Weighted Fourier inequalities via rearrangements

TL;DR

This paper revisits the use of rearrangements to establish weighted Fourier inequalities, introduces new results for q < p, compares known conditions, and provides examples and alternatives to strengthen weight conditions in specific cases.

Contribution

It provides new sufficient conditions for Fourier inequalities when q < p and clarifies the limitations of simple weight conditions in certain parameter ranges.

Findings

Established new results for q < p cases.

Compared two known sufficient conditions for Fourier inequalities.

Provided examples showing limitations of simple weight conditions and proposed alternatives.

Abstract

The method of using rearrangements to give sufficient conditions for Fourier inequalities between weighted Lebesgue spaces is revisited. New results in the case q < p are established and a comparison between two known sufficient conditions is completed. In addition, examples are given to show that a simple weight condition that is sufficient for the weighted Fourier inequality in the cases 2 < q < p and 1 < q < p < 2 is no longer sufficient in the case 1 < q < 2 < p, contrary to statements in Theorems 1 and 4 of, "Weighted Fourier inequalities: new proofs and generalizations", J. Fourier Anal. Appl. 9 (2003), 1-37. Several alternatives are given for strengthening the simple weight condition to ensure sufficiency in that case.