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

Abstract

By using optimal mass transport theory, we provide a direct proof to the sharp $L^p$-log-Sobolev inequality $(p\ge 1)$ 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 $p\ge n$ 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.

36 pages; to appear in Transactions of the American Mathematical Society