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

TL;DR

This paper investigates the monotonicity of heat flow related to the Hausdorff--Young inequality, showing it holds for even integer exponents but fails otherwise, with implications for Fourier analysis and heat equations.

Contribution

It provides explicit counterexamples demonstrating the failure of heat-flow monotonicity for non-even integer exponents, extending understanding of Fourier transform norms.

Findings

Monotonicity holds for even integer q in Fourier norms.

Counterexamples show failure of monotonicity for non-even integer q.

Results connect heat equations with Hausdorff--Young inequality extremisers.

Abstract

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.