Closure Properties of Solutions to Heat Inequalities

We prove that if u1, u2 : (0,∞)× Rd → (0,∞) are sufficiently well-behaved solutions to certain heat inequalities on Rd then the function u : (0,∞) × Rd → (0,∞) given by u1/p = u 1/p1 1 ∗ u 1/p2 2 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.