  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.
