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

We consider the $\bar{\mathrm{\partial }}$-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 $L^1({\mathbb{R}}^2)\cap L^{\mathrm{\infty }}({\mathbb{R}}^2)$ 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 $s\in (0,1)$.