$\ell^1$ mapping properties, smoothness and decay for $SU(2)$-valued nonlinear Fourier transform

TL;DR

This paper establishes an analog of Baxter's theorem for $SU(2)$-valued nonlinear Fourier transform, linking $ ext{l}^1$ properties of the potential to the Fourier coefficients of the data, and provides smoothness-decay estimates.

Contribution

It extends Baxter's theorem to $SU(2)$-valued NLFT and introduces smoothness-decay estimates inspired by linear Fourier transform properties.

Findings

Potential in $ ext{l}^1$ iff Fourier coefficients are in $ ext{l}^1$

Established smoothness-decay estimates for NLFT

Proved an analog of Baxter's theorem for $SU(2)$-valued NLFT

Abstract

We prove an analog of Baxter's theorem for $SU(2)$-valued nonlinear Fourier transform (NLFT). That is, we prove that under certain natural conditions on the NLFT data, the potential is in $\ell^1$ if and only if the linear Fourier coefficients of the NLFT data are in $\ell^1$. Furthermore, we prove some smoothness-decay estimates for the NLFT motivated by similar estimates for the linear Fourier transform.