Heat equation and the sharp Young's inequality

TL;DR

This paper demonstrates that the sharp Young's inequality can be derived from the heat equation's monotone Lyapunov functional, connecting classical inequalities with heat flow dynamics and entropy methods.

Contribution

It provides a novel proof of the sharp Young's inequality using heat equation evolution and entropy power inequality techniques, extending previous approaches.

Findings

Derivation of sharp Young's inequality from heat equation dynamics

Connection between Lyapunov functionals and convolution inequalities

Extension of Stam and Blachman's entropy power ideas

Abstract

We show that the sharp Young's inequality for convolutions first obtained by Bechner and Brascamp-Lieb can be derived from the monotone in time evolution of a Lyapunov functional of the convolution of two solutions to the heat equation, with different diffusion coefficients, first introduced by Bennett and Bez. Our proof is based on a suitable adaptation of an old idea of Stam and Blachman, used to obtain Shannon's entropy power inequality.