Closure properties of solutions to heat inequalities

TL;DR

This paper proves that certain heat inequalities are preserved under convolution operations, leading to a new proof of the sharp n-fold Young convolution inequality and its reverse, with implications for heat flow analysis.

Contribution

It establishes closure properties of solutions to heat inequalities under convolution, providing a novel heat-flow proof of the sharp Young convolution inequality.

Findings

Closure of heat inequalities under convolution operations.

Derivation of a heat-flow proof for the sharp n-fold Young inequality.

Extension to reverse Young inequalities through convolution.

Abstract

We prove that if $u_1,u_2 : (0,\infty) \times \R^d \to (0,\infty)$ are sufficiently well-behaved solutions to certain heat inequalities on $\R^d$ then the function $u: (0,\infty) \times \R^d \to (0,\infty)$ given by $u^{1/p}=u_1^{1/p_1} * u_2^{1/p_2}$ also satisfies a heat inequality of a similar type provided $\tfrac{1}{p_1} + \tfrac{1}{p_2} = 1 + \tfrac{1}{p}$. On iterating, this result leads to an analogous statement concerning $n$-fold convolutions. As a corollary, we give a direct heat-flow proof of the sharp $n$-fold Young convolution inequality and its reverse form.