The Fourier transform is an extremizer of a class of bounded operators

TL;DR

This paper characterizes when the Fourier transform is bounded between certain rearrangement spaces and fully describes weighted Fourier inequalities for radially monotone weights, solving a long-standing open problem.

Contribution

It provides a complete characterization of weighted Fourier inequalities for radially monotone weights and links Fourier boundedness to operators of joint strong type, resolving longstanding questions.

Findings

Fourier operator boundedness characterized for rearrangement spaces

Complete solution to weighted Fourier inequalities for radially monotone weights

Connection established between Fourier boundedness and joint strong type operators

Abstract

We show that, for a natural class of rearrangement admissible spaces $X$ and $Y$, the Fourier operator is bounded between $X$ and $Y$ if and only if any operator of joint strong type $(1,\infty; 2,2)$ is also bounded between $X$ and $Y$. By using this result, we fully characterize the weighted Fourier inequalities of the form $$\qquad\qquad \lVert\widehat{f}u \rVert_q \leq C \lVert fv\rVert_p,\quad 1\leq p\leq \infty,\,0<q\leq \infty,$$ for radially monotone weights $(u,v)$. This answers a long-standing problem posed by Benedetto-Heinig, Jurkat-Sampson, and Muckenhoupt. In the case of $p\le q$, such a characterization has been known since the 1980s.