Asymptotically sharp discrete nonlinear Hausdorff-Young inequalities for the SU(1,1)-valued Fourier products

TL;DR

This paper investigates the sharp constants in a discrete nonlinear Hausdorff-Young inequality related to the nonlinear Fourier transform, focusing on small sequences and those far from extremizers, advancing understanding of these inequalities.

Contribution

It provides new results on the behavior of sharp constants in the discrete nonlinear Hausdorff-Young inequality, especially for specific classes of sequences.

Findings

Sharp constants are characterized for small sequences.

Results obtained for sequences far from extremizers.

Advances understanding of nonlinear Fourier transform inequalities.

Abstract

We work in a discrete model of the nonlinear Fourier transform (following the terminology of Tao and Thiele), which appears in the study of orthogonal polynomials on the unit circle. The corresponding nonlinear variant of the Hausdorff-Young inequality can be deduced by adapting the ideas of Christ and Kiselev to the present discrete setting. However, the behavior of sharp constants remains largely unresolved. In this short note we give two results on these constants, after restricting our attention to either sufficiently small sequences or to sequences that are far from being the extremizers of the linear Hausdorff-Young inequality.