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

TL;DR

This paper establishes two-sided polynomial estimates linking Sobolev norms of solutions to the KdV equation with their nonlinear Fourier coefficients, enhancing understanding of the transform's regularity properties.

Contribution

It provides new two-sided polynomial bounds for Sobolev norms in terms of nonlinear Fourier coefficients and offers quantitative estimates in weighted Sobolev spaces.

Findings

Polynomial estimates of Sobolev norms in terms of nonlinear Fourier coefficients

Quantitative bounds in weighted Sobolev spaces

Enhanced understanding of the regularity properties of the nonlinear Fourier transform

Abstract

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.