A note on Hausdorff-Young inequalities in function spaces

TL;DR

This paper extends Hausdorff-Young inequalities to weighted Besov and Sobolev spaces with polynomial weights, broadening the scope of Fourier analysis in weighted function spaces.

Contribution

It introduces a new extension of classical inequalities to weighted Besov and Sobolev spaces with polynomial weights, complementing previous results.

Findings

Extended Hausdorff-Young inequalities to weighted spaces

Applicable to Besov and Sobolev spaces with polynomial weights

Provides a theoretical foundation for Fourier analysis in weighted function spaces

Abstract

The classical Hausdorff-Young inequalities for the Fourier transform acting between appropriate $L_p$ spaces are cornerstones of Fourier analysis. Here we extend it to weighted spaces of Besov or Sobolev type where the weight has the form $w(x)=(1+|x|^2)^{\alpha/2}$. This note is not a paper or draft but a sketchy complement to some earlier results where we dealt with mapping properties of the Fourier transform.