Closure Properties of Solutions to Heat Inequalities |
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Authors: | Jonathan Bennett Neal Bez |
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Institution: | (1) School of Mathematics, The University of Birmingham, The Watson Building, Edgbaston, Birmingham, B15 2TT, UK;(2) Department of Mathematics, University Gardens, University of Glasgow, Glasgow, G12 8QW, UK |
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Abstract: | 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.
Both authors were supported by EPSRC grant EP/E022340/1. |
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Keywords: | Heat flow Convolution inequalities |
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