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

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.