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 , Lp 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 Lp-bounds in codimension 1 under a curvature condition equivalent to that of . 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.
Michael Greenblatt. "A method for proving Lp-boundedness of singular Radon transforms in codimension 1." Duke Math. J. 108 (2) 363 - 393, 1 June 2001. https://doi.org/10.1215/S0012-7094-01-10826-0