On Two-Sided Estimates for the Nonlinear Fourier Transform of KdV

The KdV-equation $u_t=-u_{xxx}+6u{u}_x$ 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 $\mathrm{\mid }\mathrm{\mid }u\mathrm{\mid }{\mathrm{\mid }}_m$, $m\ge 0$, 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.