$L^p$ Sobolev regularity of averaging operators over hypersurfaces and the Newton polyhedron

TL;DR

This paper establishes $L^p$ to $L^p_{eta}$ boundedness for hypersurface averaging operators, with explicit smoothing estimates derived via Newton polyhedra, advancing understanding of regularity in harmonic analysis.

Contribution

It provides new $L^p$ regularity results for hypersurface averaging operators, linking smoothing effects to Newton polyhedra and covering Radon and fractional integral cases.

Findings

Proves $L^p$ to $L^p_{eta}$ boundedness for hypersurface averages.

Explicitly computes smoothing $eta$ using Newton polyhedra.

Achieves near-optimal smoothing estimates up to endpoints.

Abstract

$L^p$ to $L^p_{\beta}$ boundedness theorems are proven for translation invariant averaging operators over hypersurfaces in Euclidean space. The operators can either be Radon transforms or averaging operators with multiparameter fractional integral kernel. In many cases, the amount $\beta > 0 $ of smoothing proven is optimal up to endpoints, and in such situations this amount of smoothing can be computed explicitly through the use of appropriate Newton polyhedra.