Elementary proof of logarithmic Sobolev inequalities for Gaussian convolutions on $\mathbb{R}$

TL;DR

This paper provides a simpler, elementary proof that convolutions of compactly supported measures with Gaussian measures on the real line satisfy logarithmic Sobolev inequalities, building on previous results by the author.

Contribution

It introduces a more straightforward, elementary proof of the logarithmic Sobolev inequalities for Gaussian convolutions on , improving accessibility and understanding.

Findings

Proves that Gaussian convolutions with compactly supported measures satisfy LSIs.

Provides bounds for the optimal constants in these LSIs.

Simplifies the proof of previous results.

Abstract

In a 2013 paper, the author showed that the convolution of a compactly supported measure on the real line with a Gaussian measure satisfies a logarithmic Sobolev inequality (LSI). In a 2014 paper, the author gave bounds for the optimal constants in these LSIs. In this paper, we give a simpler, elementary proof of this result.