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

Singular Radon transforms are a type of operator combining characteristics of both singular integrals and Radon transforms. They are important in a nu mber of settings in mathematics. In a theorem of M. Christ, A. Nagel, E. St ein, and S. Wainger [3], L-P boundedness of singular Radon transforms for 1 < p < infinity is proven under a general finite-type condition using the m ethod of lifting to nilpotent Lie groups. In this paper an alternate approa ch is presented. Geometric and analytic methods are developed which allow u s to prove L-P-bounds in codimension 1 under a curvature condition equivale nt to that of [3]. We restrict consideration to the important case where th e hypersurfaces are graphs of C-infinity-functions. Our methods do not invo lve the Fourier transform, lifting, or facts about Lie groups. This might p rove useful in extending our work to related problems.