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

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.