Maximal Functions for Lacunary Dilation Structures

TL;DR

This paper studies maximal functions associated with lacunary dilations of hypersurfaces with non-vanishing curvature, establishing boundedness from a Hardy space variant to weak L^1.

Contribution

It introduces bounds for maximal functions generated by lacunary dilations of hypersurfaces with non-zero curvature, extending classical results to more general dilation structures.

Findings

Boundedness from H^1 to weak L^1 for the maximal function

Applicability to hypersurfaces with non-vanishing curvature

Extension to dilations by matrices with eigenvalues greater than 1

Abstract

If mu is a smooth density on a hypersurface in R^d whose curvature never vanishes to infinite order, and A is a d-by-d matrix whose eigenvalues all have absolute value greater than 1, then the maximal function given by convolving f with dilates of mu by powers of A, and taking the maximum, is bounded from a corresponding version of H^1 to weak L^1.