  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,
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 $\|u\|_m$, $m\ge 0$, in
terms of the weighted norms of the nonlinear Fourier transformed.
