Another approach to the equality case of the sharp logarithmic Sobolev inequality

TL;DR

This paper characterizes the equality cases of sharp weighted and unweighted logarithmic Sobolev inequalities in Euclidean spaces, introducing a novel approach that extends to various norms and p-values.

Contribution

It provides a new method to determine equality cases in sharp logarithmic Sobolev inequalities, applicable to weighted and unweighted, and different p-norm cases.

Findings

Characterization of equality cases for weighted L^2 logarithmic Sobolev inequality.

Extension of the approach to unweighted L^p inequalities for 1 < p < ∞.

Novelty in methodology even for the unweighted case.

Abstract

In this note, we characterize the equality case of the sharp $L^2$-Euclidean logarithmic Sobolev inequality with monomial weights, exploiting the idea by Bobkov and Ledoux \cite{Bob}. Our approach is new even in the unweighted case. Also, we show that the same strategy yields the equality case of the sharp $L^p$-Euclidean logarithmic Sobolev inequality for $1 < p < \infty$ with arbitrary norm on ${\mathbb R}^n$, if the inequality is unweighted.