New estimates of the nonlinear Fourier transform for the defocusing NLS equation

The defocusing NLS-equation $\mathrm{i}{u}_t=-u_{xx}+2\mathrm{\mid }u{\mathrm{\mid }}^{2}u$ on the circle admits a global nonlinear Fourier transform, also known as Birkhoff map, linearising 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 $\mathrm{\parallel }u{\mathrm{\parallel }}_m$, $m\ge 0$, in terms of the weighted norms of the nonlinear Fourier transformed.