Sobolev-to-Lipschitz property on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathsf {QCD}}$$\end{document}QCD-spaces and applications

We prove the Sobolev-to-Lipschitz property for metric measure spaces satisfying the quasi curvature-dimension condition recently introduced in Milman (Commun Pure Appl Math, to appear). We provide several applications to properties of the corresponding heat semigroup. In particular, under the additional assumption of infinitesimal Hilbertianity, we show the Varadhan short-time asymptotics for the heat semigroup with respect to the distance, and prove the irreducibility of the heat semigroup. These results apply in particular to large classes of (ideal) sub-Riemannian manifolds.


Introduction
In [20], Milman introduced the notion of quasi curvature-dimension condition QCD for a metric measure space (X , d, m), simultaneously generalizing Lott-Villani-Sturm's curvature-dimension condition CD(K , N ) with finite N [19,28,29], and the measure contraction property MCP [22,29]. As discussed in [20], the class of QCD spaces notably includes large families of (ideal) sub-Riemann manifolds, thus aiming to provide a unified perspective of (non-smooth) Riemannian, Finsler, and sub-Riemannian geometry.
In this note, we collect some metric-measure properties of a metric measure space (X , d, m) satisfying the QCD condition. As a main result, Theorem 3.6, we show the Sobolev-to-Lipschitz property, see (SL) below.
In light of recent developments in metric analysis, the property (SL) has turned out to be significant in relating differentiable and metric measure structures. For instance, • under (SL), the Bakry-Émery (synthetic Ricci) curvature (lower) bound BE is equivalent to the Riemannian curvature-dimension condition RCD, Ambrosio-Gigli-Savaré [3]. The statement is sharp, in the sense that BE without (SL) does not imply RCD, Honda [15]; • together with (SL), the BE condition implies the L ∞ -to-Lipschitz regularization of the heat semigroup, Ambrosio-Gigli-Savaré [2] (in the sub-Riemannian setting see Stefani [26]); • together with a Rademacher-type property for (X , d, m), see (Rad) below, (SL) implies the coincidence of the intrinsic distance and the given distance d, and also implies the integral Varadhan short-time asymptotic for the heat semigroup in a variety of settings (see [9,Thm. 4.25]), and furthermore, for the space of configurations (i.e., locally finite integer-valued point measures) over X , see [8,Thm. 6.10].
Apart from (SL), QCD spaces satisfy the local volume doubling, the Rademachertype property (Rad), and the local versions (Rad) loc and (SL) loc of (Rad) and (SL) after [9], see Sect. 4. When (X , d, m) is additionally infinitesimally Hilbertian, as an application of the Sobolev-to-Lipschitz property, we obtain: • the coincidence of the distance d with the intrinsic distance d Ch of the Cheeger energy Ch of (X , d, m), Theorem 4.8; • the integral Varadhan short-time asymptotic for the heat semigroup with respect to the Hausdorff distance induced by d, Theorem 4.5; • in the compact case, and assuming as well the measure contraction property MCP, the pointwise Varadhan short-time asymptotic for the heat semigroup with respect to the Hausdorff distance induced by d, Corollary 4.10; • the irreducibility of the Dirichlet form Ch, Corollary 4.6.
For these results, we make full use of the fundamental relations between Dirichlet forms and metric measure spaces developed in [9].
Regarding the irreducibility, we note that the same proof of Corollary 4.6 applies as well to RCD(K , ∞) spaces with infinite volume measure, which seems not explicitly proved in the existing literature; see Remark 4.7 for a more detailed discussion.
Our results on QCD spaces may be specialized to ideal sub-Riemannian manifolds satisfying the quasi curvature-dimension condition, such as: (ideal generalized Htype) Carnot groups, Heisenberg groups, corank-1 Carnot groups, the Grushin plane, and several H -type foliations, Sasakian and 3-Sasakian manifolds.

Milman's quasi curvature-dimension-condition
By a metric measure space (X , d, m) we shall always mean a complete and separable metric space (X , d), endowed with a Borel measure m finite on d-bounded sets and with full topological support. In order to rule out trivial cases, we assume that m is atomless, which makes X uncountable. We say that (X , d) is proper if all closed balls are compact. We let C c (X ), resp. C 0 (X ), C b (X ), be the space of continuous compactly supported, resp. continuous vanishing at infinity, continuous bounded, functions on X .
We denote by P(X ), resp. P c (X ), P m (X ), the space of all Borel probability measures on (X , d), resp. (additionally) compactly supported, (additionally) absolutely continuous w.r.t. m, and by the infimum running over all couplings π ∈ P(X ×2 ) of (μ 0 , μ 1 ). We denote by Opt(μ 0 , μ 1 ) the set of minimizers in (2.1), always non-empty. Set I := [0, 1]. We write Geo(X , d) for the space of all constant-speed geodesics in (X , d) parametrized on I , itself a complete separable metric space when endowed with the supremum distance d ∞ induced by d. By Lisini's superposition principle [18,Thm. 4] (cf. [1, Thm. 2.10]), every W 2 -absolutely continuous curve (μ t ) t∈I may be lifted to a dynamical plan π ∈ P(C(I ; X )) satisfying (ev t ) π = μ t for every t ∈ I , where ev t : γ → γ t is the evaluation map at time t. Furthermore, a curve (μ t ) t is a W 2 -geodesic if and only if π is concentrated on Geo(X , d) and in which case we say that π is an optimal dynamical plan connecting μ 0 and μ 1 . We write OptGeo(μ 0 , μ 1 ) for the set of all such plans. Whenever (Y , τ ) is a Polish space, the narrow topology τ n on the space P(Y ) of Borel probability measures on Y is defined as the topology induced by duality with continuous bounded functions on Y . Since (Y , τ ) is Polish, (P(Y ), τ n ) is Polish as well, τ n is characterized by the convergence of sequences, and a sequence (μ n ) n converges narrowly if and only if We collect here for further reference the following standard fact. Proposition 2.1 (Stability of dynamical optimality). For i = 0, 1 let μ n i n ⊂ P 2 (X ) and fix π n ∈ OptGeo(μ n 0 , μ n 1 ). If π n narrowly converges to π ∈ P(C(I ; X )), then π is concentrated on Geo(X , d), and π ∈ OptGeo (ev 0 ) π, (ev 1 ) π .
We write ρ t for the Radon-Nikodým density of μ t w.r.t. m.
Let us now collect some properties of Monge spaces.

Remark 2.3 Every geodesic
is endowed with the product of the narrow topologies, and P(Geo(X , d)) is endowed with the narrow topology.
Proof Firstly, note that is well-defined by Definition 2.2(a). Secondly, recall that the space of geodesics is a compact subset of Geo(X , d). As a consequence, P(Geo(K 0 , K 1 )) is narrowly compact metrizable, and it suffices to show the continuity of along sequences.
The following is a consequence of the sole interpolation property [16,Dfn. 4.2]. Proof Since (X , d, m) has the strong interpolation property (Dfn. 2.2(c)), it has in particular the interpolation property, and is therefore strongly non-degenerate [16,Dfn. 4.4] by [16,Lem. 4.5]. In particular, it is non-degenerate, i.e., for every Borel A ⊂ X with mA > 0 and every x ∈ X it holds that mA t,x > 0 for every t ∈ (0, 1), where Now, argue by contradiction that there exist x 0 ∈ X and r 0 > 0 with mS r 0 (x 0 ) > 0. On the one hand, since m is σ -finite we can find r ∈ (0, r 0 ) with mS r (x 0 ) = 0. On the other hand, since S r (x 0 ) = S r 0 (x 0 ) t,x 0 for t := r /r 0 ∈ (0, 1), the non-degeneracy implies mS r (x 0 ) > 0, a contradiction.
, and the first assertion follows.
We further say that (X , d, m) satisfies the regular quasi curvature-dimension condition QCD reg (Q, K , N ) if it satisfies QCD(Q, K , N ) for some Q, K , N as above and additionally the measure contraction property MCP(K , N ) for some K ∈ R and N ∈ (1, ∞).
In the following, we omit the indices Q, K , and N whenever not relevant. We refer to [20] for a thorough discussion of examples of spaces satisfying the QCD condition. We stress that they include all CD spaces (for the choice Q = 1, see [20]), and various classes of sub-Riemannian manifolds satisfying MCP (see [20,Prop. 2.4]).
Since the right-hand side of (2.4) depends on Q only by its linear dependence on the constant Q −1/N , the proof of the following result is readily adapted from the one of the analogous assertion under the curvature-dimension condition CD in [29,Thm. 2.3].
For fixed x 0 ∈ X and for every r > 0, set As customary, further define the model volume coefficient In particular, this implies that the assertion of Corollary 2.5 does not follow from (2.6) in the obvious way, which makes the Corollary non-void.

Proof of Lemma 2.7 Let
Following verbatim the proof of [29, Prop. 2.1] yields the following Q-weighted version of the generalized Brunn-Minkowski inequality: where We apply (2.7) to A 0 := B ε (x 0 ) and Thus, by (2.7), Since m does not charge spheres by Corollary 2.5, we may rewrite the above inequality as hence, making explicit the definition of the distortion coefficients,

The Sobolev-to-Lipschitz property on QCD-spaces
Let (X , d, m) be a metric measure space. We denote by L( f ) the global Lipschitz constant of a Lipschitz function f : X → R, and by Lip(d), resp. Lip bs (d), the space of all Lipschitz functions, resp. (additionally) with bounded support, on (X , d).
We briefly recall the definition of Cheeger energy of a metric measure space. For a function f ∈ Lip(d), define the slope of f at x by where, conventionally, inf ∅ := +∞. We denote the domain of Ch d,m by We recall that a metric measure space (X , d, m) is called infinitesimally Hilbertian if Ch d,m is quadratic. Introduced by Gigli in [12,Dfn. 4.19], this notion has ever since proven to be a key tool in the study of non-smooth metric measure spaces. When (X , d, m) is infinitesimally Hilbertian, Ch d,m is a strongly local Dirichlet form having square-field operator |D f | 2 w , where |D f | w is called the weak minimal upper gradient, satisfying-by construction-the Rademacher-type property: where Lip bs denotes the space of Lipschitz functions with bounded support.
The following property has been considered in a variety of non-smooth settings, including e.g. configuration spaces [25], or general metric measure spaces [11,Dfn. 4.9].

Remark 3.2
The Sobolev-to-Lipschitz property is more commonly phrased without the requirement that f ∈ L ∞ (m). In fact, this is equivalent to (SL).
Proof Let f ∈ W 1,2 with |D f | w ≤ 1, and set f r := −r ∨ f ∧ r for each r > 0. By locality of |D · | w we have that |D f r | w ≤ 1 m-a.e. for every r , hence f r has a Lipschitz m-modificationf r with L(f r ) ≤ 1, by (SL). Sincef r is continuous and m has full support, we have thatf r ≡f s everywhere on x ∈ X : |f s (x)| ≤ r for every s ≥ r . We conclude letting r → ∞. Concerning our terminology, we ought to stress that the Rademacher-type property (Rad) we defined here does not entail any (strong) differentiability nor any Gâteaux (i.e. directional) differentiability of the function involved. Indeed, even phrasing any such concept of (Gâteaux) differentiability on general metric measure spaces would be highly non-trivial. While immediate on metric measure spaces (as discussed above), the property (Rad) is non-trivial on general Dirichlet spaces when |D · | is replaced by the square field operator (or even by the energy measure) of a strongly local Dirichlet form, see e.g. [9,17,27]. In this case, proofs of strong notions of differentiability of Lipschitz functions are available in specific smooth and non-smooth settings, which invariably rely on a combination of Gâteaux differentiability along 'sufficiently many' directions together with the uniform bound on such derivatives, provided by (Rad); see e.g. [21] for Euclidean spaces, [6,10] for Wiener and Banach spaces, [25] for configuration spaces, [7] for Wasserstein spaces, etc.
In order to discuss the Sobolev-to-Lipschitz property on QCD spaces, we recall the following definition by Gigli and Han [13]. Firstly, recall from [2, Dfn. 5.1] that a dynamical plan π ∈ P(C(I ; X )) is a test plan if • it is concentrated on the family AC 2 (I ; X ) of 2-absolutely continuous curves; • it has finite 2-energy, i.e. the right-hand side of (2.2) is finite; • it has bounded compression, viz. (ev t ) π ≤ Cm for some contant C > 0 independent of t ∈ I . For i = 0, 1 and every x i ∈ there exists ε := ε(x 0 , x 1 ) > 0 such that for each ε i ∈ (0, ε] there exists a test plan π ε 0 ,ε 1 ∈ P(C(I , X )) such that: (a) the map is weakly Borel measurable, viz.
In order to show Definition 3.4(c), note that, since π ε 0 ,ε 1 ∈ OptGeo(μ ε 0 , μ ε 1 ), then, in the constant speed parametrization, which proves the assertion. It remains to show the measurability assertion in Definition 3.4(a). To this end, it suffices to note that, letting E i : Corollary 2.4 shows that the map is narrowly/narrowly continuous (hence Borel) on the image of (E 0 , E 1 ). Thus, it suffices to show that E i is Borel for i = 0, 1. We is continuous on [0, ε). The continuity at r = 0 holds since m has no atoms. For r , s > 0, we have that The second term vanishes as r → s by continuity of the measure m. As for the first term, it suffices to show that r → mB r (x i ) is continuous for r ∈ [0, ε). This follows from the Portmanteau Theorem, since all balls in X are continuity sets for m by the first assertion in Corollary 2.5.
If x 0 = x 1 , then the requirements in Definition 3.4(b)-(c) hold trivially, and Definition 3.4(a) holds as in the case x 0 = x 1 discussed above.
Combining Corollary 2.8 with Proposition 3.5 and Theorem 3.6, we conclude the Sobolev-to-Lipschitz property for QCD spaces.

Remark 3.8
It would not be difficult to show that the original proof of (SL) for CD(K , N ) spaces [11, p. 48] can be adapted as well to the case of QCD (Q, K , N ) spaces. The proof in [11] relies on an argument in [23] providing suitable test plans connecting the approximating measures μ ε i in (3.2).
The proof we presented above makes instead use of the more refined notion of measured-length space in [13]. Whereas slightly more involved, this proof makes more explicit the relation between the inequality (2.4) defining the QCD, and the upper bound on the compression for the aforementioned test plans.

Properties of RQCD spaces
In this section, we prove several applications of (SL) under the additional assumption that the Cheeger energy is quadratic. In the following, we omit the indices Q, K , and N whenever not relevant. Note that, when (X , d, m) is an RQCD space, the quadratic form induced by the Cheeger energy of (X , d, m) by polarization is a Dirichlet form, again denoted by Ch d,m and still called the Cheger energy of (X , d, m).
Recall that a Dirichlet form (E, F) on a locally compact Polish space Proof The regularity follows directly from the uniform density of Lip bs (d) in C 0 (X ) and from the norm density of Lip bs in W 1,2 . The strong locality is then a standard consequence of the locality of the weak upper gradient |D · | w .

Example 4.3 (sub-Riemannian manifolds). Let (M, H) be a sub-Riemannian manifold with smooth non-holonomic distribution H on T M.
On the one hand, by the Chow-Rashevskii Theorem, endowing M with its Carnot-Carathéodory distance d cc and with a smooth measure m turns it into a proper metric measure space, thus admitting a Cheeger energy Ch d cc ,m . On the other hand, the sub-Laplacian L induced by the distribution H generates a regular Dirichlet form (E, F) , see e.g. [5, p. 191]. In fact, for a sub-Riemannian manifold (M, d cc , m) as above, the Dirichlet form (E, F) coincides with the Cheeger energy (Ch d cc ,m , W 1,2 ) of (M, d cc , m), viz. Proof of (4.1) On the one hand, by [2, Thm. 6.2], the square field |D f | w of f ∈ W 1,2 coincides with the minimal 2-weak upper gradient of f . On the other hand, by [14,Thm. 11.7], the square field ( f ) of f ∈ F ∩ C(X ) coincides as well with the minimal 2-weak upper gradient of f . As a consequence, E( f ) = Ch d cc ,m ( f ) for every f ∈ W 1,2 ∩ C(X ) as well, and the conclusion follows since W 1,2 ∩ C 0 (X ) is a core for (Ch d cc ,m , W 1,2 ) by regularity of the latter (Prop. 4.2) and since F ∩ C 0 (X ) is a core for (E, F) by definition.
For a regular Dirichlet form (E, F) on L 2 (m), the local domain F loc of E is defined as the space of all functions f ∈ L 0 (m) so that, for each relatively compact open G ⊂ X there exists Let us define the following local versions of the Rademacher-type and Sobolev-to-Lipschitz properties: (SL) loc Again by construction, (Rad) loc holds on every metric measure space.

Proposition 4.4 Every RQCD space satisfies (SL) loc .
Proof Since we already have (Rad), by construction of Ch d,m , and (SL), by Corollary 3.7, in order to localize both properties it suffices to show the existence of good Sobolev cut-off functions, similarly to the proof of Theorem 3.9 in [3]. For every n ∈ N and fixed x 0 ∈ X set θ n : x → n ∧ 2n − d(x, x 0 ) + . Since supp θ n is bounded for every n ∈ N, we conclude by (Rad) that θ n ∈ W 1,2 ∩ C 0 (X ) and |Dθ n | w ≤ 1 m-a.e. for every n ∈ N.
Fix now f ∈ W 1,2 loc ∩ L ∞ (m) with |D f | w ≤ 1 m-a.e. Without loss of generality, up to an additive constant, we may assume that f ≥ 0. By the local property of |D · | w , we have that by assumption on f and properties of θ n , whence θ n ∧ f n ∈ W 1,2 . From (SL) we conclude that θ n ∧ f has a non-relabeled d-Lipschitz m-representative. Analogously, θ n+1 ∧ f ∈ W 1,2 and, for all n ≥ f L ∞ , we have that θ n+1 ∧ f ≡ θ n ∧ f ≡ f m-a.e. on B n (x 0 ). Since m has full topological support, we conclude that the respective d-Lipschitz m-representatives coincide everywhere on B n (x 0 ). As a consequence, (θ n ∧ f ) n is a consistent family of d-Lipschitz functions coinciding with f m-a.e. on B n (x 0 ) for all (sufficiently large) n ∈ N. The conclusion readily follows letting n → ∞.
Denote by P t : L 2 (m) → L 2 (m) the strongly continuous contraction semigroup associated to (Ch d,m , W 1,2 ), and, for open sets A, B ⊂ X , set We now give the first application of (SL).  Denote again by P t : L ∞ (m) → L ∞ (m) the extension of P t : We say that the space (X , d, m) is irreducible if every invariant set is either m-negligible or m-conegligible.
As a corollary of Theorem 4.5, we obtain the irreducibility of (X , d, m).   In the next corollary, we denote by p t the density w.r.t. m of the heat-kernel measure of the heat semigroup P t . (4.6) Proof As a consequence of the MCP condition implicit in the notation for RQCD reg , we have the validity of the local weak 2-Poincaré inequality, see [29,Cor. 6 Finally let us briefly discuss the Lipschitz regularization property of the semigroup (P t ) t≥0 . Let c : [0, +∞) → (0, +∞) be a measurable function so that locally uniformly bounded away from 0 and infinity. Following [26,Definition 3.4], we say that an RQCD space satisfies BE w (c, ∞) if for all f ∈ W 1,2 and t ≥ 0,

Conflict of interest
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