Curvature-dimension conditions under time change

We derive precise transformation formulas for synthetic lower Ricci bounds under time change. More precisely, for local Dirichlet forms we study how the curvature-dimension condition in the sense of Bakry–Émery will transform under time change. Similarly, for metric measure spaces we study how the curvature-dimension condition in the sense of Lott–Sturm–Villani will transform under time change.


Introduction
A. Bakry and Émery [5] formulated a powerful criterion for obtaining equilibration and regularity results for the Markov semigroups associated with local Dirichlet forms. Let us briefly recall their concept. A Dirichlet form E , densely defined on some L 2 (X, ) , satisfies the BE(k, N) condition with some function k ∈ L ∞ loc (X, ) and some number N ∈ [2, ∞] if for all suitable functions f and ≥ 0 on X. Here, Δ denotes the generator associated with E and Γ the carré du champ operator. Estimate (1) can be regarded as an abstract formulation of Bochner's inequality on Riemannian manifolds. Thus, in this Eulerian approach to curvature-dimension conditions, k(x) will be considered as a synthetic lower bound for the "Ricci curvature at x ∈ X " and N as an upper bound for the "dimension" of X. (1)

3
From the very beginning of this theory, the transformation formula for the Bakry-Émery condition BE(k, N) under drift transformation played a key role. Most importantly in the case N = ∞ , this states that the Dirichlet form 2 Γ Γ(f ), V denotes the lower bound for the Hessian of V at x ∈ X for any sufficiently regular function V on X.
The goal of this paper now is to analyze the transformation property of the Bakry-Émery condition under time change. That is, we will pass from the original Dirichlet form E on L 2 (X, ) to a new one defined as for some w ∈ L ∞ loc (X, ) . Our main result provides a Bakry-Émery condition for this transformed Dirichlet form provided the original Dirichlet form satisfies a Bakry-Émery condition with finite N. We remark that it is possible to weaken the curvature-dimension condition adopted in this theorem to a milder distributional condition, in the spirit of the papers [11,27].

Theorem 1
Assume that E satisfies the BE(K, ∞) condition for some K ∈ ℝ and the BE(k, N) condition for some k ∈ L ∞ loc and some N ∈ [1, ∞) , and that w ∈ D loc ( ) ∩ L ∞ loc with w = sing w + Δ ac w and sing w ≤ 0 . Then, for any N � ∈ (N, ∞] and k � ∈ L ∞ loc , the time-changed Dirichlet form E ′ on L 2 (X, � ) satisfies the BE(k � , N � ) condition provided -a.e. on X.
Corollary 1 If in addition k ′ is bounded from below, say k ′ ≥ K ′ for some K � ∈ ℝ , then the time-changed Dirichlet form E ′ and the associated heat semigroup (P � t ) t≥0 satisfy the following gradient estimate Remark 1 Generator and carré du champ operator of the time-changed Dirichlet form E ′ on L 2 (X, � ) are given by Moreover, the associated Brownian motion (ℙ � x , B � t ) (cf. Chapter 6 [12]) is given by ℙ � x = ℙ x and Note that heat semigroup (P � t ) t≥0 and Brownian motion (ℙ � x , B � t ) are linked to each other by Δ � = e −2w Δ, Γ � = e −2w Γ. (4) B. A different approach, the so-called Lagrangian approach, to synthetic lower Ricci bounds was proposed in the works of Lott and Villani [19] and Sturm [23]. Here, the objects under consideration are metric measure spaces. Such a space (X, d, ) satisfies the curvature-dimension condition CD(K, ∞)-meaning that its Ricci curvature is bounded from below by K-if the Boltzmann entropy Ent(., ) is weakly K-convex on the Wasserstein space P 2 (X) . More refined curvature-dimension conditions CD(K, N) and CD * (K, N) with finite N ∈ [1, ∞) were introduced in [23,24]. Combined with the requirement of Hilbertian energy functional, this led to the conditions RCD(K, N) and RCD * (K, N) [2], which fortunately turned out to be equivalent to each other [7].
Also from the very beginning of this theory, the transformation formula for the curvature-dimension conditions CD(K, N), CD * (K, N) RCD(K, N) under drift transformation played a key role. Most easily formulated in the case N = ∞ , it states that the condition CD(K, ∞) for a given metric measure space (X, d, ) and the L-convexity of V on X imply the condition CD(K + L, ∞) for the transformed metric measure space (X, d, e −V ) . The same holds with RCD in the place of CD.
Subject of the investigations in this paper is the time-changed metric measure space (X, d � , � ) where � = e 2w for some w ∈ L ∞ loc (X, ) and for x, y ∈ X . Assuming that w is continuous -a.e. on X this allows for a dual representation as where w(x) ∶= lim sup y→x w(y) denotes the upper semicontinuous envelope of w. Our main result provides the transformation formula for the curvature-dimension condition under time change.
Theorem 2 Let (X, d, ) be an RCD(K, N) space for some K ∈ ℝ and N ∈ [1, ∞) , and let w ∈ D loc ( ) ∩ L ∞ loc (X) be continuous -a.e. with w = sing w + Δ ac w and sing w ≤ 0 . Then, the time-changed metric measure space (X, d � , � ) satisfies the RCD(K � , N � ) condition for any N � ∈ (N, ∞) and K � ∈ ℝ such that Theorem 2 is a more or less immediate consequence of Theorem 1 and the fact that the Eulerian and the Lagrangian curvature-dimension conditions, BE(K, N) and RCD(K, N), are equivalent to each other as proven in [10].

Remark 2
The first derivation of the transformation formula for the (Eulerian) curvaturedimension condition BE(K, N) under conformal transformation as well as under time change was presented in [26] by the second author in the setting of regular Dirichlet forms admitting a nice core of sufficiently smooth functions ("Γ-calculus in the sense of Bakry-Émery-Ledoux").
Combining the techniques and results in [14,20], the first author [15,16] proved the transformation formula for the Lagrangian curvature-dimension condition RCD(K, N) under conformal transformation when the reference function w is bounded and smooth enough. Together with the well-known transformation formula for RCD(K, N) under drift transformations, this result also provides a transformation formula for RCD(K, N) under time change.
The focus of the current paper is on proving the transformation formula for the (Eulerian or Lagrangian) curvature-dimension condition under time change in a setting of great generality (Dirichlet forms or metric measure spaces) and with minimal regularity and boundedness assumptions on w.
C. One of the important applications of time-change is the "convexification" of nonconvex subsets Ω ⊂ X of an RCD(K, N)-space (X, d, ) as introduced by the second author and Lierl [18]. For sublevel sets of regular semi-convex functions V, they proved convexity after suitable conformal transformations, while control of the curvature bound under these transformations follows from the work [15] of the first author. Unfortunately, these previous results do not apply to the most natural potential, the signed distance function V = d(., Ω) − d(., X⧵Ω) due to lack of regularity. The more general results of the current paper will apply to a suitable truncation of the signed distance function and thus provide the following convexification theorem.

Time change and the Bakry-Émery condition
This section is devoted to study synthetic lower Ricci bounds under time change in the setting of Dirichlet forms. More precisely, we will derive the transformation formula for the Bakry-Émery condition under time change.

Dirichlet forms and the BE(K, N) condition
In this part, we recall some basic facts about Dirichlet form theory and the Bakry-Émery theory. Firstly we make some basic assumptions on the Dirichlet form, see also [21] for examples satisfying these conditions. Assumption 1 We assume that (a) (X, ) is a topological space, (X, B) is a measurable space and is a -finite Radon measure with full support (i.e., supp = X ); B is the -completion of the Borel -algebra generated by ; and L p (X, ) will denote the space of L p -integrable functions on (X, B, ); (b) E(⋅) ∶ L 2 (X, ) → [0, ∞] is a strongly local, quasi-regular, symmetric Dirichlet form with domain ∶= D(E) = f ∈ L 2 (X, ) ∶ E(f ) < ∞ ; denote by (P t ) t>0 the heat semigroup generated by E; (c) there exists an increasing sequence of ("cut-off") functions with compact support ( ) ≥1 ⊂ such that 0 ≤ ≤ 1 , Γ( ) ≤ C for all and → 1 , Γ( ) → 0 -a.e. as → ∞ , cf. [22]; 1 3 (d) E satisfies the Bakry-Émery condition BE(K, ∞) for some K ∈ ℝ.
To formulate the latter, recall that ∞ ∶= D(E) ∩ L ∞ (X, ) is an algebra with respect to pointwise multiplication. We say that E admits a carré du champ if there exists a quadratic continuous map Γ ∶ → L 1 (X, ) such that The Dirichlet form E induces a densely defined self-adjoint operator

Remark 3
Since by our standing assumption the Dirichlet form E satisfies BE(K, ∞) for some K ∈ ℝ , the "space of test functions" is dense in (cf. Section 2 [3] and Remark 2.5 therein). Hence, the BE(k, N) condition will follow if (5) holds true for all f ∈ TestF(E) and all non-negative ∈ D(Δ) ∩ L ∞ with Δ ∈ L ∞ . Lemma 1 For every f ∈ D(Δ) , we have Γ(f ) 1∕2 ∈ and Proof By self-improvement, the Bakry-Émery inequality BE(K, ∞) as introduced above implies the stronger L 1 -version for all f , ∈ D(Δ) with Δf ∈ , see [20]. Choosing = P t (Γ(f ) 1∕2 ) and then letting t → 0 yields the claim for f ∈ D(Δ) with Δf ∈ . Since the class of these f's is dense in D(Δ) , the claim follows. ◻

Definition 2 (i)
We say that f ∈ e if there exists a Cauchy sequence (f n ) n ⊂ w.r.t. the semi-norm E(⋅) and such that f n → f -a.e. Then, we define E(f ) ∶= lim n→∞ E(f n ) . Similarly, Γ can be extended to e . (ii) We say that f ∈ loc if for any bounded open set U, there is f ∈ such that f =f on U. Then, a function Γ(f ) ∈ L 1 loc (X, ) can be defined unambiguously by Γ(f ) ∶= Γ(f ) on U. Similarly, we define the spaces D loc (Δ) and TestF loc (E).

Definition 3 (Local weak Bakry-Émery condition)
Given a function k ∈ L ∞ loc and a number N ∈ [1, ∞] , we say that the Dirichlet form E satisfies the BE loc (k, N) condition if it admits a carré du champ and if for all f ∈ D loc (Δ) ∩ L ∞ loc with Δf ∈ loc and all non-negative ∈ ∞ with compact support and Γ( ) ∈ L ∞ .
Note that our standing assumption BE(K, ∞) implies that Γ(f ) 1∕2 ∈ loc for each f ∈ D loc (Δ) . Thus, for functions f and as above, the term − 1 Proof Assume that BE(k, N) holds true and let f and be given as in Definition 3. Choose Choose uniformly bounded, nonnegative n ∈ D(Δ) with Γ( n ), Δ n ∈ L ∞ such that n → a.e. on X and in as n → ∞ . (For instance, put n = P 1∕n .) Then, (5) implies for all n. Passing to the limit n → ∞ yields (6) with f ′ in the place of f. Since by assumption f = f � on a neighborhood of { ≠ 0} , this yields the claim (6).
Conversely, assume that BE loc (k, N) holds true and let f and be given as in Definition 1. Put n = P 1∕n and ,n = ⋅ P 1∕n with ( ) being the cut-off functions from assumption 1. According to the BE loc (k, N) assumption, (6) holds with ,n in the place of . Passing to the limit → ∞ yields (6) with n in the place of ( ∀n ). This, however, is equivalent to (5), again with n in the place of . Finally passing to the limit n → ∞ yields (5) for the given . ◻

Remark 4
From the proof of the preceding lemma, it is obvious that the class of f's to be considered for (6) can equivalently be restricted to f ∈ D loc (Δ) ∩ L ∞ loc with Δf ∈ loc ∩ L ∞ loc .

Self-improvement of the Bakry-Émery condition
The formulation of the subsequent results on the self-improvement property will require the theory of differential structures of Dirichlet forms as introduced by Gigli [14]. In order to shorten the length of the paper, we will skip the introduction of (co)tangent modules, list the results directly and ignore subtle differences. [14]) Given a strongly local, symmetric Dirichlet form E admitting a carré du champ Γ defined on e as above. Then, there exists an L ∞ -Hilbert module L 2 (TX) satisfying the following properties.
(i) L 2 (TX) is a Hilbert space equipped with the norm ‖ ⋅ ‖ such that the following correspondence (embedding) holds and for any f , g ∈ e and h ∈ L ∞ (X, ).
there exists a sequence v n = ∑ M n i=1 a n,i ∇g n,i with a n,i ∈ L ∞ and g n,i ∈ e , such that ‖v − v n ‖ → 0 as n → ∞.
By Corollary 3.3.9 [14], for any [20], Theorem 3.3.8 [14]), Hess f (⋅, ⋅) is given by the following formula: Combining Theorem 1.4.11 and Proposition 1.4.10 in [14], we obtain the following structural results. As a consequence, we can compute Hess f (⋅, ⋅) and Γ(⋅, ⋅) using local coordinates. Proposition 2 Denote by L 2 (TX) the tangent module associated with E . Then, there exists a unique decomposition (up to -null sets) {E n } n∈ℕ∪{∞} of X such that (a) For any n ∈ ℕ and any B ⊂ E n with positive measure, L 2 (TX) has an orthonormal basis where we say that a countable set Proposition 3 Let E be a Dirichlet form satisfying the BE(k, N) condition for some k ∈ L ∞ and some number N ∈ [1, ∞] and let {E n } n∈ℕ∪{∞} be the decomposition given by Proposition 2. Then, (E n ) = 0 for n > N , and for any (f , ) ∈ D(Γ 2 ) , we have The same estimate (8) also holds true for all f ∈ D loc (Δ) ∩ L ∞ loc with Δf ∈ loc and all non-negative ∈ ∞ with compact support and Γ( ) ∈ L ∞ provided E satisfies the BE loc (k, N) condition for some k ∈ L ∞ loc and N ∈ [1, ∞].

Proof
The proof for constant k = K was given in [16], Proposition 3.2 and Theorem 3.3. In fact, the proof there only relies on a so-called self-improvement technique in Bakry-Émery theory, which can also be applied to BE(k, N) case without difficulty. Also the extension via localization is straightforward. ◻ In order to proceed, we briefly recall the notion of measure-valued Laplacian as introduced in [13,20]. We say that f ∈ D( ) ⊂ e if there exists a signed Borel measure = + − − charging no capacity zero sets such that for any ∈ with quasi-continuous representative ∈ L 1 (X, | |) . If is unique, we denote it by f . If f ≪ , we also denote its density by Δf if there is no ambiguity.

Proposition 4 (See Lemma 3.2 [20]) Let E be a Dirichlet form satisfying the BE loc (k, N)
condition. Then, for any f ∈ TestF loc (E) , we have Γ(f ) ∈ D loc ( ) and In particular, the singular part of the measure Γ(f ) is non-negative.

BE(K, N) condition under time change
We define the time change of the Dirichlet form E in the following way.

Definition 4 (Time change)
Given a function w ∈ L ∞ loc (X, ) , define the weighted measure w ∶= e 2w and the time-changed Dirichlet form E w on L 2 (X, w ) by We leave it to the reader to verify the following simple but fundamental properties.

Lemma 3
(i) E w is a strongly local, symmetric Dirichlet form.
Our first main result will provide the basic estimate for the Bakry-Émery condition under time change.
Theorem 4 Let w ∈ D loc ( ) ∩ L ∞ loc be given and assume that E satisfies BE loc (k, N) condition for some N ∈ [2, ∞) and k ∈ L ∞ loc . Then, for any N � ∈ (N, ∞] , any f ∈ TestF loc (E) and any non-negative ∈ ∞ with compact support, we have where Proof By Lemma 3 we know It can also be seen that Finally, we obtain which is the thesis. ◻

Theorem 5 (BE loc (k, N) condition under time change) Let E be a Dirichlet form satis-
fying the BE loc (k, N) condition for some N ∈ [2, ∞) and k ∈ L ∞ loc (X, ) . Assume that w ∈ D loc ( ) ∩ L ∞ loc with w = sing w + Δ ac w and sing w ≤ 0 . Moreover, assume that for some N � ∈ (N, ∞] and k � ∈ L ∞ loc it holds -a.e. on X. Then, the time-changed Dirichlet form E w on L 2 (X, w ) satisfies the BE loc (k � , N � ) condition.
In particular, if there are some N � ∈ (N, ∞] and K � ∈ ℝ such that -a.e. on X, we have the following gradient estimate for all f ∈ D(E w ).
Proof Given the estimate (9) from the previous theorem, we iteratively will extend the class of functions for which it holds true.
(i) Our first claim is that (9) holds for all f ∈ D loc (Δ) ∩ L ∞ loc with Δf ∈ loc and all compactly supported, nonnegative ∈ ∞ with Γ( ) ∈ L ∞ . Indeed, given such f and , choose f � ∈ D(Δ) ∩ L ∞ with Δf ∈ such that f = f � on a neighborhood of { ≠ 0} . Choose f n ∈ TestF(E) with f n → f ′ in D(Δ) and Δf n → Δf � in . (For instance, put f n = P 1∕n f � .) Applying (9) with f n in the place of f and passing to the limit n → ∞ yields the claim. Indeed, which according to Lemma 1 for n → ∞ converges to since f n → f in D(Δ).
(ii) Our next claim is that (9) holds for all f ∈ D loc (Δ w ) ∩ L ∞ loc ( w ) with Δ w f ∈ ( w ) ∞ loc and all compactly supported, nonnegative ∈ ( w ) ∞ with Γ w ( ) ∈ L ∞ ( w ) . Indeed, the conditions on f and on will not depend on w as long as w ∈ ∞ loc which is the case by assumption. This is obvious in the case of the conditions on . For the conditions on f, note that Δ w = e −2w Δ and Γ w (Δ w ) ≤ 2e −4w Γ(Δf ) + 4(Δf ) 2 Γ(w) .
(iii) Taking into account the assumptions on w and on k ′ , according to Lemma 2 together with Remark 4 the assertion of the second claim already proves BE loc (k � , N � ).

Time change and the Lott-Sturm-Villani condition
In this section, we will study synthetic lower Ricci bounds under time change in the setting of metric measure spaces. More precisely, we will derive the transformation formula for the curvature-dimension condition of Lott-Sturm-Villani under time change.

Metric measure spaces and time change
Assumption 2 In this section we will assume that the metric measure space (X, d, ) fulfils the following conditions: ) is a complete and separable geodesic space; (ii) is a d-Borel measure and supp = X; (iii) (X, d, ) satisfies the Riemannian curvature-dimension condition RCD(K, N) for some K ∈ ℝ and N ∈ [1, ∞).
Given such a metric measure space (X, d, ) , the energy is defined on L 2 (X, ) by where lip(f )(x) ∶= lim sup y→x |f (x) − f (y)|∕d(x, y) denotes the local Lipschitz slope and |Df |(x) denotes the minimal weak upper gradient at x ∈ X . We refer to [1] for details. As a part of the definition of RCD condition, E(⋅) is a quadratic form. By polarization, this defines a quasi-regular, strongly local, conservative Dirichlet form admitting a carré du champ Γ(f ) ∶= |Df | 2 . We use the notations W 1,2 (X, d, ) = = D(E) and S 2 (X, d, ) = e .

Definition 5
Given w ∈ L 2 loc (X, ) , the time-changed metric measure space is defined as (X, d w , w ) where w ∶= e 2w and d w is given by for any x, y ∈ X.

Remark 6
There are various alternative definitions for the distance function under time change. The first of them is It is easy to see that in both of these definitions, the class of functions under consideration can equivalently be restricted to those with compact supports. In other words, Proof Assume that w is continuous at x ∈ X . Then, for each > 0 there exists > 0 such that |w(x) − w(y)| < for y ∈ B (x) . Hence, by using appropriate truncation arguments it is easy to see that for each d * ∈ d w , d w , e w ⊙ d, ew ⊙ d and all y ∈ B (x) Hence, lip * (f )(x) = e −w(x) lip(x) for the respective local Lipschitz constants associated with d * .
To obtain the respective minimal weak upper gradient for f ∈ L 2 (X, ) associated with d * , one has to consider the relaxations of lip * (f ) w.r.t. the measure w = e 2w . This, however, amounts to study the relaxations of the original lip(f ) w.r.t. the measure . Thus, the claimed identify Γ w (f ) = e −2w Γ(f ) -a.e. on X follows. ◻ In the following lemma, we show the coincidence of d w and ew ⊙ d , see [17] for related results.

Lemma 6 Assume that w is continuous a.e. on X. Then,
In particular, d w is a geodesic metric.
Proof (i) Let us first prove that d w is a geodesic metric. Since X is locally compact w.r.t. the metric d and since the metrics d w and d are locally equivalent, the space X is also locally compact w.r.t. the metric d w . Therefore, it suffices to prove that d w is a length metric. Assume this is not the case. Then, there exist points x ≠ y with d w (x, y) < 2r and B w r (x) ∩ B w r (y) = � . Put It is easy to verify that Γ w (f ) ≤ 1 and obviously f is continuous. Hence, by the very definition of d w which is in contradiction to our initial assumption.
(ii) Now let us consider the particular case where w is continuous on all of X. Then, d w = e w ⊙ d . Indeed, both metrics are geodesic metrics on X and coincide up to multiplicative pre-factors e ± on suitable neighborhoods B (x) of each point z ∈ X.
(iii) To deal with the general case, let us choose a decreasing sequence of continuous functions w n with w n ↓w as n → ∞ . Then, d w n = e w n ⊙ d for each n by the preceding case ii) and thus by monotonicity for all x, y iv) To prove the reverse estimate, for given x ∈ X observe that f = (ew ⊙ d)(x, .) is continuous and obviously lip w (f )(y) ≤ 1 in each point y of continuity of w. Thus, in particular,

Lemma 7
Assume that w is continuous a.e. on X. Then, the metric measure space (X, d w , w ) has the Sobolev-to-Lipschitz property.
Proof Assume that f ∈ loc is given with Γ w (f ) ≤ 1 w -a.e. on X. By truncation one can achieve on each bounded set B that f = f B a.e. on B for some f B with bounded support and with Γ w (f B ) ≤ 1 w -a.e. on X. Since w ∈ L ∞ loc , moreover, Γ(f B ) ≤ C -a.e. on X. By the Sobolev-to-Lipschitz property of the original metric measure space (X, d, ) it follows that In particular, f B is continuous and Γ w (f B ) ≤ 1 w -a.e. on X. Hence, for all x, y ∈ X by the very definition of the metric d w . In other words, f B ∈ Lip w (X) with Lip w (f B ) ≤ 1 . Considering these constructions for an open covering of X by such sets B, it follows that there exists f ∈ Lip w (X) with f =f -a.e. on X and Lip w (f ) ≤ 1 . ◻ Finally, we can prove the transformation formula for the RCD (K, N) condition under time change.
Theorem 6 Let (X, d, ) be an RCD(K, N) space for some K ∈ ℝ and N ∈ [1, ∞) , and let w ∈ D loc ( ) ∩ L ∞ loc (X) be continuous -a.e. with w = sing w + Δ ac w and sing w ≤ 0 . Then, the time-changed metric measure space (X, d w , w ) satisfies the RCD(K � , N � ) condition for any N � ∈ (N, ∞) and K � ∈ ℝ such that -a.e. on X A particular consequence of the theorem is that the time-changed metric measure space (X, d w , w ) satisfies the squared exponential volume growth condition: ∃C ∈ ℝ, z ∈ X: Proof From the work of [3,10], we know that the curvature-dimension condition RCD(K, N) implies the Bakry-Émery condition BE(K, N) for the Dirichlet form E on L 2 (X, ) induced by the measure space (X, d, ) . According to Theorem 5, this implies the Bakry-Émery condition BE(K � , N � ) for the Dirichlet form E w on L 2 (X, w ) . Due to Lemma 5, the latter indeed is the Dirichlet form induced by the metric measure space (X, d w , w ) . Finally, again by [3,10], BE(K � , N � ) for the Dirichlet form induced by (X, d w , w ) will imply RCD * (K � , N � ) provided the volume-growth condition (16) is satisfied and the Sobolev-to-Lipschitz property holds. The latter was proven in Lemma 7. To deal with the former, we proceed in two steps. i) Let us first consider the case w ∈ L ∞ (X) . Then, the volume-growth condition (16) for (X, d w , w ) obviously fellows from that for (X, d, ) which in turn follows from the RCD(K, N) assumption.
ii) Now let general w ∈ L ∞ loc (X) be given as well as K ′ and N ′ such that (15) is satisfied. Given z ∈ X , define w l = w ⋅ with suitable cut-off functions ( ) ∈ℕ (cf. [4], Lemma 6.7) such that for all ∈ ℕ Then, according to part i) of this proof, the metric measure space (X, d w , w ) satisfies RCD(K � − 1, N � ) . This in particular implies that there exists a constant C (which indeed can be chosen independent of ) such that

3
for all r > 0 . Since w B w r (z) = w B w r (z) for all r ≤ , this finally proves the requested volume growth condition. ◻ It might be of certain interest to analyze the validity of the volume growth condition (16) under time change without referring to curvature bounds. We will still assume that w is -a.e. continuous.

Lemma 8
Suppose there exist non-negative p, q ∈ L ∞ loc (ℝ + ) with Then, (X, d w , w ) satisfies the squared exponential volume growth condition: ∃C ∈ ℝ, x 0 ∈ X: In particular, if q is bounded and lim r→∞ p(r) r 2 < ∞ , then (X, d w , w ) satisfies the squared exponential volume growth condition.
Proof From Lemma 6, we know for any y ∈ X . Since f −1 is strictly increasing, this implies Hence, Recall that the RCD(K, ∞) condition implies the squared exponential volume growth condition, so there exist M, c > 0 such that Note that (i) implies lim sup r→∞ 1 r f −1 (r) < ∞ . Hence, together with (ii), this implies the squared exponential volume growth condition for (X, ⋅, x 0 )) -a.e. on X.

Convexity transform
Firstly we introduce the notion of local -convexity (i.e., semi-convexity) in non-smooth setting. Such notion is derived from [18] by the second author and Lierl (see Definition 2.6 and Definition 2.9 therein).
Given ell < 0 , we say that Ω is locally -convex if for each > 0 there exists

Remark 7
Assume that X is a smooth Riemannian manifold, and Ω is a bounded open subset of X with smooth boundary. It is proved in Proposition 2.10 [18] that the real-valued second fundamental form on Ω is negative and bounded from below by if and only if Ω is locally -convex.

Remark 8
If Ω = {V < 0} for some 0-geodesically convex function V (i.e., a geodesically convex function), then Ω is geodesically convex in (X, d) . In order to study convex transform, we just need to analyze the 'pure concave' part of Ω . This is the reason why we require that −V is 0-convex in Definition 7. Moreover, this condition is necessary in the proof of Lemma 11 below.
Then, we can convexify locally -convex sets using time change and the following convexification technique developed in [18] (see Theorem 2.17 therein).
Next we recall some important results concerning L 1 -optimal transport and measure decomposition. This theory has proven to be a powerful tool in studying the fine structure of metric measure spaces. We refer the readers to the lecture note [6] for an overview of this topic and the bibliography. Lemma 10 (Localization for RCD(K, N) spaces, Theorem 3.8 and Theorem 5.1 [8]) Let (X, d, ) be an essentially non-branching metric measure space with supp = X , and satisfying RCD(K, N) condition for some K ∈ ℝ and N ∈ (1, ∞) . Then, for any 1-Lipschitz function u on X and the transport set u associated with u (up to -measure zero set, u coincides with {|∇u| = 1}), there is a disjoint family of unparameterized geodesics {X q } q∈ such that and there is a probability measure on such that Furthermore, for -a.e. q ∈ , q is a Radon measure with q ≪ H 1 | X q and (X q , d, q ) satisfies RCD (K, N) . In particular, q = h q H 1 | X q for some CD(K, N) probability density h q .

Lemma 11
Let Y be a locally -convex domain in X for some ell < 0, and ( Y) = 0 .
Then, the transport set V associated with the signed distance function V has full measure in X. There is a disjoint family of unparameterized geodesics {X q } q∈ satisfying (18) and (19) in Lemma 10, and a constant r 0 > 0 such that where a q = a q (X q ), b q = b q (X q ) are the end points of X q .
Proof Firstly, recall that RCD(K, N) condition yields local compactness, so for any for some r 0 > 0 . By the main theorem of [25], for each x 0 ∈ Y 0 −r 0 there exists a unique gradient flow for V (and −V ) starting in x 0 . In particular, there is a maximal transport (geodesic) line By Theorem 10 there is a disjoint family of unparameterized geodesics {X q } q∈ such that ( V ⧵ ∪ X q ) = 0 . In addition, X q ∩ {V ≤ −r 0 } ≠ � and X q ∩ {V ≥ 0} ≠ � for any q ∈ . Therefore (X⧵ ∪ X q ) = 0 , V(a q ) ≥ 0 and V(b q ) ≤ −r 0 . ◻
on ℝ . Then, we define w ∶= (− � V) . By Convexification Theorem (cf. Theorem 2.17 [18]) we know Ω is locally geodesically convex in (X, d w ). By chain rule (cf. Proposition 4.11 [13]) and Corollary 4.16 [9] we have w ∈ D( , X⧵ Ω) , and In addition, by Corollary 4.16 [9] and the fact �� ≤ − 2 Proof Let w be the reference function obtained in Proposition 5. Denote by the trivial extension of w| X⧵ Ω on whole X. To apply Lemma 6 and Proposition 5, it suffices to show that w ∈ D( ) and w ≤ .
Given an arbitrary non-negative Lipschitz function ∈ Lip(X, d) with bounded support. For any > 0 , there exists a Lipschitz function ∈ Lip(ℝ) satisfying  Since q = h q H 1 | X q for some CD(K, N) probability density h q , we know h q and (ln h q ) � are bounded. So for any X q such that Ω − ∕2 − ∩ X q ≠ � and Ω ∕2 ∩ X q ≠ � for small enough, we have Hence by Lemma 11 and Fatou's lemma, we obtain In conclusion, we obtain Therefore, w ≤ , by Proposition 5 we know sing w ≤ 0 and Δ ac w) + ∈ L ∞ . Then, by Theorem 6 we know (Ω, d w , w ) is a RCD(K � , N � ) space. ◻