New Estimates of the Nonlinear Fourier Transform for the Defocusing NLS Equation

The defocusing NLS equation $\mathrm {i} u_t = -u_{xx} +2|u|^2u$ on the circle admits a global nonlinear Fourier transform, also known as Birkhoff map, linearizing the NLS flow. The regularity properties of $u$ are known to be closely related to the decay properties of the corresponding nonlinear Fourier coefficients. In this paper, we quantify this relationship by providing two-sided polynomial estimates of all integer Sobolev norms $\|u\|_m$, $m\geqslant 0$, in terms of the weighted norms of the nonlinear Fourier transforme