Abstract
We prove that if u 1,u 2:(0,∞)×ℝd→(0,∞) are sufficiently well-behaved solutions to certain heat inequalities on ℝd then the function u:(0,∞)×ℝd→(0,∞) 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 \(\frac{1}{p_{1}}+\frac{1}{p_{2}}=1+\frac{1}{p}\) . 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.
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Both authors were supported by EPSRC grant EP/E022340/1.
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Bennett, J., Bez, N. Closure Properties of Solutions to Heat Inequalities. J Geom Anal 19, 584–600 (2009). https://doi.org/10.1007/s12220-009-9070-2
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DOI: https://doi.org/10.1007/s12220-009-9070-2