Closure Properties of Solutions to Heat Inequalities

We prove that if u(1), u(2) : (0, infinity) x R-d -> (0, infinity) are sufficiently well-behaved solutions to certain heat inequalities on Rd then the function u : (0, infinity) x R-d -> (0, infinity) given by u(1/p) = u(1)(1/p1) * u(2)(1/p2) also satisfies a heat inequality of a similar type provided 1/p1 + 1/p2 = 1 + 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.