Well-posedness for the NLS hierarchy

We prove well-posedness for higher-order equations in the so-called NLS hierarchy (also known as part of the AKNS hierarchy) in almost critical Fourier-Lebesgue spaces and in modulation spaces. We show the $j$th equation in the hierarchy is locally well-posed for initial data in ${\widehat{H}}_r^s(\mathbb{R})$ for $s\ge \frac{j-1}{r^{\mathrm{\prime }}}$ and $1<r\le 2$ and also in $M_{2,p}^s(\mathbb{R})$ for $s=\frac{j-1}{2}$ and $2\le p<\mathrm{\infty }$. Supplementing our results with corresponding ill-posedness results in Fourier-Lebesgue spaces shows optimality. Using the conserved quantities derived in Koch-Tataru (2018) we argue that the hierarchy equations are globally well-posed for data in $H^s(\mathbb{R})$ for $s\ge \frac{j-1}{2}$. Our arguments are based on the Fourier restriction norm method in Bourgain spaces adapted to our data spaces and bi- & trilinear refinements of Strichartz estimates.