A method for proving Lp-boundedness of singular Radon transforms in codimension 1

Abstract

Singular Radon transforms are a type of operator combining characteristics of both singular integrals and Radon transforms. They are important in a number of settings in mathematics. In a theorem of M. Christ, A. Nagel, E. Stein, and S. Wainger [3], L^p boundedness of singular Radon transforms for 1<p<∞ is proven under a general finite-type condition using the method of lifting to nilpotent Lie groups. In this paper an alternate approach is presented. Geometric and analytic methods are developed which allow us to prove L^p-bounds in codimension 1 under a curvature condition equivalent to that of [3]. We restrict consideration to the important case where the hypersurfaces are graphs of C^∞-functions. Our methods do not involve the Fourier transform, lifting, or facts about Lie groups. This might prove useful in extending our work to related problems.