Heat equation and convolution inequalities

TL;DR

This paper presents new proofs for several classical convolution inequalities by analyzing the monotonicity of heat equation solutions, offering a unified approach inspired by Stam's entropy power inequality proof.

Contribution

It introduces an alternative, heat equation-based method to prove key convolution inequalities, unifying various results under a common framework.

Findings

New proofs of Young's inequality and its converse

Alternative derivations of Brascamp--Lieb and Prékopa--Leindler inequalities

Unified approach inspired by entropy power inequality

Abstract

It is known that many classical inequalities linked to convolutions can be obtained by looking at the monotonicity in time of convolutions of powers of solutions to the heat equation, provided that both the exponents and the coefficients of diffusions are suitably chosen and related. This idea can be applied to give an alternative proof of the sharp form of the classical Young's inequality and its converse, to Brascamp--Lieb type inequalities, Babenko's inequality and Pr\'ekopa--Leindler inequality as well as the Shannon's entropy power inequality. This note aims in presenting new proofs of these results, in the spirit of the original arguments introduced by Stam to prove the entropy power inequality.