On Plancherel's identity for a two-dimensional scattering transform

TL;DR

This paper extends the nonlinear Plancherel identity for a scattering transform related to the Davey-Stewartson system to a broader class of function spaces, specifically weighted Sobolev spaces with fractional order.

Contribution

It generalizes the validity of the nonlinear Plancherel identity to weighted Sobolev spaces $H^{s,s}(R^2)$ for $s$ in (0,1), broadening previous results.

Findings

Extended Plancherel identity to $H^{s,s}(R^2)$ with $s ext{ in }(0,1)$

Broadened the class of functions for which the identity holds

Connected scattering theory with fractional Sobolev spaces

Abstract

We consider the $\overline{\partial}$-Dirac system that Ablowitz and Fokas used to transform the defocussing Davey-Stewartson system to a linear evolution equation. The nonlinear Plancherel identity for the associated scattering transform was established by Beals and Coifman for Schwartz functions. Sung extended the validity of the identity to functions belonging to $L^1(\mathbb{R}^2)\cap L^\infty(\mathbb{R}^2)$ and Brown to $L^2(\mathbb{R}^2)$-functions with sufficiently small norm. More recently, Perry extended to the weighted Sobolev space $H^{1,1}(\mathbb{R}^2)$ and here we extend to $H^{s,s}(\mathbb{R}^2)$ with $s\in(0,1)$.