Improving Beckner's bound via Hermite functions

TL;DR

This paper improves Beckner's inequality for Gaussian measures, especially for intermediate p-values, and extends it to all real p, enhancing the understanding of functional inequalities.

Contribution

The authors provide a refined version of Beckner's inequality valid for p in [1,2], including an extension to all real p, with significant implications for Gaussian measures.

Findings

Improved inequality for p in [1,2]

Extension of inequality to all real p

Enhanced understanding of Gaussian functional inequalities

Abstract

We obtain an improvement of the Beckner's inequality $\| f\|^{2}_{2} -\|f\|^{2}_{p} \leq (2-p) \| \nabla f\|_{2}^{2}$ valid for $p \in [1,2]$ and the Gaussian measure. Our improvement is essential for the intermediate case $p \in (1,2)$, and moreover, we find the natural extension of the inequality for any real $p$.