[Paper Review] From Harnack inequality to heat kernel estimates on metric measure spaces and applications

Luca Tamanini|arXiv (Cornell University)|Jul 16, 2019

Geometric Analysis and Curvature Flows23 references6 citations

TL;DR

This paper establishes sharp upper Gaussian bounds for heat kernels on metric measure spaces satisfying a dimension-free Harnack inequality and an integral representation of the heat semigroup. By proving a local logarithmic Sobolev inequality as an intermediate step, it derives a heat kernel estimate of the form ${\sf r}_t[x](y) \leq \frac{1}{\sqrt{\mathfrak{m}(B_{\sqrt{t}}(x))\mathfrak{m}(B_{\sqrt{t}}(y))}}\exp\Big{(}C_\varepsilon(1+C_K t)-\frac{{\sf d}^2(x,y)}{(4+\varepsilon)t}\Big{)}$, which resolves a key gap in ${\sf RCD}(K,\infty)$ spaces where both the local logarithmic Sobolev inequality and such estimates were previously unknown.

ABSTRACT

Aim of this short note is to show that a dimension-free Harnack inequality on an infinitesimally Hilbertian metric measure space where the heat semigroup admits an integral representation in terms of a kernel is suffcient to deduce a sharp upper Gaussian estimate for such kernel. As intermediate step, we prove the local logarithmic Sobolev inequality (known to be equivalent to a lower bound on the Ricci curvature tensor in smooth Riemannian manifolds). Both results are new also in the more regular framework of $RCD(K,\infty)$ spaces.

Motivation & Objective

• To close the gap in heat kernel estimates for ${\sf RCD}(K,\infty)$ spaces, where neither the local logarithmic Sobolev inequality nor sharp upper Gaussian bounds were previously established.
• To show that a dimension-free Harnack inequality on an infinitesimally Hilbertian metric measure space with integral heat semigroup implies sharp upper Gaussian estimates for the heat kernel.
• To establish the local logarithmic Sobolev inequality as a new result in ${\sf RCD}(K,\infty)$ spaces, which was previously missing in the literature.
• To generalize known results from ${\sf RCD}^*(K,N)$ and smooth Riemannian manifolds to the broader class of ${\sf RCD}(K,\infty)$ spaces using metric-measure structure.

Proposed method

• Derives the local logarithmic Sobolev inequality from the dimension-free Harnack inequality via a duality argument involving the heat semigroup and $L^p$-norm estimates.
• Uses the $L^\infty$-${\rm LIP}$ regularization property of the heat flow to control the growth of heat kernel in terms of distance and time.
• Applies a weighted $L^2$-estimate on the heat kernel using the exponential weight $\exp(\frac{{\sf d}^2(x,\cdot)}{Dt})$ to control the mass distribution.
• Employs a scaling argument with parameter $D>2$ to derive a uniform bound independent of $D$, leading to the final Gaussian upper bound.
• Combines the local logarithmic Sobolev inequality with the $L^\infty$-${\rm LIP}$ regularization to obtain the sharp upper estimate for the heat kernel.
• Validates the sharpness of the bound via Varadhan's asymptotic formula in the short-time limit, confirming the correct $-\frac{{\sf d}^2(x,y)}{4t}$ decay rate.

Experimental results

Research questions

• RQ1 Can a dimension-free Harnack inequality on a metric measure space with integral heat semigroup imply sharp upper Gaussian estimates for the heat kernel?
• RQ2 Is the local logarithmic Sobolev inequality valid in ${\sf RCD}(K,\infty)$ spaces, and can it be derived from the Harnack inequality?
• RQ3 Does the heat kernel on ${\sf RCD}(K,\infty)$ spaces satisfy an upper bound of the form ${\sf r}_t[x](y) \leq \frac{1}{\sqrt{\mathfrak{m}(B_{\sqrt{t}}(x))\mathfrak{m}(B_{\sqrt{t}}(y))}}\exp\Big{(}C_\varepsilon(1+C_K t)-\frac{{\sf d}^2(x,y)}{(4+\varepsilon)t}\Big{)}$?
• RQ4 Can the short-time behavior of the heat kernel in ${\sf RCD}(K,\infty)$ spaces be shown to satisfy Varadhan's asymptotic formula?
• RQ5 Is the $L^\infty$-${\rm LIP}$ regularization of the heat flow a sufficient tool to bridge Harnack inequalities to heat kernel estimates?

Key findings

• The local logarithmic Sobolev inequality holds on any infinitesimally Hilbertian metric measure space satisfying the dimension-free Harnack inequality, which is a new result even in ${\sf RCD}(K,\infty)$ spaces.
• The heat kernel satisfies the upper Gaussian estimate ${\sf r}_t[x](y) \leq \frac{1}{\sqrt{\mathfrak{m}(B_{\sqrt{t}}(x))\mathfrak{m}(B_{\sqrt{t}}(y))}}\exp\Big{(}C_\varepsilon(1+C_K t)-\frac{{\sf d}^2(x,y)}{(4+\varepsilon)t}\Big{)}$ for all $x,y \in {\rm X}$ and $t>0$, with $C_K = 0$ when $K \geq 0$.
• The bound is sharp in the short-time limit, as confirmed by Varadhan's asymptotic formula: $\lim_{t \downarrow 0} t \log {\sf r}_t[x](y) = -\frac{{\sf d}^2(x,y)}{4}$ for $x \neq y$, which matches the leading-order decay.
• The proof relies on a novel $L^2$-estimate with exponential weight and a scaling argument over $D>2$, which allows uniform control of the heat kernel mass in terms of the volume of balls of radius $\sqrt{t}$.
• The $L^\infty$-${\rm LIP}$ regularization property of the heat flow is established as a key auxiliary result, enabling control of the heat kernel's pointwise behavior.
• The result extends known estimates from ${\sf RCD}^*(K,N)$ and smooth Riemannian manifolds to the more general ${\sf RCD}(K,\infty)$ framework, where such bounds were previously unavailable.