Distributional Valued Ricci Bounds for Metric Measure Spaces, Singular Time Changes, and Gradient Estimates for Neumann Heat Flows

We will study metric measure spaces $(X,d,m)$ beyond the scope of spaces with synthetic lower Ricci bounds. In particular, we introduce distributional valued lower Ricci bounds BE$_1(\kappa,\infty)$ $\bullet$ for which we prove the equivalence with sharp gradient estimates, $\bullet$ the class of which will be preserved under time changes with arbitrary $\psi\in{\mathrm Lip}_b(X)$, and $\bullet$ which are satisfied for the Neumann Laplacian on arbitrary semi-convex subsets $Y\subset X$. In the latter case, the distributional valued Ricci bound will be given by the signed measure $\kappa= k\, m_Y + \ell\,\sigma_{\partial Y}$ where $k$ denotes a variable synthetic lower bound for the Ricci curvature of $X$ and $\ell$ denotes a lower bound for the"curvature of the boundary"of $Y$, defined in purely metric terms. We also present a new localization argument which allows us to pass on the RCD property to arbitrary open subsets of RCD spaces. And we introduce new synthetic notions for boundary curvature, second fundamental form, and boundary measure for subsets of RCD spaces.

In the latter case, the distributional valued Ricci bound will be given by the signed measure κ = k m Y + σ ∂Y where k denotes a variable synthetic lower bound for the Ricci curvature of X and denotes a variable lower bound for the "curvature of the boundary" of Y , defined in purely metric terms. We introduce new synthetic notions for boundary curvature, second fundamental form, and boundary measure for subsets of RCD spaces. The distributional valued Ricci bound BE 1 (κ, ∞) with κ as above will imply a gradient estimate for the Neumann heat flow (∇P Y t ) t≥0 on Y of the type Here (P Y x , B Y t ) x∈Y,t≥0 denotes reflected Brownian motion on Y and (L ∂Y t ) t≥0 , the local time of ∂Y , is defined via Revuz correspondence as the positive continuous additive functional associated with the surface measue σ ∂Y .
Note that • for non-convex Y , no estimate of type (1) can hold true without taking into account the curvature of the boundary; • even for convex Y , estimate (1) will improve upon all previous estimates which ignore the curvature of the boundary.
For instance, for the Neumann heat flow on the unit ball of R n , the right hand side of (1) will decay as C 0 e −C 1 t for large t whereas ignoring will lead to bounds of order C 0 .
We also present a new powerful localization argument which allows us to pass on the RCD property to arbitrary open subsets of RCD spaces.
Outline. Besides this Introduction, the paper has five sections, each of them of independent interest. Let us briefly summarize them.
In Section 2, we define and analyze metric measure spaces with Ricci curvature bounded from below by distributions. Our BE 1 (κ, ∞) condition for κ ∈ W −1,∞ (X) is the first formulation of a synthetic Ricci bound with distributional valued κ which leads to a sharp gradient estimate.
Section 3 is devoted to the study of mm-spaces with variable Ricci bounds. The main result will be the proof of the equivalence of the Eulerian curvature-dimension condition (or "Bakry-Émery condition") BE 2 (k, N ) and the Lagrangian curvature-dimension condition (or "Lott-Sturm-Villani condition") CD(k, N ) -as well as four other related conditions. This provides an extension of the seminal paper [EKS15] towards variable k (instead of constant K) and of the recent paper [BHS19] towards finite N (instead of N = ∞).
In Section 4 we present two extensions of our recent work [HS19] with B. Han on transformation of the curvature-dimension condition under time-change, both of fundamental importance. Firstly, we prove that for φ ∈ Lip loc (X) ∩ D loc (∆), time change with weight 1 φ leads to a mm-space (X , d , m ) with X = {φ > 0} which satisfies RCD(k , N ) for suitable k , N . This is of major general interest since it allows for localization within the class of RCD-spaces. Secondly, we prove that for arbitrary ψ ∈ Lip b (X), time change with weight e ψ leads to a mm-space with distributional valued Ricci bound κ given in terms of the distributional valued Laplacian ∆ψ. This will be a crucial ingredient in our strategy for the proof of the gradient estimate in the final Section 6.
In Section 5 we extend the existence result and the contraction estimate for gradient flows for semiconvex functions from [Stu18a] to the setting of locally semiconvex functions. The contraction estimate for the flow will be in terms of the variable lower bound for the local semiconvexity of the potential. And we will prove the fundamental Convexification Theorem which allows us to transform the metric of a mm-space (X, d, m) in such a way that a given semiconvex subset Y ⊂ X will become locally geodesically convex w.r.t. the new metric d . Moreover, in a purely metric manner, we introduce the notion of variable lower bound for the curvature of the boundary. In the Riemannian setting, such a bound will be equivalent to a lower bound for the fundamental form of the boundary.
The paper reaches its climax in Section 6 with the proof of the gradient estimates for the Neumann heat flow on not necessarily convex subsets Y ⊂ X. The proof of these gradient estimates is quite involved. It builds on results from all other sections of the paper.
• Given a semi-convex subset Y of an RCD(k, N )-space (X, d, m), to get started, we perform a time-change with weight e ψ in order to make Y locally geodesically convex in (X, d ) := (X, e ψ d). The choice ψ = ( − ) V with V = ±d(., ∂Y ), any > 0, and being a lower bound for the curvature of ∂Y will do the job, see Section 5.
• Under the assumption that ψ ∈ D loc (∆), the transformation formula for time changes provides a RCD(k , N )-condition for the time-changed space (X, d , m ), Section 4.
• Together with the local geodesical convexity of Y this implies that also the restricted space (Y, d Y , m Y ) satisfies the RCD(k , N )-condition. Making use of the equivalence of Eulerian and Lagrangian characterizations of curvature-dimension conditions, we conclude the BE 2 (k , N )-condition for (Y, d Y m Y ), Section 3.
• To end up with (Y, d Y m Y ) requires a "time re-change", i.e. another time change, now with weight e −ψ . In general, however, ψ will not be in the domain of the Neumann Laplacian ∆ Y . Ricci bounds under time re-change thus have to be formulated as BE 1 (κ, ∞)-condition for some κ ∈ W −1,∞ (X) in terms of the distributional Laplacian ∆ Y ψ, Section 4.
• The BE 1 (κ, ∞)-condition will imply the gradient estimate ∇P Y t f ≤ P κ t ∇f for the Neumann heat flow (∇P Y t ) t≥0 on Y in terms of a suitable semigroup (P κ t ) t≥0 , Section 2.
This "taming semigroup" (P κ t ) t≥0 will be represented in terms of the Brownian motion on Y by means of the Feynman-Kac formula involving the integral t 0 k(B s )ds (taking into account the effects of the Ricci curvature in Y ), and the integral t 0 (B s )dL s (taking into account the effects of the curvature of ∂Y ).
Basic concepts and notations. Throughout this paper, (X, d, m) will be an arbitrary metric measure space, that is, d is a complete separable metric on X inducing the topology of X and m is a Borel measure which is finite on bounded sets. Moreover, we assume that (X, d, m) is infinitesimally Hilbertian.
To simplify notation, we often will write L p (X) or L p (m) or just L p instead of L p (X, m) and, similarly, Lip(X) instead of Lip(X, d). The space of Lipschitz functions with bounded support on X will be denoted by Lip bs (X) whereas as usual Lip b (X) denotes the space of bounded Lipschitz functions. The number Lipf will denote the Lipschitz constant of f .
Indeed, the bilinear form E is a quasi-regular Dirichlet form on L 2 (X, m), [Sav14]. Its generator ∆ is the "Laplacian" on the mm-space (X, d, m). The associated semigroup ("heat semigroup") (e ∆ t ) t≥0 on L 2 (X, m) will extend to a positivity preserving, m-symmetric, bounded semigroup (P t ) t≥0 on each L p (X, m) with P t L p (X,m)→L p (X,m) ≤ 1 for each p ∈ [1, ∞], strongly continuous on L p (X, m) if p < ∞. Quasi-regularity of E implies that each f ∈ W 1,2 (X) admits a quasi continuous versionf (and two such versions coincide q.e. on X). Thus in particular, for each f ∈ p∈[1,∞] L p (X, m) and t > 0, there exists a quasicontinuous versionP t f of P t f (uniquely determined q.e.). The m-reversible, continuous Markov process P x , B t x∈X,t≥0 (with life time ζ) associated with E is called "Brownian motion" on X. It is uniquely characterized by the fact that (The factor 2 arises from the fact that by standard convention, the generator of the Brownian motion is 1 2 ∆ whereas the generator of the heat semigroup in our setting is ∆.) The author would like to thank Mathias Braun, Zhen-Qing Chen, Matthias Erbar, Nicola Gigli, and Tapio Rajala for fruitful discussions and valuable contributions.
Financial support by the European Union through the ERC-AdG "RicciBounds" and by the DFG through the Excellence Cluster "Hausdorff Center for Mathematics" and through the Collaborative Research Center 1060 is gratefully acknowledged.

W −1,∞ -valued Ricci bounds
The goal of this section is to define and analyze metric measure spaces with Ricci curvature bounded from below by distributions. In particular, we will give a meaning to this extended notion of synthetic lower Ricci bounds and -most importantly -we will prove that these Ricci bounds lead to sharp estimates for the gradient of the heat flow. These results are of independent interest.
In the context of this paper, they are of particular importance since in Section 6 we will prove that the Ricci curvature of a semiconvex subset Y of an RCD-space (X, d, m) is bounded from below by the W −1,∞ (X)-distribution where k denotes a variable synthetic lower bound for the Ricci curvature of X and denotes a lower bound for the "curvature of the boundary" of Y while σ ∂Y denotes the "surface measure" on ∂Y . In particular, the Ricci curvature of Y will be bounded from below by a function if and only if Y is convex.

Taming Semigroup
In the sequel, we also need certain Banach spaces, denoted by W 1,1+ (X), W 1,∞ (X) and W −1,∞ (X). We will define these spaces tailor made for the purpose of this paper. Our concept will be based on the 2-minimal weak upper gradient |Df |.
Definition 2.1. We put denotes the truncation of f at levels ±n, and i) The precise definition of these spaces will not be so relevant for us. What we need are the following properties: W 1,1+ (X) contains all squares of functions from W 1,2 (X); W 1,∞ (X) includes Lip b (X); Γ extends to a continuous bilinear map is a Banach space. If the mm-space (X, d, m) satisfies some RCD(K, ∞)condition, according to the Sobolev-to-Lipschitz property, the space W 1,∞ (X) will coincide with the space Lip b (X) and the space W 1,∞ * (X) will coincide with the space Lip(X). iii) W 1,1+ (X) is a normed space but in general not complete. For instance, the functions f j (r) = r ∨ (1/j), j ∈ N, on X = [−1, 1] will constitute a Cauchy sequence in W 1,1+ (X) but their L 1 -limit f ∞ (r) = √ r is not contained in W 1,1+ (X). For Riemannian (X, d, m), the completion of W 1,1+ (X) will coincide with W 1,1 (X).
For general (X, d, m), the definition of W 1,1 (X) is quite sophisticated and allows for ambiguity, see e.g. [ADM14]. For a detailed study of the spaces W 1,p (X) for p ∈ (1, ∞), see [GH16].
Indeed, by locality of the minimal weak upper gradient, the family |Df [n] |, n ∈ N, is consistent in the sense that |Df [n] | = |Df [j] | m-a.e. on the set |f | ≤ min(n, j) for each n, j ∈ N. Hence, |Df [n] |, n ∈ N, is a Cauchy sequence in L 1 (X) and therefore, it admits a unique limit in L 1 (X), denoted by |Df |.
Proof. It suffices to prove the claim for f = g. Given f ∈ W 1,2 (X), put h = f 2 . Then obviously h ∈ L 1 (X). Moreover, h [n 2 ] ∈ W 1,2 (X) for each n since h [n 2 ] ≤ n f and This proves the claim.
Example 2.8. Let (X, d, m) be the standard 1-dimensional mm-space with X = R and let x n i for n ∈ N and i = 1, . . . , 2 n−1 be the centers of the intervals of length 3 −n in the mid-third construction of the Cantor set. Choose ϕ(x) = ( 1 2 −|x|) + or, more sophisticated, choose a nonnegative function ϕ ∈ C 2 (R) with {ϕ > 0} = (−1/2, 1/2) and put Then Φ ∈ W 1,∞ (R), more precisely, But ∆Φ is not a signed measure. Indeed, with C : as j → ∞ and similarly t s ∆Φ j dx → ∞ for each 0 ≤ s < t ≤ 1 for which either s is a right endpoint of one of the mid-third intervals or that t is a left endpoint of one of these intervals.
Proof of Proposition and Remark (i). The lower boundedness and more generally the form smallness easily follow from In particular, this implies that E κ (f ) ≥ −(C + C 2 ) f 2 for all f and thus by spectral calculus f, P κ t f 2 ≥ e −(C+C 2 ) t f 2 2 . According to the first Beurling-Deny criterion, the semigroup (P κ t ) t≥0 is positivity preserving if and only if Of particular interest will be to analyze the semigroup (P κ t ) t≥0 in the case where κ = −∆ψ for some ψ ∈ Lip(X). Recall that this semigroup is well understood in the "regular" case where ψ ∈ D(∆) ∩ L ∞ (X). Indeed, then For general ψ, however, this Feynman-Kac formula a priori does not make sense. We will have to find an appropriate replacement of it.
Proposition 2.11. (i) Given ψ ∈ Lip(X), put κ = −∆ψ. Then the closed, lower bounded bilinear form E κ on L 2 (X) with domain W 1,2 (X) is given by (For the last expression here we used the fact that E extends to a continuous bilinear form W 1,1+ (X) × Lip(X) → R, Lemma 2.4, and that f g ∈ W 1,1+ (X) for f, g ∈ W 1,2 (X), Lemma 2.3.) The strongly continuous, positivity preserving semigroup on L 2 (X) associated to it satisfies P κ t L 2 →L 2 ≤ e (Lip ψ) 2 t . (ii) Putm := e −2ψ m. Then the unitary transformation (= Hilbert space isomorphism) Φ : L 2 (X, m) → L 2 (X,m), f →f = e ψ f maps the quadratic form E κ , densely defined on L 2 (X, m), onto the quadratic form densely defined on L 2 (X,m) and bounded from below by −(Lipψ) 2 g 2 L 2 . (Since ψ is bounded on bounded sets, Γ coincides with the Gamma-operator for the metric measure space (X, d,m) and Φ maps W 1,2 (X, d, m) bijectively onto W 1,2 (X, d,m). Moreover,Ê is just a perturbation of the canonical energy on (X, d,m) by a bounded zeroth order term.) (iii) The semigroup (P κ t ) t≥0 on L 2 (X,m) associated with the the quadratic formÊ κ is related to the semigroup (P κ t ) t≥0 on L 2 (X, m) viâ Furthermore, it can be represented in terms of the heat semigroup (P t ) t≥0 on L 2 (X,m) by the Feynman-Kac formula with potential −Γ(ψ). Since the latter is a bounded function, the semigroup is bounded on each L p (X,m) with . This allows us to conclude that for each ψ ∈ Lip b (X), the original semigroup satisfies Proof. The norm estimate in (i) follows from the fact that and the estimate in (iii) from The rest is straightforward.
A more explicit representation for the semigroup (P κ t ) t≥0 will be possible by extending the Fukushima decomposition which in turn is an extension of the famous Ito decomposition. In the Euclidean case with smooth ψ, the latter states that This indicates a way how to replace the expression 1 2 t 0 ∆ψ(B s )ds appearing in (4) by expressions which only involve first (and zero) order derivatives of ψ.
(i) For each ψ ∈ Lip bs (X) there exists a unique martingale additive functional M ψ and a unique continuous additive functional which is of zero quadratic variation N ψ such that ψ(B t ) = ψ(B 0 ) + M ψ t + N ψ t (∀t ∈ [0, ζ)) P x -a.s. for q.e. x ∈ X (7) (ii) For each ψ ∈ Lip(X) there exists a unique local martingale additive functional M ψ such that for each z ∈ X, where M ψn denotes the martingale additive functional associated with the function ψ n = χ n · ψ ∈ Lip bs (X) according to part (i) and where χ n (.) = [1 − d(B n (z), .)] + for n ∈ N.
(iii) The quadratic variation of M ψ is given by for any choice of a Borel version of the function Γ(ψ) ∈ L ∞ (X, m).
For the defining properties of "martingale additive functionals" and of "continuous additive functionals of zero quadratic variation" (as well as for the relevant equivalence relations that underlie the uniqueness statement) we refer to the monograph [FOT11].
Proof. Assertion (i) is one of the key results in [FOT11]. Indeed, is is proven there as Theorem 5.2.2 for general quasi continuous ψ ∈ D(E) and it is extended in Theorem 5.5.1 by localization to a more general class which contains Lip(X). Also assertion (iii) for ψ ∈ Lip bs (X) is a standard result, see [FOT11], Theorem 5.2.5. Let us briefly discuss its extension to general ψ ∈ Lip(X).
Given ψ ∈ Lip(X) and z ∈ X, we define ψ n = χ n · ψ with cut-off functions χ n as above and stopping times τ λ := inf{t ≥ 0 : B t ∈ B λ (z)} for λ ∈ N. Then we put It follows that for q.e. x, the family (M λ t ) λ∈N is an L 2 -bounded martingale w.r.t. P x with Thus the limit M t := lim λ→∞ M λ t exists and is a martingale w.r.t. P x for q.e. x ∈ X.
with M ψ t as defined in the previous Lemma, part (ii). Then with (P κ t/2 ) t≥0 as defined in Proposition 2.9, for each f ∈ p∈[1,∞] L p (X, m), Proof. In order to derive the representation formula (9) with the additive functional N given by (8), we will replace the (non-existing) Feynman-Kac transformation with potential 1 2 ∆ψ by (a) a Girsanov transformation with drift −Γ(ψ, .) (b) together with a Feynman-Kac transformation with potential 1 2 Γ(ψ) (c) followed by a Doob transformation with function e ψ .
Each of these transformations provides a multiplicative factor in the representation of the semigroup which together amount to Let us perform these transformations first under the additional assumption that ψ ∈ Lip bs (X) in which case all details can be found in the paper [CZ02] since in this case ψ ∈ D(E) and µ ψ = Γ(ψ) m is a Kato class measure (indeed, it is a measure with bounded density).
(a) In the first step, we pass from the metric measure space (X, d, m) to the metric measure space (X, d,m) withm = e −2ψ m or, equivalently, we pass from the Dirichlet form E(f ) = Γ(f )dm on L 2 (X, m) to the Dirichlet formÊ(f ) = Γ(f )dm on L 2 (X,m). This amounts to pass from the heat semigroup (P t ) t≥0 to the semigroup (P t ) t≥0 given by Girsanov's formulâ with M ψ being the martingale additive functional as introduced in the previous Lemma.
(b) In the second step, we pass from the Dirichlet formÊ(f ) = Γ(f )dm on L 2 (X,m) to the Dirichlet formÊ κ (f ) = Γ(f ) − Γ(ψ) · f 2 dm on L 2 (X,m). This amounts to pass from the semigroup (P t ) t≥0 to the semigroup (P κ t ) t≥0 given by Feynman-Kac's formulâ (c) In the final step, we pass from the Dirichlet formÊ κ (f ) = Γ(f )−Γ(ψ)·f 2 dm on L 2 (X,m) to the Dirichlet form E κ (f ) = Γ(f ) + Γ(f 2 , ψ) dm on L 2 (X, m), see previous Proposition. This amounts to pass from the semigroup (P κ t ) t≥0 to the semigroup (P κ t ) t≥0 given by Doob's formula for m-a.e. x ∈ X. This proves the claim in the case ψ ∈ Lip bs (X). For general ψ ∈ Lip(X), we choose cut-off functions χ n , n ∈ N, as in the previous Lemma and put ψ n = χ n · ψ and κ n = −∆ψ n . Then by the previous argumentation, x ∈ X for each n ∈ N. It remain to prove as n → ∞ as well as To prove the latter, let us first restrict to f ∈ L p (X, m) for some p ∈ (1, ∞]. Then For every q, s ∈ (1, ∞) with 1 p + 1 q + 1 s = 1, by Hölder's inequality for m-a.e. x ∈ X. For the last estimate we used the fact that e −sM ψ x ∈ X as n → ∞. Analogously, we can estimate for m-a.e. x ∈ X as n → ∞. This proves (11) in the case f ∈ L p (X, m) for some p ∈ (1, ∞]. The claim for f ∈ L 1 (X, m) follows by a simple truncation argument and monotone convergence. To prove (10), it suffices to consider the case f ∈ L 2 (X, m). The assertion for f ∈ L p , p = 2, follows by density of L 2 ∩L p in L p and by boundedness of P κ t (as well as boundedness of P κn t , uniformly in n) on L p , cf. previous Proposition. To deduce (10) in the case p = 2, Duhamel's formula allows us to derive for all s ∈ [0, t]. Hence, t 0 X Γ P κ s g + P κ s g 2 dm ds ≤ C t · g 2 L 2 and thus uniformly in n. This proves g P κ t f − P κn t f ) dm → 0 as n → ∞ which is the claim. Corollary 2.14. (i) Given φ ∈ L ∞ (X) and ψ ∈ Lip b (X), the semigroup (P κ t ) t≥0 for κ = φ − ∆ψ is given by for each f ∈ p∈[1,∞] L p (X, m) and for m-a.e. x ∈ X.
(ii) Even more, lettingP κ t f denote a quasi continuous version of P κ t f , theñ holds true for q.e. x ∈ X.
Proof. (i) Define a semigroup (Q t ) t≥0 by the right hand side of (12), i.e. Q t/2 f (x) := . We will prove that it is associated with the quadratic form E κ . Put κ 0 = −∆ψ. From the probabilistic representations of Q t f and P κ 0 f , we easily deduce (Note that E κ is obtained from E κ 0 by perturbation with a bounded potential. Hence, both Q t and P κ 0 are strongly continuous semigroups on L 2 .) Therefore, for all f ∈ D(E κ ) and thus Q t f = P κ t f for all t and all f . (ii) follows by standard arguments for quasi-regular Dirichlet forms.

Bochner Inequality BE 1 (κ, ∞) and Gradient Estimate
For n ∈ N, we define the Hilbert space V n (X) : Of particular interest are the spaces Definition 2.16. Given κ ∈ W −1,∞ (X), we say that the Bochner inequality or Barky- for all f ∈ V 3 (X) and all nonnegative φ ∈ V 1 (X).
Note that equally well also the first integral in the above estimate can be restricted to the set {Γ(f ) > 0}.
as an absolutely continuous function on (0, t). Then where we have put φ s = P κ s φ and f s = P t−s f . The crucial point now is that the semigroup (P t ) t≥0 preserves the class V 3 (X) where f is chosen from, and that the the semigroup (P κ t ) t≥0 preserves the cone of nonnegative elements in V 1 (X) where φ is chosen from. Assuming BE 1 (κ, ∞) and applying it to f s and φ s in the place of f and φ implies that in the last integral the expression in [...] is nonpositive. This proves the claimed gradient estimate (14) for f ∈ V 3 (X). The assertion for general f then follows by approximation. b) Now let us assume that the gradient estimate holds true. Let us first derive the assertion on domain inclusion which in our formulation is requested for BE 1 (κ, ∞). Using the gradient estimate, we conclude that By spectral calculus, it is well known that for t → 0 the LHS of (15) converges monotonically to E κ (Γ(f ) 1/2 ) and To deal with the RHS of (15), first observe that Γ( Thus for t → 0, the RHS of (15) converges as follows Combining the asymptotic results for both sides of (15), we obtain This yields the domain assertion requested for BE 1 (κ, ∞). c) To derive the requested functional inequality for BE 1 (κ, ∞), we integrate the gradient estimate for f ∈ V 3 (X) w.r.t. φ δ dm and subtract φ δ Γ(f ) 1/2 dm on both sides. Here for arbitrary φ ∈ V 1 (X) and δ > 0, we put This yields In the limit t → 0, this gives One easily verifies that for δ → 0 this converges to Localization. In the sequel, we will also localize various statements. To do this, will require some care since in general X will not be locally compact. Given a space G(X) of functions (or of m-equivalence classes of functions) on X we denote by G sloc (X) the set of all functions g (or m-equivalence classes of functions, resp.) on X "which semilocally lie in G(X)" in the sense that for each bounded open subset B ⊂ X there exists a g B ∈ G(X) such that g = g B on B (or m-a.e. on B, resp.). This way, e.g. we define the spaces W 1,1+ sloc (X). We denote by W −1,∞ sloc (X) the set of all κ such that for all bounded open sets B ⊂ X there exist κ B ∈ W −1,∞ (X) which are consistent in the sense that φ, κ B W 1,1+ ,W −1,∞ = φ, κ B W 1,1+ ,W −1,∞ for all such B, B and for all φ ∈ W 1,1+ (X) with support in B ∩ B . In this case, we say that κ = κ B on B and put for all f ∈ V 3 sloc (X) and all nonnegative φ ∈ V 1 bs (X). The domain inclusion requested for BE 1 (κ, ∞) obviously implies the inclusion for the localized domains: Applying (13) to f B and φ implies (16) for f and φ.
"⇐": Given nonnegative φ ∈ V 1 (X), by partition of unity we can find countably many nonnegative φ n ∈ V 1 bs (X) such that φ = n φ n . Applying (16) to each φ n and the given f ∈ V 3 (X), and adding up these estimates yields (13) for the given φ and f .
3 Equivalence of BE 2 (k, N ) and CD(k, N ) Throughout this section, (X, d, m) will be a metric measure space, N ∈ [1, ∞) a number, and k : X → R will be a bounded, lower semicontinuous function. We present the Eulerian and the Lagrangian characterizations of "Ricci curvature at x bounded from below by k(x) and dimension bounded from above by N " and prove their equivalence.
Without loss of generality, we will assume that (X, d, m) satisfies the Riemannian curvature-dimension condition RCD(K, ∞) for some constant K ∈ R. Among others, this will guarantee that the space is infinitesimally Hilbertian, that the volume of balls does not grow faster than e Cr 2 , and that functions with bounded gradients have Lipschitz continuous versions ("Sobolev-to-Lipschitz property"). Moreover, it implies that In the sequel, as usual P 2 (X) will denote the space of probability measures µ on X with d 2 (., z) dµ < ∞ equipped with the L 2 -Kantorovich-Wasserstein distance W 2 . We say that a measure π ∈ P(Geo(X)) represents the W 2 -geodesic (µ r ) r∈[0,1] if µ r = (e r ) π for r ∈ [0, 1]. Here Geo(X) denotes the set of d-geodesics γ : [0, 1] → X and e r : Geo(X) → X, t → γ r denotes the projection or evaluation operator.
Thanks to our a priori assumption RCD(K, ∞), there exists a heat kernel defines a strongly continuous, non expanding semigroup in L p (X, m) for each p ∈ [1, ∞). For p = 2, this actually can be defined (or re-interpreted) as the gradient flow for the energy E in L 2 (X, m). Moreover, defines a semigroup on P 2 (X). The latter can be equivalently regarded as the gradient flow for the Boltzmann entropy Ent in the Wasserstein space P 2 (X). Here and in the sequel, Ent(µ) := u log u dm if µ = u m and Ent(µ) := ∞ if µ is not absolutely continuous w.r.t m.
Proof. Firstly note that the addendum follows from the uniqueness of the measure representing a W 2 -geodesics connecting a given pair of measures of finite entropy [GRS16].
and its representing measure π ∈ P(Geo(X)), apply (i) to µ s , µ s+δ in the place of µ 0 , µ 1 to deduce for a.e. s ∈ (0, 1) (where the LHS has to be understood as the distributional second derivative of a semiconvex function). Integrating this w.r.t. the measure g(s, r) ds on (0, 1) yields (iii).
(iii) ⇒ (ii): Given a W 2 -geodesic (µ r ) r∈[0,1] and its representing measure π ∈ P(Geo(X)), we add up the estimate (iii) together with its counterpart with 1 − r in the place of r to obtain Dividing by r and then letting r → 0 yields (ii).
Definition 3.3. We say that (X, d, m) satisfies the 2-Bochner inequality or 2-Bakry-Emery estimate with variable curvature bound k and dimension bound N , briefly BE 2 (k, N ), if Our first main results states that also for variable curvature bound k and finite N , the Eulerian and Lagrangian approaches to synthetic lower Ricci bounds are equivalent. For constant k, this has been proven in joint work [EKS15] of the author with Erbar and Kuwada. For variable k and N = ∞, it has been proven in joint work [BHS19] with Braun and Habermann. In particular, in the latter work a formulation of the transport estimate has been given in terms of the following quantity: where the infimum is taken over all coupled pairs of Brownian motions (B 1 s ) 0≤s≤2t and (B 2 s ) 0≤s≤2λt with initial distributions µ and ν, resp.
Proof. (i) ⇒ (ii): Using the equivalent CD(k, N ) formulation from the previous Lemma 3.2(iii) and passing there to the limit r → 0, one easily sees that (i) implies Thus the claim (ii) is an immediate consequence of the fact that [AG + 15], Thm. 6.3.
(ii) ⇒ (i). We follow the standard path of argumentation. Given two probability measures µ 0 , µ 1 of finite entropy, let (µ r ) r∈[0,1] , represented by π, denote the unique W 2 geodesic connecting them and note that the standing CD(K, ∞)-assumption implies that the µ r 's also have finite entropy. Consider the heat flow starting in µ r with observation point µ 0 as well as with observation point µ 1 . Note that This proves the CD(k, N )-estimate.
(i) ⇒ (iii): For t > 0 let φ t , ψ t denote a W 2 -optimal pair of Kantorovich potentials for the transport from P * t µ 0 = u t m to P * t µ 1 = v t m. Then following [AG + 15], Thm. 6.3 and Thm. 6.5, by Kantorovich duality for a.e. t > 0 where (µ t r ) r∈[0,1] , represented by π t , denotes the W 2 -geodesic connecting µ t 0 := P * t µ 0 and µ t 1 := P * t µ 1 . Together with (i) this implies d + dt for a.e. t and thus 1 s Passing to the limit s → 0 finally yields the claim (iii) since Ent(P * t µ 0 ) as well as Ent(P * t µ 0 ) are continuous in t and since π t weakly converges to π and k is lower semicontinuous.
(iii) loc ⇒ (i). This implication can be proven with the "trapezial argument" from [KS18]. Note that thanks to the local-to-global property of the CD(k, N )-condition, for this implication it suffices that the differential transport inequality holds locally, that is, for each z ∈ X there exist δ > 0 such that DTE(k, N ) holds true for all µ 0 , µ 1 which are supported in B δ (z).
Given µ 0 , µ 1 of finite entropy and ∈ (0, 1 2 ) as well as t > 0, note that Estimating the first and third term on the RHS by means of EVI(K, ∞) (which is true as consequence of our standing a priori assumption) and the second term by means of In the limit → 0, this gives the CD(k, N )-inequality (i).
(v) ⇒ (iii) loc : This follows similar as in [BHS19] from a localization argument.
(v) ⇔ (vi): The proof follows the standard line of argumentation via differentiating the forward-backward evolution. More precisely, for bounded, nonnegative φ ∈ D(E) and fixed t > 0, put a(s) := φ P 2k s Γ P t−s f dm. This function is absolutely continuous in s with for a.e. s ∈ [0, t] where we have put φ s := P 2k s φ and f s : Varying over φ, this yields (v). Conversely, assuming (v) yields for all bounded nonnegative φ ∈ D(E) and all sufficiently regular f .
From Lemma 1.3 we easily deduce where π λ t denotes the measure on P(Geo(X)) representing the geodesic (µ t r ) r∈[0,1] from P * t µ to P * λt ν. Adding up these inequalities and using Young's inequality we obtain Introducing the function and denoting by q λ t the W 2 -optimal coupling of P * t µ and P * λt ν, the latter estimate can be rephrased as Slightly extending the scope of [BHS19], we define where the infimum is taken over all coupled pairs of Brownian motions (B 1 s ) 0≤s≤2t and (B 2 s ) 0≤s≤2λt with initial distributions µ and ν, resp. Following the proof of Theorem 4.6 in [BHS19], from (18) we conclude To proceed, we now will make use of a subtle localization argument. Recall from [AGS08] or from [BHS19], Lemma 2.1, that we may assume without restriction that k is continuous (even Lipschitz continuous). Given z ∈ X and > 0, choose δ > 0 and K z such that K z ≤ k ≤ K z + in B 2δ (z). Then following the proof of Theorem 4.2 in [Stu18b], we conclude that for each p < 2, there exists T > 0 such that for all t, λ > 0 with t(1 + λ) ≤ T and for all µ, ν with support in B δ (z) Combining this with the previous estimate (19) yields This is very similar to the estimates (4.1) and (4.2) in [EKS15] which are used there as key ingredients for deriving gradient estimates -the main difference being now that p < 2 on the LHS of (21). Given a bounded Lipschitz function f on X and putting ,y) , following the proof of Theorem 4.3 in [EKS15], instead of their estimate (4.7) we now obtain with µ = δ x , ν = δ y and q > 2 being the dual exponent for p Choosing a sequence (y n ) n∈N such that y n → x and |∇P t f (x)| = lim sup n as in [EKS15], and putting λ n = 1 + α d(x, y n ) leads to Optimizing w.r.t. α and passing to the limit R → 0 then yields 2t Integrating this estimate w.r.t. φ(x) dm(x) with a bounded nonnegative φ ∈ Lip(X) supported in B δ (z) and then differentiating it at t = 0 yields the following perturbed, local form of the Bochner inequality . Covering the whole space by balls B δ/2 (z) of the above type, we can find a partition of unity consisting of functions φ of the above type which allows us to deduce the perturbed Bochner inequality on all of X, cf. the analogous argumentation in [BHS19]. Since > 0 and q > 2 were arbitrary we finally obtain the Bochner inequality in the following form: for all f ∈ D(∆) ∩ Lip(X) with ∆f ∈ D(E) and all bounded nonnegative φ ∈ Lip(X). Following the argumentation in the proof of Lemma 2.19, one verifies the equivalence to the Bochner inequality BE 2 (k, N ) in its standard form. This proves the claim.

Time-Change and Localization
This section is devoted to prove the transformation formula for the curvature-dimension condition under time-change. In contrast to our previous work with Han [HS19], we now also will consider weight functions e ψ where ψ is no longer in D loc (∆) but merely in Lip b (X). This will result in W −1,∞ -valued Ricci bounds involving the distributional Laplacian ∆ψ. Moreover, we deal with weight functions 1 φ = e ψ where the local Lipschitz function φ may degenerate in the sense that φ = 0 is admitted. Choosing φ to be an appropriate cut-off function, this allows us to "localize" the RCD-condition: we can restrict a given RCD-space (X, d, m) to any subset X := {φ > 0} ⊂ X.

Curvature-Dimension Condition under Time-Change
Assume that a metric measure space (X, d, m) is given which satisfies RCD(k, N ) for some lower bounded Borel function k on X and some finite number N ∈ [1, ∞). Given ii) Assume that k is lower semicontinuous, Then (X, d , m ) satisfies RCD(k , N ) for any lower bounded, lower semicontinuous function k on X and any number N ∈ (N, ∞] such that Remark 4.3. i) Let us re-formulate the previous Theorem in terms of φ := e −ψ . That is, assume that φ ∈ Lip loc (X) ∩ D loc (∆) is given with φ > 0 on X and define a metric and a measure on X by d := 1 φ d and m := 1 φ 2 m, resp. Observe that for ψ := − log φ, 2∆ψ). Thus the metric measure space (X, d , m ) satisfies RCD(k , N ) for any lower bounded, lower semicontinuous functions k on X and any number N ∈ (N, ∞] such that ii) Another remarkable way of re-formulating the previous result is in terms of Then estimate (25) can be re-written as Recall that in the case N * = 2, estimate (27) states k ≤ e −2ψ k − ∆ψ m -a.e. on X .

Localization
We are now going to relax the positivity assumption on φ, admitting φ also to vanish on subsets of X.
Theorem 4.4. (i) Given φ ∈ Lip loc (X) such that the set {φ > 0} is connected. Define a metric measure space (X , d , m ) by Then d is a complete separable metric on X and m is a locally finite Borel measure on (X , d ). The metric measure space (X , d , m ) is infinitesimally Hilbertian. The sets Lip loc (X , d) and Lip loc (X , d ) coincide. For f ∈ W 1,2 loc (X , d, m) = W 1,2 loc (X , d , m ), the minimal weak upper gradients |Df | and |D f | w.r.t. the mm-spaces (X, d, m) and (X , d , m ), resp., coincide.
(ii) Assume in addition that φ ∈ Lip loc (X) ∩ D loc (∆). Then the metric measure space To see the latter, let points x ∈ X and z ∈ ∂X be given and let (γ t ) t∈[0,1] be any absolutely continuous curve in (X, d) with γ 0 = x and γ 1 = z. Without restriction, we may assume that γ has constant speed. Let L = Lipφ. Then It is easy to check that the RCD(k , N ) condition has the local-to-global property, see [Stu15] for the proof in the case N = ∞. Therefore, it suffices to prove that X is covered by open sets B such that the Boltzmann entropy is (k , N )-convex along W 2geodesics with endpoints supported in B. We are going to verify this for B := B r (z) := {y ∈ X : d (y, z) < r} with

Given such a ball
Corollary 4.6. Assume that a metric measure space (X, d, m) is given which satisfies RCD(K, N ) for some finite numbers K, N ∈ R.
Then for any open subsets D 0 , D 1 ⊂ X with D 0 ⊂ D 1 , there exists a metric measure space (X , d , m ) satisfying RCD(K , N ) for some finite numbers K , N ∈ R such that Proof. Previous Theorem, part (ii), plus existence of cut-off functions with bounded Laplacian according to previous Lemma.

Singular Time Change
In the previous paragraph we dealt with an extension of Theorem 4.2 where ψ = − log φ is allowed to degenerate in the sense that it attains the value ∞ on closed subsets of arbitrary seize. Now we will deal with the extension towards ψ which are no longer in D loc (∆) but merely in Lip b (X). Assume that a metric measure space (X, d, m) is given which satisfies BE 2 (k, N ) for some lower bounded function k on X and some finite number N ∈ [1, ∞).
Theorem 4.7. Given ψ ∈ Lip b (X), the "time-changed" metric measure space (X, d , m ) with d := e ψ d and m := e 2ψ m satisfies BE 1 (κ, ∞) for Proof. i) Without restriction, assume that k is bounded. Choose K ∈ R + with k ≥ −K on X. Given ψ ∈ Lip b (X) ∩ D(E), we will approximate it by ψ n := P 1/n ψ. Thanks to the BE 2 (−K, N ) assumption, the heat semigroup preserves the class of Lipschitz functions and of course it always maps L 2 into D(∆). Thus ψ n ∈ D(∆) with sup n Lip ψ n < ∞ and with ψ n → ψ ∞ := ψ in D(E) and in L ∞ . To see the latter, observe that by Ito's formula, ii) For n ∈ N ∪ {∞}, consider the mm-space (X, d n , m n ) with d n := e ψn d and m n := e 2ψn m. Let (P n t ) t≥0 and (P x , B n t ) x∈X,t≥0 denote the heat semigroup and the Brownian motion, resp., associated with it. Note that B n t = B τn(t) with τ n (t) being the inverse to σ n (t) := t 0 e 2ψn (B s )ds.
Also note that due to the BE 1 (−K, ∞)-property, the lifetime of the original Brownian motion is infinite and thus also the lifetime of any of the time-changed Brownian motions. Moreover, as as n → ∞, obviously τ n (t) → τ (t) (even uniformly in ω), thus B n t → B ∞ t a.s. and for every bounded continuous function f on X and every x ∈ X.
iii) According to Theorem 4.2, for finite n, the the mm-space (X, d n , m n ) satisfies the BE 1 (k n , ∞)-condition with In particular, the associated heat semigroup (P n t ) t≥0 satisfies with the Feynman-Kac semigroup P kn t given in terms of the Brownian motion on (X, d n , m n ) by P kn t g(x) = E x e − t 0 kn(B n 2s )ds g B n 2t . As already observed before, this can be reformulated as iv) Since ψ n → ψ in D(E) as n → ∞, obviously s. for m-a.e. x. Moreover, A n is Lipschitz continuous in t, uniformly in n, and τ n (t) → τ ∞ (t) (uniformly in ω). Thus P x -a.s. for m-a.e. x A n τ n (t) → A ∞ τ ∞ (t) as n → ∞.
v) Define the taming semigroup (P κ t ) t≥0 by P κ t g(x) = E x e −A∞(τ∞(2t)+N∞(τ∞(2t) g(B τ∞(2t) )} (31) with A ∞ and N ∞ as introduced above. Then for every bounded, quasi continuous function g on X as n → ∞ for m-a.e. x ∈ X. (Note that quasi continuity of g implies that t → g(B t ) is continuous P x a.s. for m-a.e. x ∈ X.) Moreover, recall from Proposition 2.11 (iii) and estimate (51) that P kn t g(x) ≤ e C 0 +C 1 t g ∞ uniformly in n for m-a.e. x ∈ X with constants C 0 , C 1 depending only on ψ, on sup n osc(ψ n ), and on sup n Lip ψ n . For any test plan Π on X, therefore vi) Now assume that f ∈ D(∆ ∩ Lip(X). Since the mm-space (X, d, m) satisfies an RCD-condition, it implies |∇f | ∈ D(E) ∩ L ∞ (X) and in turn that |∇f | admits a quasi continuous version. Thus applying (30), (28), and (32) to a quasi continuous version g of |∇f | yields for any test plan Π on X. Therefore, P k t |∇ ∞ f | is a weak upper gradient for P ∞ t f . Hence, in particular, According to Theorem 2.17, the latter indeed is equivalent to the L 1 -Bochner inequality This proves the claim in the case ψ ∈ Lip b (X) ∩ D(E).
vii) In the general case of ψ ∈ Lip b (X), let us choose a sequence of ψ j ∈ Lip b (X) ∩ D(E), j ∈ N, with ψ j ∞ ≤ ψ ∞ , Lip ψ j ≤ Lip ψ and ψ j ≡ ψ on B j (z) for some fixed z ∈ X. For j ∈ N, let (P ψ j t ) t≥0 denote the heat semigroup on the mm-space (X, e ψ j , e 2ψ j m) and let (P κ j t ) t≥0 denote the associated taming semigroup defined as in (31) with ψ now replaced by ψ j . Then obviously as j → ∞ as well as Thus for any f ∈ D(∆ ∩ Lip(X) and any test plan Π on X, as j → ∞, Arguing as in the previous part vi), this proves that the mm-space (X, e ψ , e 2ψ m) satisfies Corollary 4.8. For ψ ∈ Lip b (X), the heat flow P t ) t≥0 on the metric measure space (X, d , m ) with d := e ψ d and m := e 2ψ m satisfies where (M ψ t ) t≥0 denotes the martingale additive functional in the Fukushima decompositions of (ψ(B t )) t≥0 .
Equivalently, this can be re-formulated as now with ∇ in the place of ∇ and with A replaced by Example 4.9. Let (X, d, m) = (R 2 , d Euc , m Leb ) be the standard 2-dimensional mm-space. Define functions ψ j , j ∈ N, and ψ : R 2 → R by with Φ, Φ j as defined in (3) for some ϕ ∈ C 2 (R) and with η ∈ C 2 (R) given by η(t) := (t 2 − 1) 3 t for t ∈ [−1, 1] and η(t) := 0 else. For each j ∈ N, the time-changed mm-space (R 2 , d j , m j ) with d j := e ψ j d Euc , m j := e 2ψ j m Leb corresponds to the Riemannian manifold (R, g j ) with metric tensor given by g j := e 2ψ j g Euc . Its Ricci tensor is bounded from below by k j = −e −2ψ j ∆ψ j cf. previous Theorem. cf. Propostion 4.2. (Note that the measure-valued Ricci bound is given by k j m j = −∆ψ j .) In the limit j → ∞, we will end up with a mm-space (R 2 , d ∞ , m ∞ ) with distributional Ricci bound given by κ = −∆ψ.

Gradient Flows for Locally Semiconvex Functions
The goal of this subsection is to extend the existence result and the contraction estimate for gradient flows for semiconvex functions from [Stu18a] to the setting of locally semiconvex functions. The contraction estimate for the flow will be in terms of the variable lower bound for the local semiconvexity of the potential.
Let (X, d, m) be a locally compact metric measure space satisfying an RCD(K, ∞)condition (cf. Definition 3.1) for some K ∈ R. Assume moreover, that we are given a continuous potential V : X → R which is weakly -convex for some continuous, lower bounded function : X → R in the following sense: for all x, y ∈ X there exists a geodesic (γ r ) r∈[0,1] connecting them such that for all r ∈ [0, 1] where g(r, s) := min{(1 − s)r, (1 − r)s} denotes the Green function on [0, 1]. We say that a curve (x t ) t∈[0,∞) in X is an EVI -gradient flow for V (where EVI stands for "evolution-variational inequality") if the curve is locally absolutely continuous in t ∈ (0, ∞) and if for every t > 0, every y ∈ X, and every geodesic (γ r ) r∈[0,1] connecting x t and y, y). Proof. Existence and uniqueness of an EVI Λ -gradient flow (Φ t (x)) t≥0 for any x ∈ X with Λ := inf follows from our previous work [Stu18a]. Following the previous proof, one also can deduce the refined EVI -property. Indeed, this will follow as before by a scaling argument from the EVI K−n -property for the heat flow on the weighted metric measure space (X, d, e −nV m) which (obviously) satisfies the RCD(K + n , ∞)-condition. If we now compare two flows, then we can apply (37) twice: first to the flow (x t ) t≥0 and with y t in the place of y; then to the flow (y t ) t≥0 and with x t in the place of y. Adding up both estimates yields (after some tedious arguments to deal with weakly differentiable functions with double dependence on the varying parameter) Alternatively, one can argue as follows: Given > 0, let X be covered by balls Thanks to the Localization Theorem 4.6, for each i there exists an RCD-spaces (X i , d i , m i ) with B r i (z i ) ⊂ X i whose local data on B r i (z i ) coincide with those of the original one. Thus as long as the flow does not leave B r i (z i ), we can consider the original flow also as an EVI-gradient flow for V on the mm-spaces (X i , d i , m i ).
Given any (X, d)-geodesic (γ t ) t∈[0,1] and r, s ∈ [0, 1] with γ r , γ s ∈ B r i (z i ) we thus conclude that Adding up these estimates for consecutive pairs of points on the geodesic (γ t ) t∈[0,1] finally gives Choosing γ optimal, we therefore obtain for arbitrary x 0 , x 1 ∈ X and for all t ≥ 0 Since > 0 was arbitrary, this yields the claim.
As pointed out in [Stu18a] in the case of constant , the existence of EVI-flows for V can be regarded as a strong formulation of semiconvexity of V .
Corollary 5.2. Every weakly -convex function is indeed strongly -convex in the sense that the inequality (36) holds for every geodesic (γ t ) t∈[0,1] in X.
A closer look on the proof of the previous Theorem 5.1 shows that appropriate reformulations of the results also hold true for flows which are defined only locally.
Theorem 5.3. Assume that continuous functions V and : Y → R are defined on an open subset Y ⊂ X and that V is -convex on Y in the sense that the inequality (36) holds for every geodesic (γ t ) t∈[0,1] contained in Y .
(i) Then for each x 0 ∈ Y there exists a unique local EVI -gradient flow (x t ) t∈[0,τ ) for V with maximal life time τ = τ (x 0 ) ∈ (0, ∞]. If τ < ∞ then x τ = lim t→τ x t exists and x τ ∈ ∂Y . (ii) For any pair of initial points x 0 , y 0 ∈ Y and their EVI -flows holds for all t ≤ T * where T * = T * (x 0 , y 0 ) denotes the first time where a geodesic connecting x t and y t will leave Y .
Proof. (i) Existence and uniqueness of a local EVI -gradient flow are straightforward.
Applying the estimate to points x 0 and y 0 := x δ proves that the flow (x t ) t has finite speed. Assuming τ < ∞, the family (x t ) t<τ will be bounded and therefore admits a unique accumulation point for t → τ , say x τ ∈ Y . Assuming that τ is the maximal life time for the flow implies that x τ ∈ ∂Y .
(ii) follows exactly as in the case of the globally defined gradient flow.

Convexification
In this subsection, we will prove the fundamental Convexification Theorem which (via time-change) allows to transform the metric of a mm-space (X, d, m) in such a way that a given semiconvex subset Y ⊂ X will become geodesically convex w.r.t. the new metric d . We will prove this in two versions: first, for closed sets Y , then for open sets Z. Throughout this section, we fix a locally compact RCD(K, ∞)-space (X, d, m).
Definition 5.4. We say that a subset Y ⊂ X is locally geodesically convex if there exists an open covering i∈I U i ⊃ X such that every geodesic (γ s ) s∈[0,1] in X completely lies in Y provided γ 0 , γ 1 ∈ Y ∩ U i for some i ∈ I.
Every geodesically convex set is locally geodesically convex but not vice versa.
Theorem 5.6. Let V, : X → R be continuous functions and assume that for each > 0 there exists a neighborhood D of the closed set Y := {V ≤ 0} such that Then for every > 0, the set Y is locally geodesically convex in (X, d ) for d = e ( − )·V d.
Remark 5.7. (i) The above Theorem provides a far reaching extension of our previous result in [LS18] which covers the case of constant negative . Now we also admit variable and 's of arbitrary signs.
(ii) Note that in the case of positive , the set Y will already be convex in the old metric space (X, d) and it will be "less convex" in the new space (X, d ).
(iii) In the above Theorem, without restriction, we may put V ≡ 0 in Y . Moreover, for both functions V and it suffices that they exist as continuous functions on D \ Y 0 for some neighborhood D of Y .
Proof. Let > 0 be given and put d = e ( − )V d.
(i) In order to prove the local convexity of Y in (X, d ), let z ∈ ∂Y be given and choose > 0 sufficiently small (to be determined later). In any case, assume that ( 1+ 1− ) 2 < 2. Choose δ > 0 such that Our proof of the local convexity of Y will be based on a curve shortening argument under the gradient flow for V : Assume that (γ a ) a∈[0,1] was a d -geodesic in B δ/3 (z) with endpoints γ 0 , γ 1 ∈ Y and γ a ∈ Y for some a ∈ (0, 1). Then we will construct a new curve (γ 0 a ) a∈[0,1] with the same endpoints but which is shorter (w.r.t. d ) than the previous one -which obviously contradicts the assumption. For each a ∈ [0, 1], we consider the gradient flow curve Φ t (γ a ) t≥0 for V starting in Φ 0 (γ a ) = γ a and we stop it as soon as the flow enters the set Y . Then we put γ 0 To get started, let us first summarize some key facts for the gradient flow for V , that is, for the solution toẋ t = −∇V (x t ) in the sense of EVI-flows.
(iii) To proceed, it is more convenient to parametrize the flow not by time (as we did before) but by "height", measured by the value of V . That is, for r ≥ 0 we put Φ r (x) = Φ Tr(x) (x) with T r (x) as above. Moreover, for x ∈ Y we put T r (x) := 0 and Φ r (x) := x for all r ≥ 0. The see [LS18], Lemma 2.13 (or, more precisely, estimate (10) in the proof of it).
Measuring the speed of the curves now in the metric d = e ( − )V d, the previous estimate yields |γ 0 a | ≤ |γ a | and, moreover, |γ 0 a | < |γ a | whenever γ a ∈ Y and |γ a | = 0. This proves the claim. In the previous Theorem, we used the gradient flow w.r.t. a function V (which shares basic properties with the distance function d( . , ∂Y )) as a path-shortening flow on the exterior of Y in order to prove that the closed set Y is locally geodesically convex w.r.t. the new metric d .
To make a given open set Z ⊂ X locally geodesically convex w.r.t. a new metric d , we will proceed in a complementary way: we will use the gradient flow w.r.t. a function V which shares basic properties with the negative distance function −d( . , ∂Z) as a pathshortening flow in the interior of Z. This is the content of the Second Convexification Theorem.
Theorem 5.8. Let V, : X → R be continuous functions and assume that for each > 0 there exists δ > 0 such that Then for every > 0, the open set Z := {V < 0} is locally geodesically convex in (X, d ) for d = e ( − )·V d.
Proof. Given > 0, we choose δ > 0 as above. Then for each r ∈ (0, δ], we can apply the First Convexification Theorem 5.6 with V as above, with the closed set Z r := {V ≤ −r} in the place of Y , and with D := Z \ Z r . This yields that the set Z r is locally geodesically convex w.r.t. the metric d := e ( − )V d. Having a closer look on the proof of Theorem 5.6, we see that we can choose an open covering i∈I U i ⊃ X, independently of r, such that every d -geodesic (γ s ) s∈[0,1] in X completely lies in Z r -and thus in particular in Z -provided γ 0 , γ 1 ∈ Z r ∩ U i for some i ∈ I. This proves the claim: every d -geodesic (γ s ) s∈[0,1] completely lies in Z provided γ 0 , γ 1 ∈ Z ∩ U i for some i ∈ I.

Bounds for the Curvature of the Boundary
The canonical choice for the function V in both of the previous Convexification Theorems is the signed distance function V = d( . , Y ) − d( . , X \ Y ) (or suitable truncated and/or smoothened modifications of it). In the Riemannian setting, a lower bound for the Hessian of this function has a fundamental geometric meaning: it is a lower bound of the fundamental form of the boundary. In the abstract setting, this observation will provide a synthetic definition for the variable lower bound curvature of the boundary. Definition 5.9. i) We say that a closed set Y ⊂ X is locally semiconvex with ∈ C(X) as exterior lower bound for the curvature of ∂Y if for each > 0 there exists a neighborhood ii) We say that an open set Z ⊂ X is locally semiconvex with ∈ C(X) as an interior lower bound for the curvature of ∂Z if for each > 0 there exists a neighborhood D of iii) We say that a closed set Y with Y = Y 0 is locally semiconvex with ∈ C(X) as lower bound for the curvature of ∂Y if for each > 0 there exists a neighborhood D of ∂Y such that the signed distance function Remark 5.10. The previous approach does not only allow us to define lower bounds for the curvature of the boundary (interpreted as lower bounds for the "second fundamental form of the boundary") but also to define the second fundamental form II ∂Y itself as well as the mean curvature ρ ∂Y : the former as the Hessian and the latter as the Laplacian of the signed distance function V = d( . , Y ) − d( . , X \ Y ). That is, provided V ∈ D loc (∆) and ρ ∂Y := ∆V.
This concept of curvature bounds for the boundary has been introduced in [LS18], restricted there to the case of constant, nonpositive . As already observed there, the two most important classes of examples are Riemannian manifolds and Alexandrov spaces. The proof of the next result is literally as in [LS18].
Proposition 5.11. Let X be a Riemannian manifold, Y a closed subset of X with smooth boundary. Then is a lower bound (or interior lower bound or exterior lower bound, resp.) for the curvature of ∂Y if and only if the real-valued second fundamental form of ∂Y satisfies II ∂Y ≥ .
Lemma 5.12. Let (X, d) be an Alexandrov space with generalized sectional curvature ≥ K. Put ρ K = π 2 √ K if K > 0 and ρ K = ∞ else. Then for each r ∈ (0, ρ K ), the curvature of the boundary of the complement of a ball with radius r is bounded from below by Proof. Given z ∈ X and r ∈ (0, ρ K ), put Y := X \ B r (z) and Then obviously {V ≤ 0} = Y and |∇V | = 1 on ∂Y . Moreover, by comparison results for Hessians of distance functions in Alexandrov spaces Thus ∇ 2 V ≥ −c K (r) on ∂Y which proves the claim.
Lemma 5.13. Let (X, d) be an CAT space with generalized sectional curvature ≤ L. Put ρ L = π 2 √ L if L > 0 and ρ L = ∞ else. Then for each r ∈ (0, ρ L ), the curvature of the boundary of the ball with radius r is bounded from below by Proof. Given z ∈ X and r ∈ (0, ρ * ), put Y := X \ B r (z) and Then obviously {V ≤ 0} = Y and |∇V | = 1 on ∂Y . Moreover, by comparison results for Hessians of distance functions in CAT spaces Thus ∇ 2 V ≥ c L (r) on ∂Y which proves the claim.
Proposition 5.14. Let (X, d) be an Alexandrov space with generalized sectional curvature ≥ K. Put ρ K = π 2 √ K if K > 0 and ρ K = ∞ else. Assume that a family of balls B r i (z i ) i∈I is given with r i ∈ (0, ρ K ) and is a neighborhood of ∂Y ("exterior ball condition"). Then with c K given by (40), is an exterior lower bound for the curvature of ∂Y .
with V r,z (x) as defined in (41). Then Analogously, we conclude Proposition 5.15. Let (X, d) be a CAT space with generalized sectional curvature ≤ L. Put ρ L = π 2 √ L if L > 0 and ρ L = ∞ else. Assume that that a family of balls B r i (z i ) i∈I is given with r i ∈ (0, ρ K ) and is a neighborhood of ∂Y ("interior ball condition"). Then with c L given by (42), is an interior lower bound for the curvature of ∂Y .

The Convexification Theorems from the previous subsection immediately yield
Theorem 5.16. i) Assume that ∈ C(X) is an exterior lower bound for the curvature of ∂Y . Then for every > 0, the set Y is locally geodesically convex in (X, d ) for ii) Assume that ∈ C(X) is an interior lower bound for the curvature of ∂Y . Then for every > 0, the set Y 0 is locally geodesically convex in (X, d ) Remark 5.17. The Convexification Theorem provides a method to make a given set convex by local changes of the geometry. By construction of this transformed geometry, the given set will be "as little convex as possible". Indeed, in regions where the set already was convex, the set will be transformed into a less convex set.
Example 5.18. Let X = R n for n ≥ 2, equipped with the Euclidean distance and the Lebesgue measure. If we apply the previous results to the complement of a ball, say Y = R n \ B r (z), then we see that Y (as well as Y 0 ) will be locally geodesically convex in (R n , e (1+ )ψ d) for any > 0 where On the other hand, applying the previous results to a ball, say Z = B r (z), then we see that Z (as well as Z) will be locally geodesically convex in (R n , e (1− )ψ d) for any > 0 with the same ψ as before. The same "convexification effect" will be achieved by choosing in a neighourhood of ∂B r (z). In the case n = 2, with this choice of ψ, in a neighorhood of ∂B r (z) the space (R n , e ψ d) will be a flat torus. In particular, ∂B r (z) will be a totally geodesic subset. This provides a simple explanation why both, B r (z) and its complement, are convex. Let a metric measure space (X, d, m) be given; assume that it is geodesic, locally compact, and infinitesimally Hilbertian. Observe that due to the local compactness, W 1,2 (X) = u ∈ W 1,2 loc (X) : X [Γ(u) + u 2 ] dm < ∞ . By restriction to a closed set Y ⊂ X, we define the mm-space (Y, d Y , m Y ). Here d Y denotes the length metric on Y induced by d and m Y denotes the measure m restricted to Y . The Cheeger energy associated with the restricted mm-space (Y, d Y , m Y ) will be denoted by E Y and its domain by F Y = D(E Y ) = W 1,2 (Y ). To avoid pathologies, throughout the sequel, we assume that The minimal weak upper gradients (and thus also the Γ-operators) w.r.t. (X, d, m) and w.r.t. (Y, d Y , m Y ) will coincide a.e. on Y 0 , i.e. F Y loc (Y 0 ) = F loc (Y 0 ) and Γ Y (u) = Γ(u) a.e. on Y 0 for all u ∈ F Y loc (Y 0 ). Moreover, for all u ∈ W 1,2 (Y ). In particular, the restricted mm-space (Y, d Y , m Y ) is also infinitesimally Hilbertian.
The heat semigroup associated with the restricted mm-space (Y, d Y , m Y ) will be called Neumann heat semigroup and denoted by (P Y t ) t≥0 . The associated Brownian motion will be called reflected Brownian motion and denoted by (P Y x , B Y t ). Remark 6.1. i) In literature on Dirichlet forms and Markov processes (in particular, in [CF12]), Chapter 7, "reflected Brownian motion" on the closure of an open set Y 0 ⊂ X is by definition (and by construction) the reversible Markov process associated with the Dirichlet form E Y given by (44) ii) In general, the sets W 1,2 (Y ) and W 1,2 (Y 0 ) do not coincide, see subsequent Example. In [LS18], Section 4.2, equality of W 1,2 (Y ) and W 1,2 (Y 0 ) was erroneously stated as a general fact. Instead, it should be added there as an extra assumption. Equality holds if Y 0 is regularly locally semiconvex, see Proposition 6.4 below, and of course also if Y 0 has the extension property W 1,2 (Y 0 ) = W 1,2 (X) Y 0 . where {x n } n∈N denotes a countable dense subset of [−1, 1]. Then W 1,2 (Y ) = W 1,2 (Y 0 ). For instance, the function u(x, y) = sign(y) belongs to W 1,2 (Y 0 ) but not to W 1,2 (Y ).
Indeed, functions in W 1,2 (Y 0 ) can have arbitrary jumps at the x-axis since Y 0 has two disconnected components, one being a subset of the open upper half plane, the other one being a subset of the open lower half plane. On the other hand, functions in W 1,2 (Y ) will be continuous along almost every vertical line which does not hit one of the small balls B 2 −n (x n , 0), n ∈ N, (which is the case for more than half of the vertical lines). Here and in the sequel, we put D cont loc (∆) := {f ∈ D loc (∆) with f, Γ(f ), ∆f ∈ C(X)} and D cont ∞ (∆) := {f ∈ D loc (∆) with f, Γ(f ), ∆f ∈ C(X) ∩ L ∞ (X)}. Note that D cont ∞ (∆) ⊂ Lip b (X) provided the Sobolev-to-Lipschitz property holds. Proposition 6.4. Assume that (X, d, m) satisfies RCD(K, N ) for some K, N ∈ R and that Y 0 is regularly locally semiconvex. Then and |D Y u| = |Du| m-a.e. on Y for every u ∈ W 1,2 (Y 0 ).
Proof. i) To simplify the subsequent presentation, we assume that the defining functions , V for the regular semiconvexity of Y 0 can be chosen to be in D cont ∞ (∆) and not just in D cont loc (∆). Under this simplifying assumption, for any > 0 also ψ := ( − ) V ∈ D cont ∞ (∆) and thus the time-changed mm-space (X, d , m ) with d = e ψ d and m = e 2ψ m will satisfy RCD(K , ∞) with some K ∈ R. The general case can be treated by a localization and covering argument.
ii iii) Now let us fix a test plan Π in (Y, d Y , m Y ). For n ∈ N, define Π n to be the "piecewise geodesic test plan" in (X, d , m ) such that (e t ) * Π n t∈[0,1] is the W 2 -geodesic which interpolates between the measures (e i/n ) * Π for i = 0, 1, . . . , n. Thanks to the RCD-property of (X, d , m ), such a piecewise geodesic interpolation indeed is a test plan.
iv) Given the compact set Y ⊂ Y 0 , there exists u ∈ W 1,2 (X, d , m ) such that u = u , |D u| = |D u | m-a.e. on a neighborhood of Y .
Since Π n is a test plan in (X, d , m ), we obtain for each n ∈ N and each > 0 In the limit → 0 this yields Since by assumption |Du| ∈ L 2 (Y, m Y ), according to the subsequent Lemma we may pass to the limit n → ∞ and finally obtain This proves the claim: u ∈ W 1,2 (Y ) with minimal weak upper gradient m-a.e. dominated by |Du|.
Lemma 6.5 (Private communication by N. Gigli). Assume that (X, d, m) satisfies RCD(K, N ) for some K, N ∈ R. Then for every test plan Π in X and every g ∈ L 2 (X, m) where Π n denotes the "piecewise geodesic test plan" such that (e t ) * Π n t∈[0,1] is the W 2geodesic which interpolates between the measures (e i/n ) * Π for i = 0, 1, . . . , n.
Proof. First of all, observe that it obviously suffices to prove the claim for test plans supported on bounded subsets of X. Secondly observe, that it suffices to consider bounded continuous functions g. Indeed, given any g ∈ L 2 (X, m) and ε > 0, there exists g ε ∈ C b (X) with g − g ε L 2 ≤ ε. Since Π n is a test plan, this implies ≤ ε · sup n C n · sup n A n for each n ∈ N ∪ {∞} with Π ∞ := Π where C n is the compression of the test plan Π n and Due to the RCD(K, ∞)-assumption, the compression of Π n is bounded by the compression of Π times a constant depending on K and the diameter of the supporting set of Π. Thus It remains to prove (45) for bounded continuous g. This will be an immediate consequence of the weak convergence dπ n (γ) := |γ t |dtdΠ n (γ) → |γ t |dtdΠ(γ) =: dπ(γ) as measures on the space X := [0, 1] × C([0, 1] → X). To prove the latter, we first observe that the total mass of the measures π n is uniformly bounded on X since Properness of X (due to the RCD(K, N )-assumption) and uniform boundedness of the supporting sets of Π n then guarantees the existence of a subsequential limit π ∞ . Lower semicontinuity of the map γ → |γ t | implies that π ∞ ≤ π. Now assume that π ∞ = π. Then in particular π ∞ = π on the set {(t, γ) : |γ t | = 0}. Once again using the lower semicontinuity of γ → |γ t | this will imply lim inf which is a contraction to (47) from above. Thus π ∞ = π and hence (46) follows and so does in turn (45).
Given a bounded continuous ψ ∈ W 1,2 (X), let us consider the mm-space Proposition 6.6. Assume that (X, d, m) satisfies the RCD(k, N )-condition for some finite number N ≥ 2 and some lower bounded, continuous function k. Moreover, assume that Y is locally geodesically convex in (X, d ) where d = e ψ d for some ψ ∈ Lip b (X) ∩ D cont loc (∆). Then for any (extended) number N > N , the mm-space (Y, d Y , m Y ) satisfies the RCD(k , N )-condition and the BE 2 (k , N )-condition with Proof. i) To get started, we first employ the equivalence of the Lagrangian and Eulerian formulation of curvature-dimension conditions as formulated in Theorem 3.4 to conclude that (X, d, m) satisfies the BE 2 (k, N )-condition. ii) Next we apply our result on time change, Proposition 4.2 or [HS19], Theorem 1.1, to conclude that (X, d , m ) satisfies the BE 2 (k , N )-condition with the given N and k .
iii) Once again referring to Theorem 3.4 for the equivalence of the Lagrangian and Eulerian formulation, we conlucde that (X, d , m ) satisfies the RCD(k , N )-condition. iv) In the Lagrangian formulation, it is obvious that a curvature-dimension condition is preserved under restriction to locally geodesically convex subsets. Since by assumption Y is locally geodesically convex in (X, d ), it follows that (Y, d Y , m Y ) satisfies the RCD(k , N )-condition.
v) In a final step, we once again employ Theorem 3.4 to conclude BE 2 (k , N ), the Eulerian version of the curvature-dimension condition.

Time Re-Change
We are now going to make a "time re-change": we transform the mm-space (Y, d Y , m Y ) into the mm-space (X, d Y , m Y ) by time change, now with −ψ in the place of ψ and with (k , N ) in the place of (k, N ).
The main challenge will arise from the two conflicting requirements: • |∇ψ| = 0 on ∂Y in order to make use of the Convexification Theorem • ψ ∈ D(∆ Y ) (which essentially requires |∇ψ| = 0 on ∂Y ) in order to control the Ricci curvature under the "time re-change".
To overcome this conflict, we have developed the concept of W −1,∞ -valued Ricci bounds which will allow us to work with the distribution ∆ Y ψ. More precisely, the crucial ingredient in our estimate will be the distribution ∆ Y ψ ∂Y : Note that this distribution indeed is supported on the boundary of Y in the sense that ψ = ψ on a neighborhood of ∂Y implies Theorem 6.7. Assume that (X, d, m) satisfies the RCD(k, N )-condition for some finite number N ≥ 2 and some lower bounded, continuous function k. Moreover, assume that Y is locally geodesically convex in (X, d ) where d = e ψ d for some ψ ∈ D cont ∞ (∆) with ψ = 0 on ∂Y . Then the mm-space (Y, d Y , m Y ) satisfies the BE 1 (κ, ∞)-condition with Proof. i) Let ψ ∈ D cont ∞ (∆), put d = e ψ d and N = 2(N − 1). Then according to Theorem 4.2, the mm-space (X, d , m ) satisfies the BE 2 (k , N )-condition with Since by assumption Y is locally geodesically convex, according to the previous Proposition, the mm-space (Y, d Y , m Y ) also satisfies the BE 2 (k , N )-condition with the same k .
On the space Y , let us now perform a time change with the weight function −ψ ("time re-change") to get back According to Theorem 4.7, the mm-space (X, d Y , m Y ) will satisfy BE 1 (κ, ∞) with ii) In a final approximation step, we now want to get rid of the term −4(N − 2)|∇ψ| 2 m Y in the previous distributional Ricci bound κ.
Given ψ as as above, we define a sequence of functions ψ n with the same properties as ψ, with ψ n = ψ on B 1/n (∂Y ), with |∇ψ n | being bounded, uniformly in n, and with |∇ψ n | → 0 m-a.e. on Y as n → ∞. This can easily be achieved by means of the truncation functions from Lemma 4.4.
Then according to (33) in the previous part of this proof, for each n ∈ N the mm-space where the last equality is due to (49). Since the mm-space under consideration does not depend on n, this obviously implies the BE 1 (κ, ∞)-condition with κ = k m Y +∆ Y ψ ∂Y .
Summary. Let us illustrate the strategy of the argumentation for the proof of the previous Theorem in a diagram.
Corollary 6.8. Under the assumptions of the previous Theorem, the ("Neumann") heat semigroup on (Y, d Y , m Y ) satisfies a gradient estimate of the type: where M Y,ψ and N Y,ψ denote the local martingale and local additive functional of vanishing quadratic variation w.r.t. (P Y x , B Y t ) in the Fukushima decomposition of ψ(B Y t ).
Now consider the restriction to the upper halfplane Y = R×R + which is convex w.r.t. d Euc but higly non-convex w.r.t. d . According to Theorem 6.7 (applied to d and d Euc in the place of of d and d , resp., and with ψ replaced by −ψ), the boundary effect amounts in an additional contribution in the Ricci bound given by Indeed, the distribution in turn can be identified with the signed measure since for sufficiently smooth f :

Boundary Measure and Boundary Local Time
Let V : X → R denote the signed distance function from ∂Y (being positive outside Y and negative in the interior of Y ), i.e., V := d(., Y ) − d(., X \ Y ).
Lemma 6.10. Assume that Y has regular boundary. i) Then the distribution −∆ Y V ∂Y is given by a nonnegative measure σ supported on ∂Y , denoted henceforth by σ ∂Y and called surface measure of ∂Y .
More precisely, there exists a nonnegative Borel measure σ on X which is supported on ∂Y and which does not charge sets of vanishing capacity such that for all bounded, quasi-continuous f ∈ D(E) ii) The local additive functional of vanishing quadratic variation N ∂Y,−V as defined in (53) (with −V in the place of ψ) coincides with the PCAF (= "positive continuous additive functional") associated to σ ∂Y via Revuz correspondence (w.r.t. the Brownian motion (P Y x , B Y t ) on Y ) which henceforth will be denoted by L ∂Y = (L ∂Y t ) t≥0 and called local time of ∂Y . In other words, ∆V f dm for nonnegative f ∈ D(E). In other words, Y Γ(V, f ) + ∆V f dm ≥ 0. This extends to all nonnegative f ∈ W 1,1+ (X) if Γ(V ), ∆V ∈ L ∞ (X). Moreover, due to the extension property which we assumed, it extends to all nonnegative f ∈ W 1,1+ (Y ). Thus for all nonnegative f ∈ W 1,1+ (Y ). According to the Riesz-Markov-Kakutani Representation theorem, the distribution −∆ Y V ∂Y therefore is given by a Borel measure on X, say σ. Obviously, this measure is supported by ∂Y . Moreover, on each set X ⊂ X of finite volume, this measure σ has finite energy: for all f ∈ D(E) which are supported in X . Thus σ does not charge sets of vanishing capacity, [FOT11] , Lemma 2.2.3.
ii) The fact that −∆ Y V ∂Y is a nonnegative measure (of finite energy) implies that ∆ Y V is a signed measure (of finite energy). Hence, ∆ Y V and N Y,V are related to each other via Revuz correspondence. And of course the signed measure ∆V m Y corresponds to the additive functional ( t 0 ∆V (B Y s )ds) t≥0 . Lemma 6.11. Assume that the "integration-by-parts formula" holds true for Y with some measure σ on ∂Y (charging no sets of vanishing capacity): ∀f ∈ D(∆), g ∈ D(E) withg andΓ(f, V ) denoting the quasi continuous versions of g and Γ(f, V ), resp. Then σ = σ ∂Y .
Note that for f ∈ D(∆ Y ), the above formula -with vanishing RHS -is trivial.
Proof. Applying the Integration-by-Parts formula to f = V yields Y Γ(V, g) dm + Y ∆V g dm = ∂Yg dσ which proves that the distribution −∆ Y V ∂Y is represented by the measure σ and thus σ = σ ∂Y Example 6.12. Let X be a n-dimensional Riemannian manifold, d be the Riemannian distance, m be the n-dimensional Riemannian volume measure, and Y be a bounded subset with C 1 -smooth boundary. Then σ ∂Y is the (n − 1)-dimensional surface measure of ∂Y .
Lemma 6.13. Assume that ψ = V with V as in Lemma 6.10 above and ∈ D cont ii) Moreover, with (N ∂Y,ψ t ) t≥0 and (N ∂V,ψ t ) t≥0 defined as in (53), Let us finally illustrate our results in the two prime examples, the ball and the complement of the ball. To simplify the presentation, we will formulate the results in the setting of RCD(0, N ) spaces for N ∈ N with the CAT(1)-property (or RCD(−1, N ) spaces with the CAT(0)-property). The extension to RCD(K, N ) spaces with sectional curvature bounded from above by K is straightforward.
Example 6.17. i) Consider Y = B r (z) for some z ∈ X and r ∈ (0, π/4) where (X, d, m) is a N -dimensional Alexandrov space with nonnegative Ricci curvature and sectional curvature bounded from above by 1. Then In particular, ii) Consider Y = X \ B r (z) for some z ∈ X and r ∈ (0, ∞) where (X, d, m) is a N -dimensional Alexandrov space with N ≥ 3, with Ricci curvature bounded from below by −1, and with nonpositive sectional curvature. Then In particular, Let us emphasize that in the latter setting, no estimate of the form can exist.
Proof. i) It remains to prove the second inequality in (56). Put with M Y,V t ≤ t and, therefore, where the last inequality follows from the fact that | 2 , we apply the Laplace comparison to the function V 2 (y) = r 2N −2 (N − 2) 2 · d 2−N (y, z) − r 2−N 2 which yields ∆V 2 (y) ≤ 2 + N N − 2 d(y, z) coth d(y, z) ≤ 5.
Therefore, taking into account that V (x) ≤ 0, Thus E x e 1 r L ∂Y t ≤ e Ct+C √ t .
Corollary 6.18. In the setting of the previous Example 6.17 i), the effect of the boundary curvature results in a lower bound for the spectral gap: Let us emphasize that without taking into account the curvature of the boundary, no positive lower bound for λ 1 will be available.
Proof. In the gradient estimate for the heat flow on the ball Y = B r (z), the boundary curvature causes an exponential decay: for each f and x ∈ Y , and P Y t ∇f 2 (x) → 1 m(Y ) Y ∇f 2 m as t → ∞. On the other hand, by spectral calculus ∇P Y t f 1 2 (x) = e −2λ 1 ∇f 1 2 (x).
for the eigenfunction f 1 corresponding to the first non-zero eigenvalue λ 1 .