Maximal averages over hypersurfaces and the Newton polyhedron

TL;DR

This paper establishes sharp L^p boundedness results for maximal operators over hypersurfaces using resolution of singularities and oscillatory integral techniques, extending previous results to a broader class of hypersurfaces.

Contribution

It introduces new methods combining resolution of singularities and oscillatory integrals to prove L^p bounds for maximal operators over hypersurfaces, generalizing prior work.

Findings

Sharp L^p bounds for maximal operators over convex hypersurfaces.

Extension of Sogge and Stein's results to hypersurfaces with non-vanishing Gaussian curvature.

Sharp estimates for Fourier transforms of surface measures.

Abstract

Using some resolution of singularities and oscillatory integral methods in conjunction with appropriate damping and interpolation techniques, L^p boundedness theorems for p > 2 are obtained for maximal operators over a wide range of hypersurfaces. These estimates are sharp in many situations, including the convex hypersurfaces of finite line type considered by Iosevich, Sawyer, and others. As a corollary, we also give a generalization of the result of Sogge and Stein that for some finite p the maximal operator corresponding to a hypersurface whose Gaussian curvature does not vanish to infinite order is bounded on L^p. Analogous estimates are proven for Fourier transforms of surface measures, and these are sharp for the same hypersurfaces as the maximal operators.