Heat-flow monotonicity related to the Hausdorff-Young inequality

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Colleges, School and Institutes


It is known that if q is an even integer, then the L-q(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.


Original languageEnglish
Pages (from-to)971-979
Number of pages9
JournalBulletin of the London Mathematical Society
Publication statusPublished - 3 Sep 2009