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

Abstract

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

arXiv comment50 pages. The bibliography has been extended. Minor changes have been performed