The weighted Fourier inequality, polarity, and reverse Hölder inequality

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 Hölder class, is used to formulate the sufficient condition.

Corrected errors to examples 2 and 3. Filled in gap in the proof of Theorem 5. Added missing condition to Theorem 8