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

摘要

We study the canonical heat flow (H-t)(t >= 0) on the cotangent module L-2(T*M) over an RCD( K, infinity) space ( M, d, m), K is an element of R. We show Hess-Schrader-Uhlenbrock's inequality and, if ( M, d, m) is also an RCD*( K, N) space, N is an element of(1, infinity), BakryLedoux's inequality for (H-t)(t >= 0) w.r.t. the heat flow (P-t)(t >= 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 (H-t)(t >= 0), p is an element of1, infinity. Then we establish explicit inclusions between the spectrum of its generator, the Hodge Laplacian ht arrow>, of the negative functional Laplacian -Delta, and of the Schrodinger operator -Delta+K. In the RCD*(K, N) case, we prove compactness of (Delta) over right arrow (-1) if Mis compact, and the independence of the L-p-spectrum of (Delta) over right arrow on p is an element of1, infinity under a volume growth condition. We terminate by giving an appropriate interpretation of a heat kernel for (H-t)(t >= 0). We show its existence in full generally without any local compactness or doubling, and derive fundamental estimates and properties of it. (C) 2022 Elsevier Inc. All rights reserved.