The nonlinear Hausdorff-Young inequality

Abstract

We prove the constant-one discrete nonlinear Hausdorff-Young inequality. As a consequence, by a discrete-to-continuous limiting argument, we obtain $$\Vert (\log |a_f{|}^2{)}^{1/2}{\Vert }_{L^{p^{\prime }}(\mathbb{R})}\le \Vert f{\Vert }_{L^p(\mathbb{R})},1\le p<2,$$ for all $f\in L^p(\mathbb{R})$, where $a_f$ denotes the transmission coefficient of the nonlinear Fourier transform. In particular, this resolves the Muscalu-Tao-Thiele uniformity problem. The proof uncovers a hidden Hilbert-space structure that reduces the nonlinear inequality to classical interpolation.

17 pages