Closure properties of solutions to heat inequalities

Abstract

We prove that if $u_1,u_2:(0,\infty )\times {\mathbb{R}}^d\to (0,\infty )$ are sufficiently well-behaved solutions to certain heat inequalities on ${\mathbb{R}}^d$ then the function $u:(0,\infty )\times {\mathbb{R}}^d\to (0,\infty )$ given by $u^{1/p}=u_1^{1/p_1}\ast u_2^{1/p_2}$ also satisfies a heat inequality of a similar type provided $\frac{1}{p_1}+\frac{1}{p_2}=1+\frac{1}{p}$. 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.

12 pages