Uniform constants in Hausdorff-Young inequalities for the Cantor group model of the scattering transform

TL;DR

This paper proves that uniform constants in Hausdorff-Young inequalities for the Dirac scattering transform can be achieved within the Cantor group model, extending previous results on the Euclidean line.

Contribution

It establishes the uniformity of constants in Hausdorff-Young inequalities for the scattering transform in the Cantor group setting, answering a question posed in prior research.

Findings

Uniform constants exist for Hausdorff-Young inequalities in the Cantor group model.

Extension of inequalities from Euclidean line to Cantor group setting.

Positive resolution of the uniformity question in a new mathematical context.

Abstract

Analogues of Hausdorff-Young inequalities for the Dirac scattering transform (a.k.a. SU(1,1) nonlinear Fourier transform) were first established by Christ and Kiselev [1],[2]. Later Muscalu, Tao, and Thiele [5] raised a question if the constants can be chosen uniformly in $1\leq p\leq 2$. Here we give a positive answer to that question when the Euclidean real line is replaced by its Cantor group model.