Functional inequalities for the heat flow on time-dependent metric measure spaces

TL;DR

This paper establishes that synthetic lower Ricci bounds in time-dependent metric measure spaces can be characterized by various functional inequalities, providing a new framework for understanding super-Ricci flows.

Contribution

It extends the characterization of Ricci bounds via functional inequalities to the setting of time-dependent metric measure spaces and super-Ricci flows.

Findings

Equivalence of Ricci bounds and functional inequalities in static spaces

Extension of these equivalences to time-dependent spaces

Characterization of super-Ricci flows through these inequalities

Abstract

We prove that synthetic lower Ricci bounds for metric measure spaces -- both in the sense of Bakry-\'Emery and in the sense of Lott-Sturm-Villani -- can be characterized by various functional inequalities including local Poincar\'e inequalities, local logarithmic Sobolev inequalities, dimension independent Harnack inequality, and logarithmic Harnack inequality. More generally, these equivalences will be proven in the setting of time-dependent metric measure spaces and will provide a characterization of super-Ricci flows of metric measure spaces.

Equations

$$\displaystyle|f_{t}(x)-f_{s}(y)|\leq L|t-s|+Ld_{t}(x,y),\qquad\Big{|}\log\frac{d_{t}(x,y)}{d_{s}(x,y)}\Big{|}\leq L|t-s|.$$

$$(X,d_t,m_t)$$

$$-{\int }_{X}u{\Delta }_{t}vd{m}_t={\mathcal{E}}_t(u,v)={\int }_X{\Gamma }_t(u,v)d{m}_t\forall u\in \mathcal{D}({\mathcal{E}}_t),v\in \mathcal{D}({\Delta }_t).$$

$$e^{-2L|t-s|}{\Gamma }_s(u)\le {\Gamma }_t(u)\le e^{2L|t-s|}{\Gamma }_s(u),$$

$${\partial }_t{u}_t={\Delta }_t{u}_t\text{ on }(s,T)\times X\text{ with }{u}_s=u,$$

$${\partial }_s{v}_s={\Delta }_s^{\ast }{v}_s:=-{\Delta }_s{v}_s+v_s{\dot{f}}_s\text{ on }(0,t)\times X\text{ with }{v}_t=v.$$

$$\int ud({\hat{P}}_{t,s}\mu )=\int P_{t,s}ud\mu \forall u\in {\mathcal{C}}_b(X),\forall \mu \in \mathcal{P}(X).$$

$$\partial_{a}S_{t}(\mu^{a})\big{|}_{a=1}-\partial_{a}S_{t}(\mu^{a})\big{|}_{a=0}\geq-\frac{1}{2}\partial_{t}W_{t}^{2}(\mu^{0},\mu^{1}).$$

$$W_s({\hat{P}}_{t,s}\mu ,{\hat{P}}_{t,s}\nu )\le W_t(\mu ,\nu )$$

$$\Gamma_{t}(P_{t,s}u)\leq P_{t,s}\big{(}\Gamma_{s}(u)\big{)}$$

$${\mathbf{\Gamma }}_{2,r}(u_r)(g_r)\ge \frac{1}{2}\int {\stackrel{\bullet }{\Gamma }}_r(u_r)g_{r}d{m}_r$$

$${\bf\Gamma}_{2,r}(u_{r})(g_{r}):=\int\Big{[}\frac{1}{2}\Gamma_{r}(u_{r})\Delta_{r}g_{r}+(\Delta_{r}u_{r})^{2}g_{r}+\Gamma_{r}(u_{r},g_{r})\Delta_{r}u_{r}\Big{]}dm_{r}$$

$$\stackrel{{\scriptstyle\bullet}}{{\Gamma}}_{r}(u_{r}):=\mbox{w-}\lim_{\delta\to 0}\ \frac{1}{\delta}\Big{(}\Gamma_{r+\delta}(u_{r})-\Gamma_{r}(u_{r})\Big{)}$$

$${\mathrm{Ric}}_t\ge -\frac{1}{2}{\partial }_t{g}_t.$$

$$\int\Big{[}\Gamma_{t}(u)\Delta_{t}g+2(\Delta_{t}u)^{2}g+2\Gamma_{t}(u,g)\Delta_{t}u-\partial_{t}\Gamma_{t}(u)g\Big{]}dm_{t}\geq 0.$$

$$\big{(}\Gamma_{t}(P_{t,s}u)\big{)}^{1/2}\leq P_{t,s}\big{(}\Gamma_{s}(u)^{1/2}\big{)}$$

$$P_{t,s}(u^2)(x)-(P_{t,s}u{)}^2(x)\le$$

$$P_{t,s}(u^2)(x)-(P_{t,s}u{)}^2(x)\ge$$

$$P_{t,s}(u\log u)-P_{t,s}u\log P_{t,s}u\le$$

$$P_{t,s}(u\log u)-P_{t,s}u\log P_{t,s}u\ge$$

$$(P_{t,s}u{)}^{\alpha }(y)\le P_{t,s}(u^{\alpha })(x)\exp \left\{\frac{\alpha d_t^2(x,y)}{4(\alpha -1)(t-s)}\right\}.$$

$$P_{t,s}(\log u)(x)\le \log (P_{t,s}u)(y)+\frac{d_t^2(x,y)}{4(t-s)}.$$

$$\frac{{\Gamma }_t(P_{t,s}u)}{P_{t,s}u}\le P_{t,s}\left(\frac{{\Gamma }_s(u)}{u}\right),$$

$$\displaystyle\frac{\Gamma_{t}(P_{t,s}u)}{P_{t,s}u}\leq\frac{\big{(}P_{t,s}\sqrt{\Gamma_{s}(u)}\big{)}^{2}}{P_{t,s}u}\leq P_{t,s}\left(\frac{\Gamma_{s}(u)}{u}\right).$$

$$P_t(u^2)(x)-(P_{t}u{)}^2(x)\le$$

$$P_t(u^2)(x)-(P_{t}u{)}^2(x)\ge$$

$$P_t(u\log u)-P_{t}u\log P_{t}u\le$$

$$P_t(u\log u)-P_{t}u\log P_{t}u\ge$$

$$(P_{t}u{)}^{\alpha }(y)\le P_t(u^{\alpha })(x)\exp \left\{\frac{\alpha K{d}^2(x,y)}{2(\alpha -1)(1-e^{-2Kt})}\right\}.$$

$$P_t(\log u)(x)\le \log (P_{t}u)(y)+\frac{K{d}^2(x,y)}{2(1-e^{-2Kt})}.$$

$$-{\int }_s^{\tau }{\mathcal{E}}_r(u_r,w_r)dr={\int }_s^{\tau }⟨{\partial }_r{u}_r,w_r⟩dr$$

$${\int }_{\sigma }^t{\mathcal{E}}_s(v_s,w_s)ds+{\int }_{\sigma }^t\int v_s{w}_s{\partial }_s{f}_{s}d{m}_{s}ds={\int }_{\sigma }^t⟨{\partial }_s{v}_s,w_s⟩ds$$

Full text

More generally, these equivalences will be proven in the setting of time-dependent metric measure spaces and will provide a characterization of super-Ricci flows of metric measure spaces.

1. 1 Introduction

2. 1.1 Setting

3. 1.2 Summary of the main results

4. 1.3 Preliminaries

5. 2 The local and the reverse local Poincaré inequalities

6. 2.1 From $L^{2}$-gradient estimate to local and reverse local Poincaré inequalities

7. 2.2 From reverse local Poincaré inequality to dynamic Bochner inequality

8. 2.3 From local Poincaré inequality to dynamic Bochner inequality

9. 2.4 From dynamic Bochner inequality (‘integrated in time’) to dynamic Bochner inequality pointwise in time

10. 3 The local logarithmic Sobolev inequalities

11. 3.1 From $L^{1}$-gradient estimate to local logarithmic Sobolev inequality

12. 3.2 From local logarithmic Sobolev inequalities to dynamic Bochner inequality

13. 4 The dimension independent Harnack inequality

14. 4.1 From $L^{1}$-gradient estimate to dimension independent Harnack inequality

15. 4.2 From dimension independent Harnack inequality to local logarithmic Sobolev inequality

16. 5 The logarithmic Harnack inequality

1 Introduction

1.1 Setting

Huge research interest and extensive literature is devoted to the study of functional inequalities for the heat equation, both on Riemannian manifolds and on more abstract spaces. Of particular importance are functional inequalities which are equivalent to a uniform lower bound on the Ricci curvature, say $\mathrm{Ric}_{g}\geq K\cdot g$. In F.-Y. Wang’s monograph [22], Theorem 2.3.3., an impressive collection of 15 equivalent properties is listed.

In principle, all these properties and equivalences should hold – and indeed most of them do hold – in much more general settings. Many of them have been reformulated and proven in the setting of Markov diffusion semigroups and $\Gamma$-calculus, initiated by the seminal work of Bakry & Émery [6] and culminating now in the monograph [7] of Bakry, Gentil and Ledoux, see Theorems 4.7.2, 5.5.2, 5.5.5, 5.6.1 and Remark 5.6.2 in [7].

Another, more recent, important setting for the study of heat equations and functional inequalities are metric measure spaces, in particular, such mm-spaces which are infinitesimally Hilbertian and which satisfy a synthetic lower Ricci bound as introduced in the foundational works of Sturm [19] and Lott & Villani [16]. In a series of ground breaking papers, Ambrosio, Gigli & Savaré [2, 3, 4] introduced and analyzed the heat flow on such spaces and derived various functional inequalities. In particular, they proved that both the Bochner inequality (without dimensional term) and the $L^{2}$-gradient estimate are equivalent to the synthetic Ricci bound CD$(K,\infty)$; and they deduced the local Poincaré inequality and the logarithmic Harnack inequality. Savaré [18] extended the powerful self-improvement property of Bochner’s inequality to mm-spaces and utilized it to deduce the $L^{1}$-gradient estimate; based on the latter, H. Li [15] proved the dimension-independent Harnack inequality which in turn implies the logarithmic Harnack inequality.

Only recently, some of these properties and equivalences have been extended to the heat flow on time-dependent Riemannian manifolds, e.g. by Cheng & Thalmaier [9], Haslhofer & Naber [12], McCann & Topping [17], and Cheng [8]. The authors of the current paper had been the first to study the heat flow on time-dependent metric measure spaces [14], to introduce the time-dependent counterpart of synthetic lower Ricci bounds, and to derive various functional inequalities equivalent to it.

Here and throughout this paper, the setting will be as follows. $(X,d_{t},m_{t})_{t\in I}$ is a time-dependent metric measure space where $I=(0,T)$ and $X$ is a topological space. The Borel measures $m_{t}=e^{-f_{t}}m$ and the geodesic distances $d_{t}$ are assumed to be logarithmic Lipschitz continuous in time. Moreover, the maps $x\mapsto f_{t}(x)$ are assumed to be bounded and Lipschitz continuous. That is, there exists a constant $L>0$ such that for all $x,y\in X$ and $s,t\in I$

[TABLE]

Furthermore, for some $K\in\mathbb{R}$ and each $t\in I$ the static mm-space

[TABLE]

The static mm-space $(X,d_{t},m_{t})$ defines a Dirichlet form $\mathcal{E}_{t}$, a Laplacian $\Delta_{t}$, and a square field operators $\Gamma_{t}$ related to each other via

[TABLE]

The domains ${\mathcal{D}}(\mathcal{E}_{t})$ define Hilbert spaces with scalar products $\int uv\,dm_{t}+\mathcal{E}_{t}(u,v)$. Note that the scalar products are mutually equivalent, since we have uniform ellipticity by (A1.a), i.e.

[TABLE]

for some constant $C>0$, and for all $s,t\in I$. We fix an arbitrary $t$ and set $\mathcal{F}={\mathcal{D}}(\mathcal{E})$ as a reference Hilbert space. We have the dense and continuous embeddings $\mathcal{F}\subset L^{2}(X,m)\subset\mathcal{F}^{*}$, where $\mathcal{F}^{*}$ denotes the dual space of $\mathcal{F}$. Similarly $L^{p}(X)=L^{p}(X,m)$ will serve as a reference $L^{p}$-space.

The family of mm-spaces $(X,d_{t},m_{t})_{t\in I}$ defines a 2-parameter family of heat propagators $(P_{t,s})_{s\leq t}$ and adjoint propagators $(P^{*}_{t,s})_{s\leq t}$ on $L^{2}(X,m)$, see [14] for details. The heat flow $t\mapsto u_{t}=P_{t,s}u$ provides solutions to the heat equation

[TABLE]

whereas $s\mapsto P_{t,s}^{*}v$ provides solutions to the adjoint heat equation

[TABLE]

By duality, the propagator $(P_{t,s})_{s\leq t}$ acting on bounded continuous functions induces a dual propagator $(\hat{P}_{t,s})_{s\leq t}$ acting on probability measures as follows

[TABLE]

The main result of our previous paper is the characterization of super-Ricci flows of mm-spaces in terms of the heat flow on them. For $t\in I$, let $W_{t}$ denote the $L^{2}$-Kantorovich-Wasserstein metric with respect to $d_{t}$ and let $S_{t}(\mu)=\int\log(d\mu/dm_{t})\,d\mu$ denote the relative Boltzmann entropy with respect to $m_{t}$.

Theorem 1.1** ([14]).**

The following assertions are equivalent:

1. (i)

For a.e. $t\in(0,T)$ and every $W_{t}$-geodesic $(\mu^{a})_{a\in[0,1]}$ in $\mathcal{P}(X)$ with $\mu^{0},\mu^{1}\in{\mathcal{D}}(S)$

[TABLE] 2. (ii)

For all $0<s<t<T$ and $\mu,\nu\in\mathcal{P}(X)$

[TABLE] 3. (iii)

For all $u\in\mathcal{F}$ and all $0<s<t<T$

[TABLE] 4. (iv)

For all $0<s<t<T$ and for all $u_{s},g_{t}\in\mathcal{F}$ with $g_{t}\geq 0$, $g_{t}\in L^{\infty}(X,m)$, $u_{s}\in\mathrm{Lip}(X)$ and for a.e. $r\in(s,t)$

[TABLE]

where $u_{r}=P_{r,s}u_{s}$ and $g_{r}=P^{*}_{t,r}g_{t}$.

Here

[TABLE]

denotes the distribution valued $\Gamma_{2}$-operator (at time $r$) applied to $u_{r}$ and tested against $g_{r}$ and

[TABLE]

denotes any subsequential weak limit of \frac{1}{2\delta}\big{(}\Gamma_{r+\delta}-\Gamma_{r-\delta}\big{)}(u_{r}) in $L^{2}((s,t)\times X)$.

We say that a one-parameter family of mm-spaces $(X,d_{t},m_{t})_{t\in I}$ is a super-Ricci flow – or that it evolves as a super-Ricci flow – if it satisfies one/each assertion of the previous Theorem. This is a canonical extension of the notion of super-Ricci flows of Riemannian manifolds $(M,g_{t})$ defined through the tensor inequality

[TABLE]

Property (i) above is called dynamic convexity of the Boltzmann entropy. This concept has been introduced by the second author in [20]; it provides a canonical generalization of the synthetic Ricci bound CD$(0,\infty)$ defined in terms of the semiconvexity of the Boltzmann entropy in the static setting.

Property (iv) is the appropriate generalization of Bochner’s inequality or, in other words, of the Bakry-Émery condition to the time-dependent setting. It will be called dynamic Bochner inequality (integrated in time).

In contrast to that, we say that the dynamic Bochner inequality pointwise in time holds if $\forall t\in I$, $\forall u,g\in{\mathcal{D}}(\Delta)\cap L^{\infty}(X,m)$ with $\Gamma_{t}(u)\in L^{\infty}(X,m)$ and $g\geq 0$

[TABLE]

In the static case, Bochner’s inequality has the remarkable and powerful ‘self-improvement property’ which allows to deduce improved versions of the assertions in the previous Theorem, in particular, to derive the $L^{1}$-gradient estimate. This self-improvement strategy in the time-dependent case requires additional time regularity of the involved quantities. It was carried out by the first author in [13] and can be reformulated with the notation from the current paper as follows.

Theorem 1.2** ([13]).**

Assume (A2.a+c), see Section 2. Then the $L^{2}$-gradient estimate (E3) is equivalent to the $L^{1}$-gradient estimate: for all $u\in F$ and all $0<s<t<T$

[TABLE]

Moreover, the dynamic Bochner inequality (integrated in time) implies the dynamic Bochner inequality pointwise in time which in turn implies the $L^{1}$-gradient estimate as formulated above.

Additional assumptions on time regularity (e.g. continuity of $t\mapsto\Delta_{t}P_{t,s}u$ in appropriate spaces) will be also requested for various results of the current paper; we will formulate these assumptions tailor-made in the subsequent sections.

1.2 Summary of the main results

Let us summarize the main results of the current paper. To simplify and unify the presentation here in the introduction, we will restrict ourselves to the case $m_{t}(X)<\infty$ and in addition to our standing assumptions (A1.a+b) we will request now all the assumptions which ever will be made in the sequel. Besides our standing assumptions (A1.a+b), these are assumptions (A2.a-c) formulated in Section 2, (A3) formulated in Section 3, and assumptions (A5.a+b) formulated in Section 5. We emphasize that all these extra assumptions are always fulfilled in the static case and they are also satisfied in the case of Riemannian manifolds with metric tensors which smoothly depend on time.

Theorem 1.3**.**

Under the previously mentioned assumptions, the following assertions are equivalent:

1. (i)

$(X,d_{t},m_{t})_{t\in I}$* is a super-Ricci flow.* 2. (ii)

One/each of the local Poincaré inequalities holds

[TABLE] 3. (iii)

One/each of the local logarithmic Sobolev inequalities holds

[TABLE] 4. (iv)

The dimension independent Harnack inequality holds for one/each $\alpha\in(1,\infty)$

[TABLE] 5. (iv)

The logarithmic Harnack inequality holds

[TABLE]

The formulation “one/each” in particular means that one of the respective properties implies each of the respective properties.

Remark 1.4**.**

a) Upper and lower local Poincaré inequalities together obviously imply the $L^{2}$-gradient estimate (E3). Upper and lower local logarithmic Sobolev inequality together imply

[TABLE]

which is a prioiri weaker than the $L^{1}$-gradient estimate (E6). Indeed the $L^{1}$-gradient estimate together with Jensen’s inequality applied to the function $\beta(z,w)=z^{2}/w$ imply

[TABLE]

b) The dimension independent Harnack inequality for $\alpha_{1}$ and for $\alpha_{2}$ implies the dimension independent Harnack inequality for $\alpha_{1}\cdot\alpha_{2}$, [22], Thm. 1.4.2. The dimension independent Harnack inequality for a sequence $\alpha_{n}\to\infty$ implies the log-Harnack inequality. In particular, the dimension independent Harnack inequality for some $\alpha\in(1,\infty)$ implies the dimension independent Harnack inequality for all $k\alpha,k\in{\mathbb{N}}$, and thus the log-Harnack inequality, [22], Cor. 1.4.3.

The proof of the above theorem will be presented in the subsequent sections, decomposed into a variety of theorems devoted to individual implications. In these theorems, we also specify in detail the spaces of functions $u$ for which the respective inequalities are supposed to hold. In Section 2 we prove the implications (E3) $\Rightarrow$ (E7) $\Rightarrow$ (E4) and (E3) $\Rightarrow$ (E8) $\Rightarrow$ (E4) as well as the implication (E4) $\Rightarrow$ (E5). Section 3 is devoted to the proof of the implications (E6) $\Rightarrow$ (E9) $\Rightarrow$ (E5) and (E6) $\Rightarrow$ (E10) $\Rightarrow$ (E5). In Section 4 we prove the implications (E6) $\Rightarrow$ (E11) $\Rightarrow$ (E10) and in Section 5 the implication (E12) $\Rightarrow$ (E5). This completes the proof of our theorem since (E11) $\Rightarrow$ (E12) according to the previous remark, (E5) $\Rightarrow$ (E6) according to Theorem 1.2, and trivially (E6) $\Rightarrow$ (E3).

The previous characterizations of super-Ricci flows easily extend to characterizations of $K$-super-Ricci flows for any $K\not=0$ by considering reparametrized mm-spaces $(X,\tilde{d}_{t},\tilde{m}_{t})_{t\in\tilde{I}}$ with $\tilde{d}_{t}=e^{-K\tau(t)}d_{\tau(t)}$, $\tilde{m}_{t}=m_{\tau(t)}$, and $\tilde{I}=\{t:\tau(t)\in I,2Kt<C\}$ where $C\in\mathbb{R}$ and $\tau(t)=-\frac{1}{2K}\log(C-2Kt)$, see Theorem 1.11 in [14]. Let us restrict ourselves to formulate this in the most simple case of static mm-spaces.

Corollary 1.5**.**

Let $(X,d,m)$ be a mm-space satisfying the RCD$(-L,\infty)$ condition for some constant $L>0$. Then the following assertions are equivalent:

1. (i)

$(X,d,m)$* satisfies RCD$(K,\infty)$.* 2. (ii)

One/each of the local Poincaré inequalities holds

[TABLE] 3. (iii)

One/each of the local logarithmic Sobolev inequalities holds

[TABLE] 4. (iv)

The dimension independent Harnack inequality holds for one/each $\alpha\in(1,\infty)$

[TABLE] 5. (v)

The logarithmic Harnack inequality holds

[TABLE]

Remark 1.6**.**

So far, in the setting of mm-spaces only the implications (i) $\Rightarrow$ (iib), (i) $\Rightarrow$ (iiib) (i) $\Rightarrow$ (v), and (i) $\Rightarrow$ (iv) were known (Thm. 6.8 in [1], Cor. 4.4 in [21], Lemma 4.6 in [4], and Thm. 3.1 in [15]). The implications (i) $\Rightarrow$ (iia) and (i) $\Rightarrow$ (iiib) are new also in the static case. In particular, none of the reverse implications (iia) $\Rightarrow$ (i), (iib) $\Rightarrow$ (i), (ii), (iii), (iv), or (v) $\Rightarrow$ (i) was proven before for mm-spaces.

Also so far, for the implication (v) $\Rightarrow$ (i) no proof exists in the setting of $\Gamma$-calculus for diffusion semigroups.

1.3 Preliminaries

Let us recall some basic properties of the heat propagators $P_{t,s}$ and their adjoints $P^{*}_{t,s}$, see Section 3 in [14]. We call $u$ a solution to the heat equation on $(s,\tau)\times X$ if $u\in L^{2}((s,\tau);\mathcal{F})\cap H^{1}((s,\tau);\mathcal{F}^{*})$ and

[TABLE]

for all $w\in L^{2}((s,\tau);\mathcal{F})$, where $\langle\cdot,\cdot\rangle$ denotes the dual pairing between $\mathcal{F}$ and $\mathcal{F}^{*}$. Note that the solution $u$ lies in $\mathcal{C}([s,\tau];L^{2}(X))$ so that the values at $t=s$ and $t=\tau$ exist. For all $h\in L^{2}(X)$ there exists a unique solution $u_{t}=P_{t,s}h$ with $u_{s}=h$.

We call $v$ a solution to the adjoint heat equation on $(\sigma,t)\times X$ if $v\in L^{2}((\sigma,t);\mathcal{F})\cap H^{1}((\sigma,t);\mathcal{F}^{*})$ and

[TABLE]

for all $w\in L^{2}((s,\tau);\mathcal{F})$. Again the solution $v$ lies in $\mathcal{C}([\sigma,t];L^{2}(X,m))$. For each $g\in L^{2}(X)$ there exists a unique solution $v_{s}=P_{t,s}^{*}g$ with $v_{t}=g$.

The relation between the heat flow and its adjoint is given by

[TABLE]

We further collect the following properties from [14].

Lemma 1.7** ([14], Prop. 2.14).**

For all $u\in L^{2}(X,m)$ and all $s<t$, $p\in[1,\infty)$

$u\geq 0\Longrightarrow P_{t,s}u\geq 0$,  $u\leq M\Longrightarrow P_{t,s}u\leq M$. 2. 2.

$v\geq 0\Longrightarrow P^{*}_{t,s}v\geq 0$,  $v\leq M\Longrightarrow P^{*}_{t,s}v\leq Me^{L(t-s)}$. 3. 3.

$\|P_{t,s}u\|_{L^{p}(m_{t})}\leq e^{L(t-s)/p}\cdot\|u\|_{L^{p}(m_{s})}$,   $\|P^{*}_{t,s}v\|_{L^{p}(m_{s})}\leq e^{L(t-s)(1-1/p)}\cdot\|v\|_{L^{p}(m_{t})}$.

These estimates allow to extend the propagators $P_{t,s}$ and their adjoints $P^{*}_{t,s}$ in the canonical way from operators on $L^{2}(X,m)$ to operators on $L^{p}(X,m)$ for any $p\in[1,\infty]$.

Proposition 1.8** ([14], Theorem 2.12).**

The following properties hold.

Let $u_{t}=P_{t,s}u$. Then $u_{t}\in{\mathcal{D}}(\Delta_{t})$ for a.e. $t>s$ and if $u_{s}\in\mathcal{F}$

[TABLE]

where $s<\tau<T$ and $C>0$ only depends on the Lipschitz constants of $t\mapsto f_{t}$ and $t\mapsto\log d_{t}$. Moreover

[TABLE]

in $L^{2}(X)$ for a.e. $t>s$. 2. 2.

Let $v_{s}=P^{*}_{t,s}v$. Then $v_{s}\in{\mathcal{D}}(\Delta_{s})$ for a.e. $s<t$ and if $v_{t}\in\mathcal{F}$

[TABLE]

where $0<\sigma<t$ and $C>0$ only depends on the Lipschitz constants of $t\mapsto f_{t}$ and $t\mapsto\log d_{t}$. Moreover

[TABLE]

in $L^{2}(X)$ for a.e. $s<t$.

2 The local and the reverse local Poincaré inequalities

For later purposes it will be convenient to present the notion of semigroup mollification introduced in [4, Sec. 2.1].

Definition 2.1**.**

Let $t\in(0,T)$ and $\kappa\in\mathcal{C}_{c}^{\infty}(0,\infty)$ with $\kappa\geq 0$ and $\int_{0}^{\infty}\kappa(r)\,dr=1$. Let $(H^{t}_{r})_{r\geq 0}$ denote the heat semigroup in the static mm-space $(X,d_{t},m_{t})$. For $\varepsilon>0$ and $\psi\in\mathcal{F}\cap L^{\infty}(X)$ we define

[TABLE]

It is immediate to verify that $\psi_{\varepsilon},\Delta_{t}\psi_{\varepsilon}\in{\mathcal{D}}(\Delta_{t})\cap\mathrm{Lip}_{b}(X)$ and $\psi_{\varepsilon}\to\psi$ in $\mathcal{F}$ as $\varepsilon\to 0$, see e.g. [4, Sec 2.1].

2.1 From $L^{2}$-gradient estimate to local and reverse local Poincaré inequalities

Theorem 2.2**.**

Suppose that for all $u\in\mathcal{F}$ and all $s<t$ the $L^{2}$-gradient estimate

[TABLE]

holds. Then we have for all $u\in\mathcal{F}$, $s<t$

[TABLE]

and for all $u\in L^{2}(X)$, $s<t$

[TABLE]

In particular, for $u\in L^{2}(X)\cap L^{\infty}(X)$

[TABLE]

Proof.

Let $u=u_{s}$ and $g=g_{t}$ be both elements in $\mathcal{F}\cap L^{\infty}(X)$ and consider on $(s,t)\times X$ the solutions to the heat equation and adjoint heat equation

[TABLE]

Due to Proposition 1.8 we have $u_{r},g_{r}\in H^{1}((s,t);L^{2}(X))$. Since $u_{r},g_{r}\in H^{1}((s,t);L^{2}(X))$ and $e^{-f_{r}}\in\mathrm{Lip}((s,t);L^{\infty}(X))$ we deduce that the function $r\mapsto\int u_{r}^{2}g_{r}\,dm_{r}$ is locally absolutely continuous. The almost everywhere derivative can be computed as

[TABLE]

where the last equality holds since $\frac{g_{r+h}-g_{r}}{h}\to\partial_{r}g_{r},\frac{u_{r+h}-u_{r}}{h}\to\partial_{r}u_{r}$ in $L^{2}(X)$ for almost every $r$ and since the mapping $z\mapsto z^{2}\in\mathcal{C}^{2}({\mathbb{R}})$.

Then, by the defining properties of the heat equation (3), (4)

[TABLE]

This proves

[TABLE]

Applying (6) to $\Gamma_{r}(P_{r,s}u)$ on the right hand side gives

[TABLE]

and applying (6) to $P_{t,r}\Gamma_{r}$ gives

[TABLE]

Since $g$ is arbitrary, this proves the first two claims of the theorem in the case of bounded $u\in\mathcal{F}$. The claim (8) for bounded $u\in L^{2}(X)$ follows by applying the latter estimate with $s+\delta$ in the place of $s$ to the function $P_{s+\delta,s}u$ as $\delta\to 0$, which lies in $\mathcal{F}$ and from $\int gP_{t,s+\delta}((P_{s+\delta,s}u)^{2})dm_{t}\to\int gP_{t,s}(u^{2})dm_{t}$ which in turn is a consequence of the continuity of $\delta\mapsto P_{t,s+\delta}^{*}g$ and of $\delta\mapsto P_{s+\delta,s}u$ in $L^{2}$ and the uniform boundedness of the latter in $L^{\infty}$.

Thanks to the monotonicity (w.r.t. $C\mapsto u\wedge C$ or $C\mapsto u\vee-C$) of all the involved quantities, the claims for unbounded $u$ will follow by a simple truncation argument. Indeed, $u\wedge C\vee-C\to u$ in $L^{2}$ and thus, since $g$ is bounded, $\int g(P_{t,s}u\wedge C\vee-C)^{2}dm_{t}\to\int g(P_{t,s}u)^{2}dm_{t}$ as well as $\int(u\wedge C\vee-C)^{2}P_{t,s}^{*}g\,dm_{s}\to\int u^{2}P_{t,s}^{*}g\,dm_{s}$. Moreover, under the heat flow the initial $L^{2}$-convergence will be improved to a $\mathcal{F}$-convergence. Thus

[TABLE]

Finally, for the remaining term it suffices to observe that

[TABLE]

∎

2.2 From reverse local Poincaré inequality to dynamic Bochner inequality

Theorem 2.3**.**

*Suppose that the reverse local Poincaré inequality holds:

for all $s<t$ and for all $u\in\mathcal{F}\cap L^{\infty}(X)$*

[TABLE]

*Then the dynamic Bochner inequality (E4) holds true (‘integrated in time’):

$\forall S,T\in I$, $\forall u,g\in\mathcal{F}$ with $g\in L^{\infty}(X)$, $u\in\mathrm{Lip}(X)$ and for a.e. $q\in(S,T)$*

[TABLE]

where $u_{q}:=P_{q,S}u,g_{q}=P^{*}_{T,q}g$.

Proof.

Given $u\in\mathcal{F}\cap L^{\infty}(X)$ and nonnegative $g\in L^{1}(X)\cap L^{\infty}(X)$ we have shown in (10) that for all $s<t$

[TABLE]

Approximation by truncated $u$’s easily allows to extend the assertion to all $u\in\mathcal{F}$. The local Poincaré inequality, therefore, implies

[TABLE]

Now let us fix $S,T\in I$ and choose $g_{T},u_{S}\in\mathcal{F}$ with $g_{T}\in L^{\infty}$ and $u_{S}\in\mathrm{Lip}(X)$. Given $s,t$ with $S<s<t<T$, we put

[TABLE]

and apply the previous estimate with $g_{t},u_{s}$ in the place of $g,u$. Then

[TABLE]

where we defined

[TABLE]

Following the proof of Theorem 5.7 in [14] we have

[TABLE]

and hence

[TABLE]

Since this holds for all $(s,t)\subset(S,T)$, it implies (by Lebesgue’s density theorem) that

[TABLE]

for a.e. $q\in(S,T)$. This is the claim, namely the dynamic Bochner inequality (E4). ∎

2.3 From local Poincaré inequality to dynamic Bochner inequality

For the proof of the following implication, we will make the additional a priori assumption that

[TABLE]

for each $u\in\mathrm{Lip}(X)$. Note that this assumption is always fullfilled in the time-independent case thanks to the RCD$(K,\infty)$-condition as one of our standing assumptions.

Theorem 2.4**.**

*Suppose (A2.a) and that the local Poincaré inequality holds:

for all $s<t$ and for all $u\in\mathcal{F}\cap L^{\infty}(X)$*

[TABLE]

Then the dynamic Bochner inequality (E4) holds true (‘integrated in time’).

Proof.

The proof is very similar to that of the previous theorem. Now the a priori assumption is required to guarantee appropriate integrability of the involved quantities (which in the previous case was a simple consequence of the assumption, cf. estimate (9)).

Then, as in the proof of Theorem 2.3, the local Poincaré inequality implies

[TABLE]

where $\Psi$ is defined in (11). Consequently, arguing as in the proof of Theorem 2.3,

[TABLE]

Again by Lebesgue’s density theorem this implies that

[TABLE]

for a.e. $q\in(S,T)$. ∎

2.4 From dynamic Bochner inequality (‘integrated in time’) to dynamic Bochner inequality pointwise in time

In addition to our standing assumptions, let us now assume that

• •

the domains ${\mathcal{D}}(\Delta_{t})$ are independent of $t\in(0,T)$ and for $u,g\in{\mathcal{D}}(\Delta)$ with $\Delta_{t}u,\Delta_{t}g\in L^{\infty}(X)$ the functions

[TABLE]

are continuous in $L^{2}(X)$ and bounded in $L^{\infty}(X)$;

• •

for $u\in\mathcal{F}$ the function $\partial_{s}\Gamma_{s}(u)$ exists in $L^{1}(X)$ and the map

[TABLE]

is continuous in $L^{1}(X)$.

Note that all these assumptions are trivially satisfied in the static case.

Lemma 2.5**.**

The assumption (A2.b) implies that for $u,g\in{\mathcal{D}}(\Delta)$ with $\Delta_{t}u,\Delta_{t}g\in L^{\infty}(X)$ the functions

[TABLE]

*are continuous in $\mathcal{F}$. *

Proof.

This follows from integration by parts. ∎

Theorem 2.6**.**

*Under the previous assumptions, the dynamic Bochner inequality (E4) implies the following ‘dynamic Bochner inequality pointwise in time’:

$\forall t\in I$, $\forall u,g\in{\mathcal{D}}(\Delta)\cap L^{\infty}(X)$ with $\Gamma_{t}(u)\in L^{\infty}(X)$ and $g\geq 0$*

[TABLE]

Proof.

Given $t\in I$, $u,g\in{\mathcal{D}}(\Delta)\cap L^{\infty}(X)$ with $\Gamma_{t}(u),\Delta_{t}u,\Delta_{t}g\in L^{\infty}(X)$ and $g\geq 0$, choose $s<t$ and define $u_{q,s}:=P_{q,s}u,g_{q}=P^{*}_{t,q}g$ for $q\in[s,t]$. Then the dynamic Bochner inequality in its integrated version and (A2.c) imply that the function

[TABLE]

is nonnegative for a.e. $q$. Moreover, according to (A2.b), Lemma 2.5 and (A2.c), this function is continuous. Thus, in particular, it is nonnegative for $q=s$, i.e.

[TABLE]

Now finally we consider the limit $s\to t$ which implies $P^{*}_{t,s}g\to g$ in $L^{2}(X)$ as well as $\Delta_{s}P^{*}_{t,s}g\to\Delta_{t}g$ by (A2.b). According to Lemma 2.5, $P^{*}_{t,s}g\to g$ in $\mathcal{F}$. Therefore,

[TABLE]

To obtain the estimate for general $u,g$, we approximate them using the static $(X,d_{t},m_{t})$-heat semigroup mollifier from Definition 2.1. ∎

3 The local logarithmic Sobolev inequalities

3.1 From $L^{1}$-gradient estimate to local logarithmic Sobolev inequality

Theorem 3.1**.**

Suppose that the $L^{1}$-gradient estimate

[TABLE]

holds for every $s<t$, $u\in\mathcal{F}$ and $m$-a.e.. Then for every $s<t$ and $u\geq 0$ such that $u\in{\mathcal{D}}(S)$ and $\sqrt{u}\in\mathcal{F}$ we have $m$-a.e. on $X$

[TABLE]

Estimate (15) holds more generally for all nonnegative $u\in{\mathcal{D}}(S)\cap L^{1}(X)$.

Proof.

Define for $s<r<t$, $g\in L^{1}(X)\cap L^{\infty}(X)\cap\mathcal{F}$ such that $g\geq 0$ and $u\in{\mathcal{D}}(S)\cap L^{\infty}(X)$ such that $M\geq u\geq 0$ for some constant $M$ and $\sqrt{u}\in\mathcal{F}$

[TABLE]

where $g_{r}=P_{t,r}^{*}g$ and $u_{r}=P_{r,s}u$ and where $\psi_{\varepsilon}(z)\colon[0,\infty)\to\mathbb{R}$ by setting $\psi_{\varepsilon}^{\prime}(z)=\log(z+\varepsilon)+1$ and $\psi_{\varepsilon}(0)=0$.

Since $g_{t},u_{s}\in\mathcal{F}$ we have by virtue of Proposition 1.8 that $g_{r},u_{r}\in H^{1}((s,t);L^{2}(X))$. Since $g_{r},u_{r}\in H^{1}((s,t);L^{2}(X))$ and $e^{-f_{r}}\in\mathrm{Lip}((s,t);L^{\infty}(X))$, we deduce that the map $r\mapsto\Psi_{\varepsilon}(r)$ is locally absolutely continuous. Then, since $\psi_{\varepsilon}^{\prime}\in\mathrm{Lip}_{b}([0,M])$, we compute similarly as in the proof of Theorem 2.2

[TABLE]

Using the Cauchy-Schwarz inequality and (13) we find for the integrand

[TABLE]

Integration over $(s,t)$ yields

[TABLE]

Since $u\in{\mathcal{D}}(S)$ we have by Proposition 2.8 in [14] that $P_{t,s}u\in{\mathcal{D}}(S)$ and we find by dominated convergence that the left hand side converges as $\varepsilon\to 0$ to

[TABLE]

while by monotone convergence the right hand side converges to

[TABLE]

and hence

[TABLE]

By taking $u^{n}:=u\wedge n$ and letting $n\to\infty$ we obtain (17) for general $u\in{\mathcal{D}}(S)$ with $\sqrt{u}\in\mathcal{F}$, since $u^{n}\to u$ and $P_{t,s}u^{n}\to P_{t,s}u$ in $L^{1}(X)$, and $\Gamma(u^{n})=\Gamma(u)1_{\{u<n\}}$ a.e..

Since $g$ is arbitrary we find for a.e. $x\in X$

[TABLE]

To obtain the reverse bound (15) we apply Jensen’s inequality to the functions $\eta(z)=z^{2}$ and $\beta(z,w)=z^{2}/w$, which amounts to

[TABLE]

A similar argumentation as above yields the desired estimate. ∎

3.2 From local logarithmic Sobolev inequalities to dynamic Bochner inequality

For this subsection we will additionally assume that (A2.a-c) hold. Moreover, we assume that $m_{t}(X)<\infty$ for some (hence all) $t\in(0,T)$ and that

• •

for all fixed $s\in(0,T)$ and all $u\in{\mathcal{D}}(\Delta)\cap L^{\infty}(X)$ such that $\Delta_{s}u\in L^{\infty}(X)$

[TABLE]

Note that (A3) is always satisfied for the usual heat flow $(P_{t})_{t\geq 0}$ on RCD$(K,\infty)$-spaces, see Lemma 5.3.

We show the following.

Theorem 3.2**.**

Assume that one of the local logarithmic Sobolev inequalities, (14) or (15), holds. Then the pointwise dynamic Bochner holds for $t$, i.e. for all $v\in{\mathcal{D}}(\Delta_{t})\cap L^{\infty}(X)$ such that $\Gamma_{t}(v)\in L^{\infty}(X)$ and all $g\in{\mathcal{D}}(\Delta_{t})\cap L^{\infty}(X)$ with $g\geq 0$ holds

[TABLE]

Proof.

Let $v,\Delta_{t}v\in{\mathcal{D}}(\Delta_{t})\cap\mathrm{Lip}_{b}(X)$. Define $u=e^{v}$. Then $u\in{\mathcal{D}}(S)\cap\mathrm{Lip}_{b}(X)\cap{\mathcal{D}}(\Delta_{t})$ with $\Delta_{t}u\in L^{\infty}(X)\cap\mathcal{F}$ and there exists constants $M,c$ such that $M\geq u\geq c>0$. Let $g\in{\mathcal{D}}(\Delta_{t})\cap L^{\infty}(X)$ with $g\geq 0$.

Then we know from the proof of Theorem 3.1 that

[TABLE]

Together with (14) we find

[TABLE]

where $u_{r}=P_{r,s}u$ and $g_{r}=P_{t,r}^{*}g$.

We now claim that the map

[TABLE]

is absolutely continuous. To this end we compute for a.e. $r_{1},r_{2}\in(s,t)$ with $r_{1}<r_{2}$

[TABLE]

where we applied (2) for the third term on the right hand side.

The first three terms are finite by virtue of (A2.a) and Proposition 1.8. For the last one we further compute

[TABLE]

This proves absolute continuity and together with (19) we obtain

[TABLE]

The almost everywhere derivative is given by

[TABLE]

where we used $u\geq c$, $u_{q},g_{q}\in H^{1}((s,t);L^{2}(X))$, Proposition 1.8, (A3), (A2.a-c), and Lemma 2.5.

Together with (20) we get

[TABLE]

Define

[TABLE]

We want to show that $\Phi\colon[s,t]\to\mathbb{R}$ defines a continuous function. In order to do so, we consider each term separately.

The first term $q\mapsto\int\Delta_{q}(g_{q})\frac{\Gamma_{q}(u_{q})}{u_{q}}\,dm_{q}$ is continuous since $q\mapsto\Delta_{q}g_{q}$, and $q\mapsto u_{q}^{-1}$ are continuous in $L^{2}(X)$ by (A2.b) and (3), $q\mapsto\Gamma_{q}(u_{q})$ is weak∗ continuous in $L^{\infty}(X)$ by Lemma 2.5 and (A2.a).

The second term $q\mapsto\int\Delta_{q}(u_{q})\frac{\Gamma_{q}(u_{q})}{u^{2}_{q}}g_{q}\,dm_{q}$ is continuous since $q\mapsto\Delta_{q}u_{q}$ is continuous in $L^{2}(X)$ by (A2.b), $q\mapsto\frac{g_{q}}{u_{q}^{2}}$ is continuous in $L^{\infty}(X)$ by (3) and (A3), and $q\mapsto\Gamma_{q}(u_{q})$ is weak∗ continuous in $L^{\infty}(X)$ by Lemma 2.5 and (A2.a).

The third term $q\mapsto\int(\Delta_{q}u_{q})^{2}\frac{g_{q}}{u_{q}}\,dm_{q}$ is continuous since $q\mapsto\Delta_{q}u_{q}$ is continuous in $L^{2}(X)$ by (A2.b), and $q\mapsto\Gamma_{q}(u_{q})$ is weak∗ continuous by (A2.a) and Lemma 2.5, and $q\mapsto\frac{g_{q}}{u_{q}}$ are continuous in $L^{\infty}(X)$ by (A3).

The fourth term $q\mapsto\int\Gamma_{q}(u_{q},\frac{g_{q}}{u_{q}})\Delta_{q}u_{q}\,dm_{q}$ is continuous since $q\mapsto\Delta_{q}u_{q}$ is continuous in $L^{2}(X)$ and weak∗-continuous in $L^{\infty}(X)$ by (A2.b), and $q\mapsto\Gamma_{q}(u_{q},\frac{g_{q}}{u_{q}})$ is continuous in $L^{1}(X)$ by (A3.a) and Lemma 2.5.

The last term $q\mapsto\int\frac{g_{q}}{u_{q}}(\partial_{q}\Gamma_{q})(u_{q})\,dm_{q}$ is continuous since $q\mapsto\frac{g_{q}}{u_{q}}$ is continuous in $L^{\infty}(X)$ by (A3) and $q\mapsto(\partial_{q}\Gamma_{q})(u_{q})$ is continuous in $L^{1}(X)$ by (A2.c) and $q\mapsto e^{-f_{q}}$ is continuous in $L^{\infty}(X)$.

Then it holds by Lebesgue differentiation

[TABLE]

where we used the chain rule in the last equation.

Similarly as before we let $s\to t$ and obtain after choosing $\tilde{g}=e^{-v}g\in{\mathcal{D}}(\Delta_{t})\cap L^{\infty}(X)$ and obtain recalling $u=e^{v}$

[TABLE]

for all $v,\Delta_{t}v\in{\mathcal{D}}(\Delta_{t})\cap\mathrm{Lip}_{b}(X)$ and $\tilde{g}\in{\mathcal{D}}(\Delta_{t})\cap L^{\infty}(X)$ with $\tilde{g}\geq 0$. The result for general $v\in{\mathcal{D}}(\Delta_{t})\cap L^{\infty}(X)$ such that $\Gamma_{t}(v)\in L^{\infty}(X)$ and all $\tilde{g}\in{\mathcal{D}}(\Delta_{t})\cap L^{\infty}(X)$ with $\tilde{g}\geq 0$ follows by approximation with the semigroup mollifier from Definition 2.1.

Similarly one deduces Bochner from the reverse local logarithmic Sobolev bound. Indeed by (15) it holds by the same argument as above

[TABLE]

and since $q\mapsto\int g_{q}u_{q}\Gamma_{q}(\log u_{q})\,dm_{q}$

[TABLE]

which is the same as in line (20). ∎

4 The dimension independent Harnack inequality

4.1 From $L^{1}$-gradient estimate to dimension independent Harnack inequality

This section will be devoted to derive the following result.

Theorem 4.1**.**

Fix $\alpha>1$. Suppose that the $L^{1}$-gradient estimate (13) holds. Then for all $u\in L^{2}(X)$ such that $u\geq 0$, $t>s$ and $m$-a.e. $x,y\in X$ we have

[TABLE]

Before starting with the proof of this results, let us recall the notion of regular curves as introduced in [4] and refined in [5], as well as the notion of velocity densities taken from [5]. A curve $(\mu_{r})_{r\in[0,1]}$ with $\mu_{r}=\rho_{r}m$ is called regular if the following are satisfied:

• •

$\mu\in\mathrm{Lip}([0,1];(\mathcal{P}_{2}(X),W))\cap\mathcal{C}^{1}([0,1];L^{1}(X))$

• •

There exists a constant $R>0$ such that $\rho_{r}\leq R$ $m$-a.e. for every $s\in[0,1]$

• •

$\sqrt{\rho_{r}}\in{\mathcal{D}}(\mathcal{E})$ such that $\mathcal{E}(\sqrt{\rho}_{r})\leq E$ for every $s\in[0,1]$.

We recall the following result (Lemma 12.2 in [5]).

Lemma 4.2**.**

For every geodesic $(\mu_{r})_{r\in[0,1]}$ there exist regular curves $\mu^{n}$ such that $\mu_{r}^{n}\to\mu_{r}$ in $L^{2}$-Kantorovich sense for all $r\in[0,1]$ and

[TABLE]

A regular curve $\mu$ admits a velocity density $v\in L^{2}(X\times[0,1],\int\mu_{t}\,dt)$ in the sense that for every $\varphi\in\mathcal{F}$

[TABLE]

and there exists a unique velocity density with minimal $L^{2}(X\times[0,1],\int\mu_{t}\,dt)$-norm satisfying

[TABLE]

see Theorem 6.6 and Lemma 8.1 in [5].

Proof of Theorem 4.1.

Let $u\in L^{2}(X)\cap L^{\infty}(X)$, with $u\leq M$ $m$-a.e.. Fix $s<t$ and define for all $s<r<t$

[TABLE]

where $\mu_{r}=\rho_{r}m_{t}$ is a regular curve in $\mathcal{P}_{2}(X)$, and $\omega_{\varepsilon},\eta_{\varepsilon}$ are functions on $\mathbb{R}$ given

[TABLE]

Note that $\eta_{\varepsilon},\eta_{\varepsilon}^{\prime},\omega_{\varepsilon},\omega^{\prime}_{\varepsilon}\in\mathrm{Lip}_{b}([0,M])$ and

[TABLE]

Then we readily find that

[TABLE]

by (3), (4), (5), Lemma 1.7 and $\eta_{\varepsilon}\in\mathrm{Lip}_{b}([0,M])$.

We claim that $r\mapsto\Psi^{\varepsilon}(r)$ is locally absolutely continuous. To see this we write

[TABLE]

where $\zeta,\xi\in(r,r+h)$ and $v$ is the unique velocity density of $\mu$. The first term we estimate by

[TABLE]

where we used (3) and (4), the 2-absolute continuity of $r\mapsto P^{*}_{t,r}g$, $r\mapsto P_{r,s}u$ by Proposition 1.8, the Lipschitz continuity of $\eta_{\varepsilon}$, and the Lipschitz continuity of $r\mapsto f_{r}$. A calculation shows that this term is finite for almost all $s<r<h$.

For the second term in (25) note that $|\omega_{\varepsilon}^{\prime}(\psi_{\varepsilon}^{r}(u))|$ is uniformly bounded for almost all $s<r<t$, and by virtue of the $L^{1}$-gradient estimate (13)

[TABLE]

which is an $L^{2}$-function on $(s,t)\times X$. All in all this proves the locally absolute continuity of $\Psi^{\varepsilon}$.

In the next step we calculate the derivative of $\Psi^{\varepsilon}(r)$. We compute

[TABLE]

where we used (22) for the last term. Taking the limit $h\to 0$, by Proposition 1.8 we get for the first term on the right hand side in (LABEL:eq:_diff)

[TABLE]

Note that the last term is equal to [math]. Indeed, on the one hand $\frac{1}{h}(\frac{\rho_{r+h}}{\psi_{\zeta}^{\varepsilon}(u)+\varepsilon}-\frac{\rho_{r}}{\psi_{r}^{\varepsilon}(u)+\varepsilon})$ is a converging sequence in $L^{1}(X)$ due to $\rho\in\mathcal{C}^{1}([0,1];L^{1}(X))$, $\omega_{\varepsilon}^{\prime}\in\mathrm{Lip}_{b}([0,M])$, and (24). On the other $P_{t,r+h}(\eta_{\varepsilon}(P_{r,s}u))-P_{t,r}(\eta_{\varepsilon}(P_{r,s}u))\to 0$ weakly∗ in $L^{\infty}(X)$ due to (4), (5), Lemma 1.7, and the Banach-Alaoglu theorem.

For the second term on the right hand side in (LABEL:eq:_diff) it holds

[TABLE]

for a.e. $r$, since $\eta^{\prime}_{\varepsilon}(P_{r,s}u)$ in $\mathrm{Lip}_{b}([0,M])$, $\frac{1}{h}(P_{r+h,s}u-P_{r,s}u)\to\Delta_{r}P_{r,s}u$ in $L^{2}(X)$ for a.e. $r$ and $P_{t,r+h}^{*}(\frac{\rho_{r}}{\psi_{r}^{\varepsilon}(u)+\varepsilon})\to P^{*}_{t,r}(\frac{\rho_{r}}{\psi_{r}^{\varepsilon}(u)+\varepsilon})$ weakly∗ in $L^{\infty}(X)$ due to the uniform boundedness. The last term is equal to [math] since $P_{t,r+h}^{*}\left(\frac{\rho_{r+h}}{\psi_{\zeta}^{\varepsilon}(u)+\varepsilon}-\frac{\rho_{r}}{\psi_{r}^{\varepsilon}(u)+\varepsilon}\right)\to 0$ weakly∗ in $L^{\infty}(X)$ by the Banach Alaoglu theorem and since $\frac{\rho_{r+h}}{\psi_{\zeta}^{\varepsilon}(u)+\varepsilon}-\frac{\rho_{r}}{\psi_{r}^{\varepsilon}(u)+\varepsilon}\to 0$ in $L^{1}(X)$ similarly as above, and $P_{t,r}^{*}$ is a continuous operator on $L^{1}(X)$ (Lemma 1.7).

For the third term in (LABEL:eq:_diff) we apply Young’s inequality and (13) and note that $|\omega_{\varepsilon}^{\prime}(\psi_{q}^{\varepsilon}(u))|$ and $P_{t,q}(|\eta_{\varepsilon}^{\prime}(P_{q,s}u)|^{2})$ are uniformly bounded on $(s,t)\times X$. Moreover by virtue of the local Poincaré inequality (Theorem 2.2)

[TABLE]

is a locally integrable function on $(s,t)\times X$. Then the Lebesgue differentiation theorem applies and thus

[TABLE]

for a.e. $s<r<t$.

Summarizing we find by taking the limit in (LABEL:eq:_diff)

[TABLE]

where we used integration by parts and the chain rule in the last line.

Applying the gradient estimate (13), using the chain rule twice, and inserting the definitions we compute

[TABLE]

Calculating the supremum and using (23) further yields

[TABLE]

where we used that $v$ is the minimal velocity density for $\mu$.

Due to the local absolute continuity, integrating from $s$ to $t$ yields

[TABLE]

Hence, by approximating $W_{t}^{2}$-geodesics with regular curves and taking the scaling into account we end up with

[TABLE]

We get for $m$-a.e. $x,y\in X$, after letting $\mu_{s}\to\delta_{x}$ and $\mu_{t}\to\delta_{y}$ with respect to $L^{2}$-Kantorovich distance,

[TABLE]

Now we let $\varepsilon\to 0$. Since $\eta_{\varepsilon}(P_{t,s}u)\to(P_{t,s}u)^{\alpha}$, and $P_{t,s}\eta_{\varepsilon}(u)\to P_{t,s}(u^{\alpha})$ a.e. by monotone convergence we find

[TABLE]

which is the result for $u\in L^{2}(X)\cap L^{\infty}(X)$. The result for general $u$ follows by a truncation argument. ∎

4.2 From dimension independent Harnack inequality to local logarithmic Sobolev inequality

We assume in this section that $m_{t}(X)<\infty$ for some and thus for all $t\in(0,T)$.

Theorem 4.3**.**

Assume that the Harnack inequality (21) holds. Then for all $u\in{\mathcal{D}}(S)\cap L^{1}(X)$ such that $u\geq 0$ the local logarithmic Sobolev inequality holds

[TABLE]

Proof.

Let $u\in L^{1}(X)\cap L^{\infty}(X)$ with $u\geq c>0$. From the Harnack inequality it follows that

[TABLE]

holds for each probability measures $\mu,\nu$ which are absolutely continuous with respect to $m_{t}$. This follows from integrating (21) with respect to an optimal transport plan.

Now choose $\mu=gm_{t}$ with $g\geq 0$ and $g\in\mathcal{F}\cap L^{\infty}(X)$. Consider the associated Dirichlet form $\mathcal{E}^{g}(u):=\int\Gamma_{t}(u)g\,dm_{t}$ with heat semigroup $(H_{r}^{g})_{r\geq 0}$ and generator $\Delta^{g}$. We introduce for fixed $\varepsilon>0$ the function

[TABLE]

where $\kappa\in\mathcal{C}_{c}^{\infty}(0,\infty)$ with $\kappa\geq 0$ and $\int_{0}^{\infty}\kappa(r)\,dr=1$ and $\psi_{0}\in{\mathcal{D}}(\mathcal{E}^{g})\cap L^{\infty}(gm_{t})$. Note that $||\Delta^{g}\psi||_{\infty}\leq M$ for some $M\geq 0$ and hence $\mu_{\tau}:=g(1-\tau\Delta^{g}\psi)m_{t}$ is a probability measure for all $\tau<1/2M$. First we will show that

[TABLE]

using the Hopf-Lax semigroup $(Q_{r})_{r\geq 0}$ with respect to $d_{t}$. For $\varphi\in\mathcal{C}_{b}(X)$ we find for $r\leq\tau$

[TABLE]

Integrating on $[0,\tau]$, taking the supremum over all $\varphi$, dividing by $\tau$ and letting $\tau\to 0$ yields (28). For $\alpha=1+\tau$, $\tau>0$ (27) reads as

[TABLE]

We divide by $\tau>0$ and let $\tau\to 0$. By (28) the right hand side can be estimated from above by

[TABLE]

We claim that together with the left hand side this amounts to

[TABLE]

Indeed, it is straight forward to check that $r\mapsto\int\log P_{t,s}u^{1+r}\,d\mu_{r}$ is absolutely continuous with derivative

[TABLE]

Since $u\geq c>0$ we see that $r\mapsto\Psi(r)$ is continuous. Hence

[TABLE]

Together with (29) this yields (30).

Letting $\varepsilon\to 0$ we conclude

[TABLE]

Now we may choose $\psi_{0}=-2(t-s)\log(P_{t,s}u)$ and obtain

[TABLE]

Since this holds for all $\mu=gm_{t}$, we recover the local logarithmic Sobolev inequality

[TABLE]

for all $u\in L^{1}(X)\cap L^{\infty}(X)$ with $u\geq c>0$. We obtain the estimate for all nonnegative $u\in{\mathcal{D}}(S)\cap L^{1}(X)$ by a truncation argument. ∎

5 The logarithmic Harnack inequality

We already noted in Remark 1.5, that the dimension-independent Harnack inequality (for some exponent $\alpha$) implies the logarithmic Harnack inequality.

This section is devoted to prove that the logarithmic Harnack inequality implies the dynamic Bochner inequality. To do so, in addition to our standing assumptions, in particular, the validity of a RCD$(K,\infty)$-condition for each $(X,d_{t},m_{t})$ and a log-Lipschitz dependence on $t$ for $d_{t}$ and $m_{t}$, we have to impose various continuity assumptions (all of which are satisfied in the static case).

We assume that $m_{t}(X)<\infty$ for $t\in(0,T)$, (A2.a-c), and (A3) hold. Moreover, writing $u_{q,s}=P_{q,s}u$, we assume that

• •

for $u\in\mathcal{F}\cap{\mathcal{D}}(\Delta)$ the functions

[TABLE]

are continuous in $\mathcal{F}\cap L^{1}(X)$;

• •

for $w,w_{q}\in{\mathcal{D}}(\Delta)$ as $q\to t$, and $\Delta_{t}w_{q}\to\Delta_{t}w$ in $L^{1}(X)$

[TABLE]

Let us emphasize that (A5.a+b) are always satisfied in the static case.

Theorem 5.1**.**

If for all nonnegative $u\in L^{1}(X)\cap L^{\infty}(X)$ and $s<t$ the logarithmic Harnack inequality

[TABLE]

holds for $m$-a.e. $x,y\in X$, then the pointwise dynamic Bochner inequality holds at time $t$, i.e.

[TABLE]

for all $f\in{\mathcal{D}}(\Delta_{t})\cap L^{\infty}(X)$ such that $\Gamma_{t}(f)\in L^{\infty}(X)$ and all nonnegative $g\in{\mathcal{D}}(\Delta_{t})\cap L^{\infty}(X)$.

Proof.

Let us introduce some function $g$ satisfying $C\geq g\geq c>0$. Moreover we will assume that $g\in{\mathcal{D}}(\Delta_{t})\cap\mathrm{Lip}(X)$ such that $\Delta_{t}g\in\mathcal{F}$. We define the Cheeger energy $\frac{1}{2}\mathcal{E}^{g}_{t}$ associated with $d_{t}$ and finite measure $gm_{t}$. The operator $\Gamma_{t}(f)$ is invariant under this perturbations, hence $\Gamma_{t}^{g}(f)=\Gamma_{t}(f)$ and ${\mathcal{D}}(\mathcal{E}^{g}_{t})=\mathcal{F}$. We refer to [2, Section 4] for these facts. This leads to the following integral representation of $\mathcal{E}^{g}_{t}$

[TABLE]

which makes it a symmetric bilinear form. We denote the associated (Markovian) semigroup by $P_{s}^{g}$ and its generator by $\Delta_{t}^{g}$, which satisfies the following integration by parts formula

[TABLE]

for all $f\in{\mathcal{D}}(\Delta_{t}^{g})$ and $h\in{\mathcal{D}}(\mathcal{E}_{t}^{g})$. Since $\log g\in\mathcal{F}$ this can be rewritten into

[TABLE]

and thus ${\mathcal{D}}(\Delta_{t})\subset{\mathcal{D}}(\Delta_{t}^{g})$.

Let $f,\Delta_{t}f\in D(\Delta_{t})\cap\mathrm{Lip}_{b}(X)$. Then by Lemma 5.3 $u:=e^{f}\in{\mathcal{D}}(\Delta_{t})\cap\mathrm{Lip}_{b}(X)$ with $\Delta_{t}e^{f},\Delta^{g}_{t}e^{f}\in L^{\infty}(X)\cap\mathcal{F}$ and $u\geq e^{-||f||_{\infty}}=:\varepsilon>0$.

For $s\leq t$ we set

[TABLE]

Note that $v_{s}\in{\mathcal{D}}(\Delta_{t}^{g})\cap L^{\infty}(X)$ for all $s\leq t$ by Lemma 5.3. Without restriction, we may assume that $\mu_{t}$, and hence $\mu_{s}$ for every $s<t$, is a probability measure. Otherwise, simply replace $f$ by $f+C$ for a suitable constant $C$.

Assume that the logarithmic Harnack inequality holds for the function $u=e^{f}$. We integrate the inequality w.r.t. the $W_{t}$-optimal coupling of $\mu_{t}$ and $\mu_{s}$ to obtain for any $s<t$

[TABLE]

Consider the map $r\mapsto\int P_{t,r}\log P_{r,s}u\,d\mu_{r}$. This map is absolutely continuous since for a.e. $s<r_{1}<r_{2}<t$

[TABLE]

Hence for the left hand side of (32) we find by differentiation

[TABLE]

and for the right hand side Kuwada’s Lemma ([2, Lemma 6.1]) yields

[TABLE]

Hence (32) can be rewritten as follows

[TABLE]

Now let us consider the map

[TABLE]

From Lemma 5.2 we know that the map $r\mapsto III(r)$ is absolutely continuous with derivative

[TABLE]

For $I$ we calculate for a.e. $r_{1}<r_{2}$

[TABLE]

The second term of this subdivision can be estimated as follows

[TABLE]

For the almost everywhere derivative we obtain by eventually using Proposition 1.8, Lemma 1.7, Lemma 2.5, (A2.a), (A2.b), (A2.c), and (A3).

[TABLE]

Finally for $II$ we argue similarly as for $I$ and prove local absolute continuity by

[TABLE]

For the almost everywhere derivative we obtain by eventually using Proposition 1.8 and Lemma 1.7

[TABLE]

Thus $r\mapsto I(r)+II(r)+III(r)$ is absolutely continuous and we rewrite (33) as

[TABLE]

where the right hand side comes from the boundary term $I(t)+II(t)+III(t)$.

Recall that $\mu_{q}=v_{q}\,g\,m_{t}$. Then the term on the LHS of (LABEL:infdimHarnackb2c) takes the form

[TABLE]

We decompose $\Psi$ into five terms and verify the continuity of each of them. For the first one,

[TABLE]

continuity follows from the fact that $q\mapsto\Gamma_{q}(\log u_{q,s})$ is weak∗-continuous in $L^{\infty}(X)$ by (A5.a) and (A2.a), and $q\mapsto\Delta_{q}P_{t,q}^{*}(v_{q}g)$ is continuous in $L^{1}(X)$ by assumption (A5.b) together with the fact that $q\mapsto\Delta_{t}(v_{q}\,g)$ is continuous in $L^{1}(X)$.

Continuity of the second one,

[TABLE]

follows from $L^{1}$-continuity of $q\mapsto\partial_{q}\Gamma_{q}(\log u_{q,s})$, as requested in assumption (A2.c), (A3), and the weak∗-continuity of q\mapsto P_{t,q}^{*}\big{(}v_{q}\,\,g\big{)} in $L^{\infty}(X)$, resulting from (A5.b) together with the uniform boundedness in $L^{\infty}(X)$.

For the third one,

[TABLE]

assumptions (A2.b), (A3) and (A5.a) yield continuity of q\mapsto\Gamma_{q}\Big{(}\log u_{q,s},\frac{1}{u_{q,s}}\Delta_{q}u_{q,s}\Big{)} in $L^{1}(X)$ combined with (A2.a) and dominated convergence. Together with the weak∗-continuity of q\mapsto P_{t,q}^{*}\big{(}v_{q}\,\,g\big{)} in $L^{\infty}(X)$, this yields the claim.

The fourth term,

[TABLE]

is continuous since $q\mapsto P_{t,q}\log u_{q,s}$ is continuous in $\mathcal{F}$ by (A5.a) and (A2.a), and $q\mapsto\Delta^{g}_{t}v_{q}$ is continuous in $\mathcal{F}$ by Lemma 5.3.

The final term

[TABLE]

is always continuous in $q$ without any extra assumption.

Similarly one computes the right hand side of (LABEL:infdimHarnackb2c). Recalling that $\log v_{t}=-2f$:

[TABLE]

Note that by the continuity of $q\mapsto\log u_{q}$ in $\mathcal{F}$ and the continuity of $q\mapsto\frac{\Delta_{q}u_{q}}{u_{q,s}}$ in $\mathcal{F}$ by virtue of (A5.a), (A2.a) and the fact that $u\geq\varepsilon$, the map q\mapsto\int\Gamma_{t}\Big{(}\log u_{q,s}-f,\frac{\Delta_{q}u_{q,s}}{u_{q,s}}\Big{)}\,d\mu_{t} is continuous. Then by the Lebesgue differentiation theorem and the continuity discussion above we deduce from (LABEL:infdimHarnackb2c) that (recalling that $u=e^{f}$)

[TABLE]

Then, letting $s\to t$, by continuity we have (recalling that $v_{t}=e^{-2f}$)

[TABLE]

Choose $g=(\tilde{g}+\varepsilon)e^{2f}$, where $\tilde{g}\in\mathrm{Lip}_{b}(X)\cap{\mathcal{D}}(\Delta_{t})$ with $\Delta_{t}\tilde{g}\in\mathcal{F}$. Then $g\in{\mathcal{D}}(\Delta_{t})\cap\mathrm{Lip}(X)$ such that $\Delta_{t}g\in\mathcal{F}$ by Lemma 5.3 and [18, Theorem 3.4], and there exists constants $c,C$ such that $0<c\leq g\leq C$. With this choice we obtain

[TABLE]

for all $f,\Delta_{t}f\in D(\Delta_{t})\cap\mathrm{Lip}_{b}(X)$ and nonnegative $\tilde{g}\in\mathrm{Lip}_{b}(X)\cap{\mathcal{D}}(\Delta_{t})$ with $\Delta_{t}\tilde{g}\in\mathcal{F}$. The result for general $f\in D(\Delta_{t})\cap\mathrm{Lip}_{b}(X)$ and nonnegative $\tilde{g}\in L^{\infty}(X)\cap{\mathcal{D}}(\Delta_{t})$ follows by approximation with the standard $t$-semigroup mollifier from Definition 2.1. ∎

Lemma 5.2**.**

Let $(X,d,m)$ be an RCD$(K,\infty)$-space. Let $g\in\mathrm{Lip}_{b}(X)$ satisfying $C\geq g\geq c>0$. Let $v\in\mathrm{Lip}_{b}(X)\cap{\mathcal{D}}(\Delta)$ such that $\Delta^{g}v\in L^{\infty}(X)\cap\mathcal{F}$. Moreover let $\psi\in\mathcal{C}^{2}(\Im(v))$. Then for $v_{r}=P_{r}^{g}v$ the map $r\mapsto\int\Gamma(u_{r})\psi(u_{r})g\,dm$ is absolutely continuous and

[TABLE]

for a.e. $r\geq 0$.

Proof.

Let $0<s<t$. Then it is well-known that, see e.g. [10, Theorem 4.8] or [11, Theorem 4.6],

[TABLE]

which shows $r\mapsto\int\Gamma(v_{r})\psi(v_{r})g\,dm$ is absolutely continuous. We compute the a.e. derivative as follows

[TABLE]

Taking the limit $h\to 0$ we verify that it holds a.e.

[TABLE]

Applying the Leibniz and the chain rule we find that for a.e. $r\geq 0$

[TABLE]

∎

Lemma 5.3**.**

Let $(X,d,m)$ be an RCD$(K,\infty)$-space. Let $g\in{\mathcal{D}}(\Delta)\cap\mathrm{Lip}(X)$ such that $\Delta g\in\mathcal{F}$ and $C\geq g\geq c>0$. Let $f,\Delta f\in{\mathcal{D}}(\Delta)\cap\mathrm{Lip}_{b}(X)$. Then $e^{f}\in{\mathcal{D}}(\Delta)\cap\mathrm{Lip}_{b}(X)$ with $\Delta e^{f},\Delta^{g}e^{f}\in L^{\infty}\cap\mathcal{F}$ and $e^{f}\geq c$ for some $c>0$.

Moreover the functions $t\mapsto P_{t}e^{f}$ and $t\mapsto P_{t}^{g}e^{f}$ are continuous in $L^{\infty}(X)$.

Proof.

Since $f$ is bounded, $e^{f}$ is bounded as well and $e^{f}\geq e^{-\lvert\lvert f\rvert\rvert_{\infty}}>0$. By the chain rule we have $\Gamma(e^{f})=e^{2f}\Gamma(f)\in L^{\infty}(X)$ and

[TABLE]

which belongs to $L^{2}(X)\cap L^{\infty}(X)$. Next we show that $\Delta e^{f}\in\mathcal{F}$. For this note that

[TABLE]

is bounded and

[TABLE]

is bounded as well. In the last step we used [18, Lemma 3.2] to bound $\mathcal{E}(\Gamma(f))$. Summing $\mathcal{E}(e^{f}\Delta f)$ and $\mathcal{E}(e^{f}\Gamma(f))$ yields that $\Delta e^{f}\in\mathcal{F}$.

Similarly we show that $\Delta^{g}e^{f}\in{\mathcal{D}}(\mathcal{E}^{g})$. Recall first that ${\mathcal{D}}(\Delta)\subset{\mathcal{D}}(\Delta^{g})$ and

[TABLE]

which is an $L^{\infty}(X)$-function. Moreover note that

[TABLE]

For the first summand we know already that it is bounded. For the second summand we use [18, Theorem 3.4] and obtain

[TABLE]

where $\gamma_{2}(\log g),\gamma_{2}(e^{f})$ are $L^{1}$-functions, since $\log g$ and $e^{f}$ belong to $\mathrm{Lip}_{b}(X)\cap{\mathcal{D}}(\Delta)$ with $\Delta\log g,\Delta e^{f}\in\mathcal{F}$.

For the last claim, note that

[TABLE]

where the last integral has to be understood as a Bochner integral. Hence

[TABLE]

The other statement follows analogously. ∎