Sharp weighted log-Sobolev inequalities: characterization of equality cases and applications

TL;DR

This paper proves sharp weighted log-Sobolev inequalities using optimal mass transport, characterizes the equality cases, and applies these results to hypercontractivity estimates for the Hopf-Lax semigroup.

Contribution

It introduces a new proof method for sharp weighted log-Sobolev inequalities and characterizes equality cases for a broader class of functions and weights.

Findings

Characterization of extremal functions for weighted log-Sobolev inequalities.

New equality case characterizations for $p extgreater n$ in unweighted settings.

Sharp hypercontractivity estimates for the Hopf-Lax semigroup.

Abstract

By using optimal mass transport theory, we provide a direct proof to the sharp $L^p$-log-Sobolev inequality $(p\geq 1)$ involving a log-concave homogeneous weight on an open convex cone $E\subseteq \mathbb R^n$. The perk of this proof is that it allows to characterize the extremal functions realizing the equality cases in the $L^p$-log-Sobolev inequality. The characterization of the equality cases is new for $p\geq n$ even in the unweighted setting and $E=\mathbb R^n$. As an application, we provide a sharp weighted hypercontractivity estimate for the Hopf-Lax semigroup related to the Hamilton-Jacobi equation, characterizing also the equality cases.