Generating monotone quantities for the heat equation

Abstract

The purpose of this article is to expose and further develop a simple yet surprisingly far-reaching framework for generating monotone quantities for positive solutions to linear heat equations in euclidean space. This framework is intimately connected to the existence of a rich variety of algebraic closure properties of families of sub/super-solutions, and more generally solutions of systems of differential inequalities capturing log-convexity properties such as the Li-Yau gradient estimate. Various applications are discussed, including connections with the general Brascamp-Lieb inequality and the Ornstein-Uhlenbeck semigroup.To Michael Cowling on his 65th birthday.

Regularized inverse Brascamp--Lieb inequalities

“…and thus f 0 can be thought of as a super type M function. The fact that u 0 is a supersolution as above follows from [11,Theorem 3.7] but the argument crucially rests on the proof of [15,Proposition 8.9]. We also remark that the condition (2.10) is natural and arises when considering the case when the Brascamp-Lieb constant is attained on gaussian inputs.…”

“…Section 3 contains several preliminary observations, mostly related to the heat-flow monotonicity approach that we will use to prove our main result in Theorem 2.1. Although the main body of the proof of Theorem 2.1 is given in Section 5, the key heat-flow result needed for the proof has been isolated in Theorem 4.1 in Section 4; we take the opportunity to present this result in the form of a "closure property" for sub/supersolutions to certain heat equations and thus contribute to the emerging theory of such closure properties in, for example, [1,10,11]. After the proof of Theorem 2.1 in Section 5, in Section 6 we present some further applications and remarks.…”

Logarithmic-Sobolev and multilinear Hölder’s inequalities via heat flow monotonicity formulas

“…This approach, that is combining heat flow and entropy formula, appeared more simplified than using either heat flow or entropy formula independently. The results obtained here can be compared with existing ones [5,11,18,25,26] . Another motivation for this work is that these inequalities can be easily seen as the consequence of the ability of functionals involving powers of smooth solution to the heat equation to approach their extremal values as time grows infinitely.…”

Stability of hypercontractivity, the logarithmic Sobolev inequality, and Talagrand's cost inequality

“…under appropriate smoothness and decay conditions on v. Observations of this nature emerged in [15,16] in the context of the sharp Young convolution inequality and Brascamp-Lieb inequalities. We follow these papers and refer to (3.36) as a closure property (for supersolutions of Fokker-Planck equations) associated with the operation v t → v t .…”

“…Lemma 2.2. Let p > 1, β > 0, and twice differentiable function v : R n → (0, ∞) 16). Then for a solution v t to (2.4), we have…”