Analysis of the quadratic term in the backscattering transformation

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 $C_0^\infty({\mathsf R}^n)\times C_0^\infty({\mathsf R}^n)$. In this paper we prove that $B_2$ extends to certain Sobolev spaces with weights and show that it improves both regularity and decay.