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

Abstract

We study the canonical heat flow (H ) ≥0 on the cotangent module 2 ( * ) over an RCD( , ∞) space ( , d, m), ∈ ℝ. We show Hess-Schrader-Uhlenbrock's inequality and, if ( , d, m) is also an RCD * ( , ) space, ∈ (1, ∞), Bakry-Ledoux's inequality for (H ) ≥0 w.r.t. the heat flow (P ) ≥0 on 2 ( ). A crucial tool is that the dimensional vector 2-Bochner inequality is self-improving, entailing a dimensional vector 1-Bochner inequality-a version of which is also available in the dimension-free case-as a byproduct. 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 -properties of (H ) ≥0 , ∈ [1, ∞].Then we establish explicit inclusions between the spectrum of its generator, the Hodge Laplacian ⃗ Δ, of the negative functional Laplacian −Δ, and of the Schrödinger operator −Δ + .In the RCD * ( , ) case, we prove compactness of ⃗ Δ −1 if is compact, and the independence of the -spectrum of ⃗ Δ on ∈ [1, ∞] under a volume growth condition.We terminate by giving an appropriate interpretation of a heat kernel for (H ) ≥0 . We show its existence in full generality without any local compactness or doubling assumptions, and derive fundamental estimates and properties of it.

Vector calculus for tamed Dirichlet spaces

“…In the subsequent lemma, all terms where N ′ is infinite are interpreted as being zero. Similar proofs can be found in [16,45,53].…”

“…23. Moreover, we address various points that have not been treated in [45], but rather initiated in [16,53], among others…”

“…in proving the above mentioned semigroup domination results. On RCD(K, ∞) spaces, K ∈ R -on which (0.11) has been proven in [32] in order to find "quasi-continuous representatives" of vector fields -this has been observed in [16].…”

“…The latter point of view is not relevant in this work, but is crucial e.g. in obtaining a spectral representation of the heat kernel on 1-forms on compact RCD * (K, N ) spaces [16,Thm. 6.11]…”

“…for X ∈ Reg(T M ), which in turn allows us to define, even for X ∈ H 1,2 ♯ (T M ), • the Ricci curvature of M , see Definition 8. 16, by Ric ≪ (X, X), and • the second fundamental form of M , see Definition 8.23, by II(X, X) := Ric ⊥ (X, X).…”