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

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 ${\mathrm{\ell }}^1$ if and only if the linear Fourier coefficients of the NLFT data are in ${\mathrm{\ell }}^1$. Furthermore, we prove some smoothness-decay estimates for the NLFT motivated by similar estimates for the linear Fourier transform.