(x):=(\sqrt{2u(x)},\Pr (x)). \end{aligned}$$

(A.29)

It is immediate from the definition that $S(A_{a,b})=A^Y_{a,b}$ and $T(A^Y_{a,b})=A_{a,b}$ for every $t_0\le a<b\le T_0$.

Moreover it is clear from the definition of $\Pr$, (A.15) and the fact that $F_{-t}=F_t^{-1}$, that S and T are one the inverse of the other.

The following result follows from the definitions, Proposition A.14, Proposition A.20 and Proposition A.7.

Proposition A.22

The maps $S: A_{t_0,T_0}\rightarrow A^Y_{t_0,T_0}$ and $T: A^Y_{t_0,T_0}\rightarrow A_{t_0,T_0}$ are measure preserving local isometries.

We can now give the last details which complete the proof of the main theorem.

Proof of Theorem 6.1

We already know from above, that the maps S and T are measure preserving local isometries from $A_{t_0,T_0}$ to $A^Y_{t_0,T_0}$ and viceversa.

Pick now any $t_0',T_0'\in ({\textbf {u}}_0,\infty )$ such that $t_0'<t_0<T<T_0<T_0'$, then we can repeat all the arguments in the previous sections with $t_0',T_0'$ in place of $t_0,T_0$ (but using the same T to define $X'$ as in subsect. A.4) to obtain a map $S':A_{t_0',T_0'}\rightarrow A^Y_{t_0',T_0'}$ that is a local isometry, with an inverse $T'$, which is also a local isometry. The key observation is that $S'$ agrees with S in $A_{t_0,T_0}$. Indeed from (A.29) we deduce that $S'$ on $A_{t_0,T_0}$ depends only on the value of the function $u'$ and the map $\Pr '$ on $A_{t_0,T_0}$. From the construction it is clear that $u'$ agrees with u on $A_{t_0,T_0}$, since both agree with ${\textbf {u}}$ on this set. Therefore we need to show that the two projection maps $\Pr , \Pr ' : A_{t_0,T_0}\rightarrow S_T$ agree. Suppose they do not, i.e. there exists $x \in A_{t_0,T_0}$ such that $\Pr (x)\ne \Pr '(x)$. Recall that $\Pr (x)=F_{\frac{1}{2} \log \frac{u(x)}{T}}(x)$ and that the curve $\gamma ^1_t=F_t(x)$ for $t \in [0,\frac{1}{2} \log \frac{u(x)}{T}]$ is (up to a reparametrization) a minimizing geodesic joining x to $\Pr (x)$ and with values in $A_{t_0,T_0}$, as shown in Proposition A.4. With the same argument we deduce the existence of a geodesic $\gamma ^2$ joining x and $\Pr '(x)$ with values in $A_{t_0,T_0}.$ Moreover from (A.14) we have that $\textsf {d} (x,\Pr (x))=\textsf {d} (x,\Pr '(x))=\sqrt{2}|\sqrt{T}-\sqrt{u(x)}|$, in particular $\gamma ^1,\gamma ^2$ are geodesics with same length. Since S is a local isometry we have that the curves $S(\gamma _t^i)$ are both geodesics in Y with the same length. In particular $\textsf {d} _Y(S(x),S(\Pr (x)))=\textsf {d} _Y(S(x),S(\Pr '(x)))$ which using the expression for S gives

$$\begin{aligned} \sqrt{2}|\sqrt{T}-\sqrt{u(x)}|= & {} \textsf {d} _Y((\sqrt{2u(x)},\Pr (x)),(\sqrt{2T},\Pr (x)))\\= & {} \textsf {d} _Y((\sqrt{2u(x)},\Pr (x)),(\sqrt{2T},{\Pr }'(x))). \end{aligned}$$

However, recalling that $\Pr (x)\ne \Pr '(x)$ and from the definition of $\textsf {d} _Y$, we easily deduce that the rightmost term in the above identity is strictly bigger than $\sqrt{2}|\sqrt{T}-\sqrt{u(x)}|$, which is a contradiction.

We can now send $t_0\rightarrow {\textbf {u}}_0$ and $T_0\rightarrow + \infty$ and obtain a map ${\textbf {S}}: \{{\textbf {u}}>{\textbf {u}}_0\} \rightarrow Y\setminus \overline{B_{\sqrt{2{\textbf {u}}_0}}}(O_Y)$ which is a surjective and measure preserving local isometry. Moreover extending analogously the maps $T: A^Y_{t_0,T_0} \rightarrow A_{t_0,T_0}$, which are the inverses of the maps S, we obtain a map ${\textbf {T}}: Y\setminus \overline{B_{\sqrt{2{\textbf {u}}_0}}}(O_Y)\rightarrow U$, which is the inverse of ${\textbf {S}}$ and a local isometry as well.

Observe now that, since ${\textbf {S}}$ and ${\textbf {T}}$ are a local isometries, they send geodesics to geodesics. This easily implies that

$$\begin{aligned} \textsf {d} (x,\partial \{{\textbf {u}}>{\textbf {u}}_0\})=\textsf {d} _Y({\textbf {S}}(x),B_{\sqrt{2{\textbf {u}}_0}}(O_Y))=\sqrt{2u(x)}-\sqrt{2{\textbf {u}}_0}, \end{aligned}$$

from which (6.1) follows.

We are now in position to apply Proposition 6.3 to obtain that Y is an $\textrm{RCD}(0,N)$ space, which is the unique tangent cone at infinity to $\textrm{X}$. Moreover from the fact that Y is an $\textrm{RCD}(0,N)$ and from (2.18) it follows that $(Z,\textsf {d} _Z,{\mathfrak {m}}_Z)$ is an $\textrm{RCD}(N-2,N-1)$ space satisfying $\text {diam}(Z)\le \pi .$

Suppose now that $\text {diam}(Z)=\pi$, then again from Proposition 6.3 we obtain that $\textrm{X}$ is isomorphic to Y.

The fact that $\textrm{X}$ has Euclidean volume growth was already proved in Corollary A.8.

It remains to prove the first part of ii). Let $r=\sqrt{2{\textbf {u}}_0}$ and $r_Z$ as in the statement. It is enough to show that for every couple of points $y_1,y_2 \in Y$ such that $\textsf {d} _Y(y_i,O_Y)>r_Z$, $i=1,2$, all the geodesics connecting them are contained in $\{\textsf {d} (.,O_Y)>r\}.$ Moreover we can clearly restrict ourselves to consider points $y_1,y_2$ of the form $y_i=(t,z_i)$, with $z_i \in Z$, $i=1,2$ and $t>r_Z.$ For such points we have that

$$\begin{aligned} \textsf {d} _Y(y_1,y_2)=t\sqrt{2-2\cos (\textsf {d} (z_1,z_2))}\le t\sqrt{2-2\cos (\text {diam}(Z))} \end{aligned}$$

Let now $\gamma$ be a geodesic between $y_1$ and $y_2$, then by the triangle inequality

$$\begin{aligned} \textsf {d} (\gamma _t,O_Y)\ge t-\frac{\textsf {d} _Y(y_1,y_2)}{2}>r_Z \left( 1-\sqrt{\frac{1-\cos (\text {diam}(Z))}{2}}\right) =r, \quad \forall t \in [0,1], \end{aligned}$$

where the last identity follows from the definition of $r_Z.$ $\square$

Appendix: Obstacle Problem in $\textsf {RCD}$

1.1 Relative Capacitary Potential for Sets with Cap-fat boundary

This appendix is devoted to the proof of the existence (and uniqueness) of a relative capacitary potential in $\textrm{RCD}$ space and we will mainly focus on boundary regularity. The results contained here are needed only in the proof of Theorem 8.4.

Let us say that we are not proving anything substantially new, since all the results were essentially already present in [19]. Let us also mention that the results concerning boundary regularity and Wiener criterion for harmonic functions originally appeared in [17, 18, 21]. However, the results we needed were spread in many different chapters of [19] and often the language used there (for example for some type of Sobolev spaces) does not coincide with the one we use in this note. For this reason we decided to gather here in a self-contained exposition all the results that we required. Finally let us say that working in the context of $\textrm{RCD}$ will allow to simplify some of the arguments in [19].

Along all this appendix $(\textrm{X},\textsf {d} ,{\mathfrak {m}})$ is an $\textsf {RCD} (K,N)$ m.m.s., $N<+\infty$. Even if we will only apply the result below for $K=0$, we will consider arbitrary K for generality. We only remark that every time a constant will depend on some radius (or diameter of a set), in the case $K=0$ this dependence can be dropped. This is a consequence of the fact that $\mathrm{RCD(0,N)}$ spaces are uniformly doubling.

Our main goal is to prove the following (see below for the definition of relative Capacity).

Theorem B.1

Let $E\subset \textrm{X}$ be an open set and B be a ball such that $E\subset \subset B.$ Suppose also that E has $\textsf {Cap}$-fat boundary. Then there exists $u \in W ^{1,2}_{0}(B)\cap C(B)$, superharmonic in B and harmonic in $B\setminus {{\bar{E}}}$ with $0\le u \le 1$, $u=1$ in ${{\overline{E}}}$ and

$$\begin{aligned} \textsf {Cap} (E,B)=\int _B |\nabla u|^2\,{\mathrm d}{\mathfrak {m}}. \end{aligned}$$

Moreover we have the following continuity estimate: for every $x \in \partial E$ it holds

$$\begin{aligned} 1-u(y)\le C_x\textsf {d} (y,x)^{\alpha _x}, \quad \forall y\in B_{r_x/2}(x)\cap B, \end{aligned}$$

for some positive constants $C_x=C_x(r_x,c_x,K,N,\delta )$ $\alpha _x=\alpha (r_x,c_x,K,N,\delta )>0$, where $r_x,c_x$ are the $\textsf {Cap}$-fatness parameters of x and $\delta >0$ is such that $\textsf {d} (E,B^c)\ge \delta .$

Finally u satisfies the following comparison principle: for every $v \in W ^{1,2}(B)$ superharmonic and such that $v\ge {\chi }_{E}$ ${\mathfrak {m}}$-a.e. in B,  it holds that

$$\begin{aligned} u \le v, \quad {\mathfrak {m}}\text {-a.e. in } B. \end{aligned}$$

Definition B.2

(Variational 2-Capacity) Let $E\subset X$ and $\Omega$ open containing E. We define

$$\begin{aligned}{} & {} \textsf {Cap} (E,\Omega )\\{} & {} \quad =\inf \, \left\{ \int _\Omega |\nabla u|^2\, {\mathrm d}{\mathfrak {m}}\ : \ u \in W ^{1,2}_0(\Omega ) \text { and } u\ge 1\, {\mathfrak {m}}\text {-a.e. in a neighbourhood of } E\right\} \end{aligned}$$

Definition B.3

($\textsf {Cap}$-fat boundary points) We say that an open set E is $\textsf {Cap}$-fat at a point $x \in \partial E$ if there exists $r,c>0$ such that

$$\begin{aligned} \frac{\textsf {Cap} (B_{s}(x)\cap E,B_{2s}(x))}{\textsf {Cap} (B_{s}(x),B_{2s}(x))}\ge c \, , \quad \forall \, s \in (0,r). \end{aligned}$$

Moreover we say that E has (uniformly) $\textsf {Cap}$-fat boundary if it is $\textsf {Cap}$-fat at every point $x \in \partial E$ (with global parameters $c,r>0$).

A geometric condition that is enough to ensure $\textsf {Cap}$-fatness of the boundary is the following interior corkscrew condition. This follows essentially from the doubling property of the measure and the Poincaré inequality (see for example [19, Prop. 6.16]).

Definition B.4

(Corkscrew condition) Let $\lambda \in (0,1)$ and $r>0$. We say that E satisfies the (interior) $(\lambda ,r)$-corkscrew condition at $x \in \partial E$ if for every $s\in (0,r)$ there exists an ball of radius $\lambda s$ contained in $B_s(x)\cap E$.

It is easily verified that any ball of radius $>\delta$ satisfies the (interior) $(1/4,\delta )$-corkscrew condition. Moreover arbitrary unions of sets satisfying the (interior) $(\lambda ,r)$-corkscrew condition still satisfies the (interior) $(\lambda ,r)$-corkscrew condition. In particular union of balls with radius uniformly bounded below satisfies the interior corkscrew condition. It follows that any $\varepsilon$-enlargements of a set, i.e. a set of the form $S^{\varepsilon }=\{x \ : \ \textsf {d} (x,S)<\varepsilon \}$, with $\varepsilon >0$ and S an arbitrary set, satisfies the interior corkscrew condition.

1.2 Preliminaries

We will need the following variational characterization of sub(super)harmonic functions (see [59, Theorem 2.5] and also [48, 53]).

Proposition B.5

Let $\Omega \subset \textrm{X}$ be open. A function $u\in W ^{1,2}(\Omega )$ is superharmonic (resp. subharmonic) in $\Omega$ if and only if

$$\begin{aligned} \int _{\Omega } |\nabla u|^2 {\mathrm d}{\mathfrak {m}}\le \int _{\Omega } |\nabla (u+\varphi )|^2 {\mathrm d}{\mathfrak {m}}, \end{aligned}$$

for every $\varphi \in \textsf {LIP} _{\mathrm c}(\Omega )$ with $\varphi \ge 0$ (resp. $\varphi \le 0$) or equivalently for every $\varphi \in W ^{1,2}_0(\Omega )$ with $\varphi \ge 0$ (resp. $\varphi \le 0$) ${\mathfrak {m}}$-a.e..

Since, as shown in [74], $\textrm{RCD}(K,N)$ spaces support a (1,1) Poincarè inequality and they are also (uniformly) locally doubling, a class of Sobolev embeddings can be shown to hold (see for example [64] and also [19, Chap. 4–5 ]). Therefore a Moser iteration can be performed to obtain the following Harnack inequalities (see for example [65] for the case $K=0$).

Proposition B.6

For every $R_0>0$ there exists two positive constants $C_i=C_i(R_0,K^-,N)$, $i=1,2$, such that the following hold for any $R<R_0$

• 1. if u is subharmonic function in a ball $B_{2R}(x)$, then

  

• 2. if u is a nonnegative superharmonic function in a ball $B_{2R}(x)$, then

  

The above Harnack inequalities imply that harmonic functions have a locally Hölder continuous representative (which is actually locally Lipschitz by [66]) and that superharmonic functions have a lower semicontinuous representative (see for example [19, Theorem 8.22]). From now on we will always tacitly consider these special representatives.

Lastly, we will need the following technical lemma, whose simple proof is omitted.

Lemma B.7

Let $u \in W ^{1,2}_{0}(\Omega )$, then $u^+\in W ^{1,2}_{0}(\Omega )$.

Let $v \in W ^{1,2}(\Omega )$, $u \in W ^{1,2}_{0}(\Omega )$ be such that $0\le v\le u$, then $v\in W ^{1,2}_{0}(\Omega )$

1.3 The Obstacle Problem

Given a ball $B \subset \textrm{X}$ and a (Borel) set $E\subset \subset B$ we consider the following minimization problem

$$\begin{aligned} Obs(E,B):=\inf _{u \in {\mathcal {F}}_{E,B}} \int _B |\nabla u|^2\, {\mathrm d}{\mathfrak {m}}, \end{aligned}$$

(O)

where ${\mathcal {F}}_{E,B}=\{u \in W ^{1,2}_0(B) \ : u \ge {\chi }_E \,\, {\mathfrak {m}}\text {-a.e. in } B\}.$

It is clear that if E is open, then

$$\begin{aligned} Obs(E,B)=\textsf {Cap} (E,B). \end{aligned}$$

The proof of the following result is a straightforward application of the direct method of the calculus of variations, recalling that the embedding $W ^{1,2}_0(B)\hookrightarrow L^2(\textrm{X})$ is compact (see for example [54, Theorem 6.3]) and from the lower semi continuity and (strict) convexity of the Cheeger energy.

Proposition B.8

There exists a unique minimizer to (O). Moreover this minimizer is superharmonic in E.

We now show the two main properties of the minimizers of (O): the first is that u is harmonic far from the obstacle E and the second says that u is essentially the smallest superharmonic function which stays above ${\chi }_E.$

Proposition B.9

Let u be the minimum of (O) for some $E\subset \subset B.$ Then $u=1$ ${\mathfrak {m}}$-a.e. in E and the following hold:

• 1. u is harmonic in $B\setminus {{\bar{E}}}$,

• 2. comparison principle: for every $v \in W ^{1,2}(B)$ superharmonic and such that $v\ge {\chi }_{E}$, ${\mathfrak {m}}$-a.e., it holds that

  $$\begin{aligned} u \le v, \quad {\mathfrak {m}}\text {-a.e. in } B. \end{aligned}$$

Proof

We start by showing that $u\le 1$ ${\mathfrak {m}}$-a.e. in B. Indeed $u\wedge 1 \in {\mathcal {F}}_{E,B}$ and $\int _B |\nabla (u\wedge 1)|^2\le \int _B |\nabla u|^2{\mathrm d}{\mathfrak {m}}$, from which the claim follows. Since $u \ge {\chi }_E,$ ${\mathfrak {m}}$-a.e. it also follows that $u=1$ ${\mathfrak {m}}$-a.e. in E.

We pass to the harmonicity. Fix $\varphi \in \textsf {LIP} _c(B\setminus \bar{U})$. Clearly $(u+\varphi )^+\in {\mathcal {F}}_{E,B}$, therefore

$$\begin{aligned} \int _{B\setminus {{\bar{E}}}}| \nabla (u+\varphi )|^2\, {\mathrm d}{\mathfrak {m}}&\ge \int _{B\setminus {{\bar{E}}}}| \nabla (u+\varphi )^+|^2\, {\mathrm d}{\mathfrak {m}}\\&= \int _{B}| \nabla (u+\varphi )^+|^2\, {\mathrm d}{\mathfrak {m}}-\int _{{{\bar{E}}}} |\nabla u|^2\, {\mathrm d}{\mathfrak {m}}\ge \int _{B\setminus {{\bar{E}}}}|\nabla u|^2 {\mathrm d}{\mathfrak {m}}, \end{aligned}$$

where in the equality step we have used that $\varphi =0$ in ${{\bar{E}}}$ and the locality of the gradient. This and Proposition B.5 prove the claimed harmonicity.

It remains to prove the comparison principle. We start claiming that $(u-v)^+\in W ^{1,2}_0(B)$ and $\min (u,v)\in {\mathcal {F}}_{E,B}$. Indeed we have that $0\le (u-v)^+\le u,$ ${\mathfrak {m}}$-a.e. in B and ${\chi }_E\le \min (u,v)\le u$ ${\mathfrak {m}}$-a.e. in B, therefore the claim follows applying Lemma B.7.

Observe that $\max (u,v)=v+(u-v)^+$, hence from the superharmonicity of v, Proposition B.8, and the locality of the gradient we have

$$\begin{aligned} \int _{\{u>v\}}|\nabla u|^2\, {\mathrm d}{\mathfrak {m}}\ge \int _{\{u>v\}}|\nabla v|^2\, {\mathrm d}{\mathfrak {m}}. \end{aligned}$$

Therefore from the locality of the gradient it follows that

$$\begin{aligned} \int _B |\nabla \min (u,v)|^2\, {\mathrm d}{\mathfrak {m}}\le \int _B |\nabla u|^2\,{\mathrm d}{\mathfrak {m}}, \end{aligned}$$

that combined with $\min (u,v)\in {\mathcal {F}}_{E,B}$ and the uniqueness of the solution to (O) implies that $\min (u,v)=u$ ${\mathfrak {m}}$-a.e. in B. $\square$

We conclude this part with the following technical result.

Lemma B.10

Let u be the minimum of (O) for some $E\subset \subset B.$ Then for every $m\in (0,1]$, the function $\frac{u}{m}\wedge 1$ is the minimum of (O) in B with $E=\{u>m\}$.

Proof

Set $u_m=\frac{u}{m}\wedge 1$ and fix $v \in {\mathcal {F}}_{{\{u> m\}},B}$. Observe that $u_m\ge {\chi }_{\{u> m\}}$ and that $u_m \in W ^{1,2}_{0}(B)$ by Lemma B.7, hence $u_m \in {\mathcal {F}}_{\{u> m\},B}.$ Define the function ${{\bar{u}}}:=u +m(v-u_m)$ and observe that ${\bar{u}}\in W ^{1,2}_{0}(E)$. Moreover ${{\bar{u}}} \ge u \ge 0$ ${\mathfrak {m}}$-a.e. in B and, since from Proposition B.9$u=1$ ${\mathfrak {m}}$-a.e. in E, we also have that ${{\bar{u}}}=1$ ${\mathfrak {m}}$-a.e. in E. Therefore ${{\bar{u}}} \in {\mathcal {F}}_{E,B}.$ This and the fact that ${\bar{u}}=mv$ ${\mathfrak {m}}$-a.e. in $\{u\le m\}$ and ${{\bar{u}}}=u$ ${\mathfrak {m}}$-a.e. in $\{u> m\}$ gives

$$\begin{aligned} \int _B|\nabla v|^2\, {\mathrm d}{\mathfrak {m}}\ge \int _{\{u\le m\}}|\nabla v|^2\, {\mathrm d}{\mathfrak {m}}\ge \frac{1}{m^2} \int _{\{u\le m\}}|\nabla u|^2\, {\mathrm d}{\mathfrak {m}}= \int _B|\nabla u_m|^2\, {\mathrm d}{\mathfrak {m}}. \end{aligned}$$

Since $v \in {\mathcal {F}}_{{\{u>m\}},B}$ was arbitrary we conclude. $\square$

1.4 Proof of Theorem B.1

Proposition B.11

For every $r_0<4 \text {diam}(\textrm{X})$ there exists $C=C(r_0,K,N)>0$ such that the following holds. Let $E\subset B_r(x)$ be open, $r<r_0$ let $2B=B_{2r}(x)$ and let u be the solution to (O) for E in 2B. Then

$$\begin{aligned} u \ge C \frac{\textsf {Cap} (E,2B)}{\textsf {Cap} (B,2B)}, \quad {\mathfrak {m}}\text {-a.e. in } B_r(x). \end{aligned}$$

Proof

Set $B'=B_{\frac{3}{2} r}(x)$ and observe that, since $r_0<4 \text {diam}(\textrm{X})$, $\partial B'\ne \emptyset .$ Define $m :=\max _{\partial B'} u,$ which exists because u is continuous in $\partial B'.$ We claim that $m>0$. Indeed if $m=0$, from the maximum principle (Proposition 2.15) we would have that $u=0$ in the ball 2B (recall that balls are connected), and thus $u=0$ in E, which contradicts the fact that $u=1$ ${\mathfrak {m}}$-a.e. in E with E open. We claim that

$$\begin{aligned} u\le m, \quad \text { in } 2B\setminus B'. \end{aligned}$$

(B.1)

To see this let $m'>m$ and observe that $(u-m')^+\le u^+$ hence by Lemma B.7$(u-m')^+\in W ^{1,2}_{0}(2B)$. Moreover, from the continuity of u and the definition of m,we have that $(u-m')^+=0$ in a neighbourhood of $\partial B'$. These two observations together imply that $(u-m')^+\in W ^{1,2}_0(2B\setminus \bar{B}').$ Observe that $\min (u,m')=u-(u-m')^+$ hence from harmonicity of u we deduce that

$$\begin{aligned} \int _{2B\setminus {{\bar{B}}}'} |\nabla u|^2\, {\mathrm d}\ {\mathfrak {m}}\le \int _{2B\setminus {{\bar{B}}}'}|\nabla \min (u,m') |^2\, {\mathrm d}{\mathfrak {m}}, \end{aligned}$$

which combined with the locality of the gradient gives that $|\nabla u|=0$ ${\mathfrak {m}}$-a.e. in $\{2B\setminus {{\bar{B}}}'\}\cap \{u\ge m'\}$. Therefore again by locality $|\nabla (\max (u,m'))|=0$ ${\mathfrak {m}}$-a.e. in $\{2B\setminus {{\bar{B}}}'\}$ and thus $u \le m'$ in $2B\setminus {{\bar{B}}}'$. Since $m'>m$ was arbitrary (B.1) follows.

Define the functions $u_1=\frac{u}{m}\wedge 1$, $u_2=\frac{u-mu_1}{1-m}$ and observe that $u_1,u_2 \in {\mathcal {F}}_{E,2B}$. In particular for every $t \in (0,1)$ $tu_1+(1-t)u_2\in {\mathcal {F}}_{E,2B}$ and

$$\begin{aligned} \int _{2B}|\nabla u|^2\, {\mathrm d}{\mathfrak {m}}\le t^2I_1\, {\mathrm d}{\mathfrak {m}}+(1-t)^2I_2, \end{aligned}$$

where $I_i=\int _{2B}|\nabla u_i|^2\,{\mathrm d}{\mathfrak {m}}.$ Optimizing in t we obtain that

$$\begin{aligned} \frac{1}{\int _{2B}|\nabla u|^2 \, {\mathrm d}{\mathfrak {m}}}\ge \frac{1}{I_1}+\frac{1}{I_2}. \end{aligned}$$

(B.2)

Observe now that $u_2=0$ in $\{u\le m\}$ and $u_2=(1-m)^{-1}(u-1)$ in $\{u>m\}$ , therefore $|\nabla u_2|={\chi }_{\{u>m\}}|\nabla u|(1-m)^{-1}$ ${\mathfrak {m}}$-a.e. in 2B. In particular $I_2\le (1-m)^{-2}\int _{2B}|\nabla u|^2\, {\mathrm d}{\mathfrak {m}},$ that combined with (B.2) gives

$$\begin{aligned} \textsf {Cap} (E,2B) =\int _{2B}|\nabla u|^2\, {\mathrm d}{\mathfrak {m}}\le (2m-m^2)I_1\le 2mI_1. \end{aligned}$$

This combined with Lemma B.10 gives

$$\begin{aligned} \textsf {Cap} (E,2B)\le & {} 2m\, Obs(\{u> m\},2B)\nonumber \\\le & {} 2m\, Obs(B',2B)=2m\textsf {Cap} (B',2B), \end{aligned}$$

(B.3)

where in the second inequality we have used (B.1).

From the definition of m, there exists a ball $B''=B_{r/2}(y)$ with $y \in \partial B'$ such that $\sup _{B''}u\ge m$. Applying twice the Harnack inequality, recalling that u is harmonic in $B''$ and superharmonic in B, and using the doubling property, we obtain that

$$\begin{aligned} m\le \sup _{B''} u\le C(r_0,K,N) \mathop {\textrm{ess}\,\textrm{inf}}\limits _{B} u. \end{aligned}$$

The conclusion then follows from (B.3) and recalling that thanks to the doubling condition and the Poincaré inequality we have $\textsf {Cap} (B',2B)\le c\textsf {Cap} (B,2B)$, for some constant c depending only on $r_0, K$ and N (see [19, Prop. 6.16]) $\square$

Theorem B.12

For every $r_0<4 \text {diam}(\textrm{X})$ there exists $C=C(r_0,K,N)>0$ such that the following holds. Let $E\subset B_r(x)$ be open, $r<r_0,$ and set $B_i:=B_{2^{1-i}r}(x)$ for $i\in {\mathbb {N}}\cup \{0\}.$ Let u be the capacitary potential for E in $B_0$, then for every $i \ge 1$ it holds that

$$\begin{aligned} 1-u\le \exp \left( -C \sum _{j=1}^i \frac{\textsf {Cap} (E\cap B_j,B_{j-1})}{\textsf {Cap} (B_j,B_{j-1})}\right) , \quad {\mathfrak {m}}\text {-a.e. in } B_i. \end{aligned}$$

Proof

Let $u_i$ be the solution to (O) for $E\cap B_i$ in $B_{i+1}$ (in particular $u=u_1$) and define $a_i:=\frac{\textsf {Cap} (E\cap B_i,B_{i-1})}{\textsf {Cap} (B_i,B_{i-1})}$ for $i\in {\mathbb {N}}$. Proposition B.11 ensures that

$$\begin{aligned} \text {ess}\inf _{B_i} u_i \ge Ca_i\ge 1-e^{-Ca_i}. \end{aligned}$$

(B.4)

Define the functions $v_i\in W ^{1,2}_{0}(B_0)$ inductively as $v_1=u_1$ and $v_i=1-e^{Ca_{i-1}}(1-v_{i-1}),$ for $i \ge 2.$ Observe that, since $u_1$ is superharmonic in $B_0$, $v_i$ is superharmonic in $B_0$ for all $i\ge 1$. We claim that

$$\begin{aligned} v_i \ge 0, \quad {\mathfrak {m}}\text {-a.e. in } B_{i-1}. \end{aligned}$$

(B.5)

We will actually show the stronger estimate $v_i\ge u_i$ ${\mathfrak {m}}$-a.e. in $B_{i-1}$. We proceed by induction. By definition $v_1=u_1$, now suppose that $v_i\ge u_i$ in $B_{i-1}$. It follows from (B.4) that $v_{i+1}\ge 1-e^{Ca_{i}}(1-u_{i})\ge 0$ ${\mathfrak {m}}$-a.e. in $B_{i}.$ Moreover, since $u_1=1$ in $E\cap B_1$, evidently $v_{i+1}=1$ ${\mathfrak {m}}$-a.e. in $E\cap B_{i+1}.$ Combining these two observations we obtain that $v_{i+1}\ge {\chi }_{E\cap B_{i+1}}$ ${\mathfrak {m}}$-a.e. in $B_{i}.$ Recalling that $v_i$ is superharmonic in B (and thus also on $B_i$) we can apply the comparison principle of Proposition B.9 to deduce that $v_{i+1}\ge u_{i+1}$ ${\mathfrak {m}}$-a.e. in $B_i$. This proves the claim. Therefore from (B.5)

$$\begin{aligned} 1-u=1-v_1=e^{-C(a_1+\cdots +a_{i-1})}(1-v_i)\le e^{-C(a_1+\cdots +a_{i-1})}, \quad {\mathfrak {m}}\text {-a.e. in } B_{i-1}, \end{aligned}$$

that concludes the proof. $\square$

Proof of Theorem B.1

Fix $x \in \partial E$ and let c, r be its $\textsf {Cap}$-fat parameters. Let $B':=B_{r_0}(x)$ with $r_0:=(\delta \wedge r)/4$ and let u to be the solution to (O) for E in $2B'$. Fix $y\in B'\setminus {{\bar{E}}}$ with $\textsf {d} (y,x)<r/2$. There exists $i \in {\mathbb {N}}_0$ such that $2^{-i-1}r_0<\textsf {d} (x,y)<2^{-i}r_0<r$. Therefore from Theorem B.12 and the continuity of u in $B'\setminus {{\bar{E}}}$ we have

$$\begin{aligned} 1-u(y)\le & {} (e^{-i})^{c\cdot C}\le (2^{-i})^{c\cdot C}\le (2r_0^{-1})^{c\cdot C} \textsf {d} (x,y)^{c\cdot C}\nonumber \\= & {} (8\delta \wedge r)^{-c\cdot C}\textsf {d} (x,y)^{c\cdot C} . \end{aligned}$$

(B.6)

Let now ${{\bar{u}}}$ to be the solution of (O) for E in B (where B is as in the hypotheses). Fix $x \in \partial E$ and let u as in the previous part of the proof. Since $B_{2r_0}(x)\subset B$, from the comparison principle of Proposition B.9, we have that ${{\bar{u}}}\ge u$ ${\mathfrak {m}}$-a.e. in $B_{r_0}(x)$ and since both ${{\bar{u}}}$ and u are continuous in $B_{r_0}(x)\setminus {{\bar{E}}}$ we have that (B.6) holds for ${{\bar{u}}}$ and every $y\in B\setminus {{\bar{E}}}$ with $\textsf {d} (y,x)<r/2$. This proves that $\lim _{B_{r_0}(x)\setminus {{\bar{E}}} \ni y \rightarrow x}{{\bar{u}}}(y)=1$ for every $x\in \partial E$ (recall that ${{\bar{u}}}\le 1$) and since ${{\bar{u}}}$ is also lower semicontinuous we deduce that ${{\bar{u}}}=1$ in ${{\bar{E}}}.$

The comparison principle is already contained in Proposition B.9. $\square$