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