  The KdV-equation $u_t = -u_{xxx} + 6uu_x$ on the circle admits a global
nonlinear Fourier transform, also known as Birkhoff map, linearizing the KdV
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 obtain 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, which are linear in the highest order. We further obtain
quantitative estimates of the nonlinear Fourier transformed in arbitrary
weighted Sobolev spaces.
