$L^p$ Sobolev regularity for a class of Radon and Radon-like transforms of various codimension

TL;DR

This paper extends $L^p$ Sobolev regularity results to Radon-like transforms over surfaces of arbitrary codimension using novel resolution of singularities techniques, broadening the scope beyond hypersurfaces.

Contribution

It introduces a new approach employing resolution of singularities to establish Sobolev regularity for Radon-like transforms of various codimensions.

Findings

Proves $L^p$ Sobolev regularity for a broad class of Radon transforms

Uses novel resolution of singularities techniques

Extends previous results from hypersurfaces to arbitrary codimension surfaces

Abstract

In the paper [G1] the author proved $L^p$ Sobolev regularity results for averaging operators over hypersurfaces and connected them to associated Newton polyhedra. In this paper, we use rather different resolution of singularities techniques to prove $L^p$ Sobolev regularity results for a class of averaging operators over surfaces which can be of any codimension.