A variational nonlinear Hausdorff-Young inequality in the discrete setting

TL;DR

This paper establishes a variational Hausdorff-Young inequality for a discrete nonlinear Fourier transform on the Lie group SU(1,1), linking the variation of discrete curves to their linearized versions and extending classical harmonic analysis results.

Contribution

It introduces a novel variational inequality for the nonlinear Fourier transform in the discrete setting, connecting Lie group curves with their Lie algebra linearizations.

Findings

Proves a variational Hausdorff-Young inequality for discrete nonlinear Fourier transforms.

Links the variation of discrete curves on SU(1,1) to their linearized versions.

Extends classical harmonic analysis inequalities to a nonlinear, discrete context.

Abstract

Following the works of Lyons and Oberlin, Seeger, Tao, Thiele and Wright, we relate the variation of certain discrete curves on the Lie group $\text{SU}(1,1)$ to the corresponding variation of their linearized versions on the Lie algebra. Combining this with a discrete variational Menshov-Paley-Zygmund theorem, we establish a variational Hausdorff-Young inequality for a discrete version of the nonlinear Fourier transform on $\text{SU}(1,1)$.