The weighted Fourier inequality, polarity, and reverse H\"older inequality

TL;DR

This paper investigates the conditions under which the Fourier transform maps weighted Lebesgue spaces into each other, revealing geometric and reciprocal relationships involving Mahler's measure and reverse H"older weights.

Contribution

It establishes necessary and sufficient conditions for Fourier transform boundedness between weighted Lebesgue spaces, linking geometric polarity and reverse H"older weight classes.

Findings

Necessary conditions relate to Mahler's measure and convex set polarity.

Sufficient conditions involve weights in reverse H"older classes.

The work uncovers geometric and reciprocal structures underlying Fourier inequalities.

Abstract

We examine the problem of the Fourier transform mapping one weighted Lebesgue space into another, by studying necessary conditions and sufficient conditions which expose an underlying geometry. In the necessary conditions, this geometry is connected to an old result of Mahler concerning the the measure of a convex set and its geometric polar being essentially reciprocal. An additional assumption, that the weights must belong to a reverse H\"older class, is used to formulate the sufficient condition.