Analysis of the quadratic term in the backscattering transformation

TL;DR

This paper investigates the quadratic component of the backscattering transformation in odd dimensions, demonstrating its extension to Sobolev spaces and its regularity and decay improvements.

Contribution

It provides a rigorous analysis of the quadratic term, showing how it extends to Sobolev spaces and enhances regularity and decay properties.

Findings

The quadratic term extends to certain Sobolev spaces with weights.

It improves regularity of functions.

It enhances decay properties of functions.

Abstract

The quadratic term in the Taylor expansion at the origin of the backscattering transformation in odd dimensions $n\ge 3$ gives rise to a symmetric bilinear operator $B_2$ on $C_0^\infty({\mathbb R}^n)\times C_0^\infty({\mathbb R}^n)$. In this paper we prove that $B_2$ extends to certain Sobolev spaces with weights and show that it improves both regularity and decay.