Heat-flow monotonicity related to the Hausdorff--Young inequality

Details Heat-flow monotonicity related to the Hausdorff--Young inequality Heat-flow monotonicity related to the Hausdorff--Young inequality

2008-06-26

arxiv.org/abs/0806.4329v1

Mathematics

Classical Analysis and ODEs

10 pages

Scientific paper

It is known that if $q$ is an even integer then the $L^q(\mathbb{R}^d)$ norm of the Fourier transform of a superposition of translates of a fixed gaussian is monotone increasing as their centres "simultaneously slide" to the origin. We provide explicit examples to show that this monotonicity property fails dramatically if $q > 2$ is not an even integer. These results are equivalent, upon rescaling, to similar statements involving solutions to heat equations. Such considerations are natural given the celebrated theorem of Beckner concerning the gaussian extremisability of the Hausdorff--Young inequality.

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