Heat flow on 1-forms under lower Ricci bounds. Functional inequalities, spectral theory, and heat kernel

TL;DR

This paper investigates heat flow on 1-forms in metric measure spaces with Ricci curvature bounds, establishing functional inequalities, spectral properties, and heat kernel existence, extending classical analysis to non-smooth settings.

Contribution

It introduces new inequalities and spectral analysis for heat flow on 1-forms in RCD spaces, including heat kernel existence without local compactness assumptions.

Findings

Proved Hess-Schrader-Uhlenbrock's inequality for RCD spaces.

Established spectral inclusions and compactness results for the Hodge Laplacian.

Demonstrated existence and fundamental estimates of the heat kernel in general metric measure spaces.

Abstract

We study the canonical heat flow $(\mathsf{H}_t)_{t\geq 0}$ on the cotangent module $L^2(T^*M)$ over an $\mathrm{RCD}(K,\infty)$ space $(M,\mathsf{d},\mathfrak{m})$, $K\in\boldsymbol{\mathrm{R}}$. We show Hess-Schrader-Uhlenbrock's inequality and, if $(M,\mathsf{d},\mathfrak{m})$ is also an $\mathrm{RCD}^*(K,N)$ space, $N\in(1,\infty)$, Bakry-Ledoux's inequality for $(\mathsf{H}_t)_{t\geq 0}$ w.r.t. the heat flow $(\mathsf{P}_t)_{t\geq 0}$ on $L^2(M)$. Variable versions of these estimates are discussed as well. In conjunction with a study of logarithmic Sobolev inequalities for $1$-forms, the previous inequalities yield various $L^p$-properties of $(\mathsf{H}_t)_{t\geq 0}$, $p\in [1,\infty]$. Then we establish explicit inclusions between the spectrum of its generator, the Hodge Laplacian $\smash{\vec{\Delta}}$, of the negative functional Laplacian $-\Delta$, and of the Schr\"odinger operator $-\Delta+K$. In the $\mathrm{RCD}^*(K,N)$ case, we prove compactness of $\smash{\vec{\Delta}^{-1}}$ if $M$ is compact, and the independence of the $L^p$-spectrum of $\smash{\vec{\Delta}}$ on $p \in [1,\infty]$ under a volume growth condition. We terminate by giving an appropriate interpretation of a heat kernel for $(\mathsf{H}_t)_{t\geq 0}$. We show its existence in full generality without any local compactness or doubling, and derive fundamental estimates and properties of it.