Stochastic analysis of Beckner's and related functional inequalities

TL;DR

This paper employs stochastic methods to improve Beckner's inequality under Gaussian measures for certain parameters, providing explicit error bounds and related inequalities involving entropy and variance.

Contribution

It introduces an improved version of Beckner's inequality for Gaussian measures when 4/3 ≤ p < 2, with explicit error bounds and new entropy-variance inequalities.

Findings

Improved Beckner's inequality with explicit error bounds for p=3/2

Derived a Hölder-type inequality among entropy, variance, and related functionals

Established the inequality holds for Gaussian measures when 4/3 ≤ p < 2

Abstract

Beckner's inequality is a family of inequalities that interpolates the two fundamental functional inequalities, the logarithmic Sobolev and Poincar\'e's inequalities. It is parametrized by exponent $p\in (1,2]$ and it implies the logarithmic Sobolev inequality as $p\to 1$ and agrees with Poincar\'e's inequality when $p=2$. In this paper, employing a stochastic method, we prove an improvement of Beckner's inequality under the Gaussian measure when $4/3\le p<2$; in particular, when $p=3/2$, the error bound is expressed in terms of the entropy functional. A similar reasoning to the derivation of the improvement also enables us to obtain a H\"older-type inequality that holds among the entropy, variance and related functionals.