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We prove that if u 1,u 2:(0,∞)×ℝ d →(0,∞) are sufficiently well-behaved solutions to certain heat inequalities on ℝ d  then the function u:(0,∞)×ℝ d →(0,∞) given by    also satisfies a heat inequality of a similar type provided   . 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.

Closure Properties of Solutions to Heat Inequalities