Oscillatory integral operators in scattering theory

Abstract

We consider a particular Fourier integral operator with folding canonical relations, which arises in scattering theory: the Radon Transform of Melrose and Taylor. We obtain the regularity properties of this operator when the obstacle admits tangent planes with contact of precise order k (Theorem 1.1 and its Corollary). For these purposes, we derive asymptotic estimates for oscillatory integral operators in Wn with folding canonical relations (Theorem 2.2). Asymptotics correspond to vanishing principal curvature of a fold of one of the projections from the canonical relation, and to small support of the localization of oscillatory integral operator.