Functional inequalities for Gaussian convolutions of compactly supported measures: explicit bounds and dimension dependence

TL;DR

This paper derives explicit functional inequalities for Gaussian convolutions of compactly supported measures, emphasizing dimension dependence, and improves bounds for Poincaré and logarithmic Sobolev inequalities.

Contribution

It provides dimension-free bounds for Poincaré inequalities and linear-in-dimension bounds for logarithmic Sobolev inequalities for Gaussian convolutions.

Findings

Poincaré inequality with a dimension-free constant

Logarithmic Sobolev inequality with linear dimension dependence

Transport-entropy inequalities for various costs

Abstract

The aim of this paper is to establish various functional inequalities for the convolution of a compactly supported measure and a standard Gaussian distribution on Rd. We especially focus on getting good dependence of the constants on the dimension. We prove that the Poincar{\'e} inequality holds with a dimension-free bound. For the logarithmic Sobolev inequality, we improve the best known results (Zimmermann, JFA 2013) by getting a bound that grows linearly with the dimension. We also establish transport-entropy inequalities for various transport costs.