Improved endpoint bounds for the lacunary spherical maximal operator

TL;DR

This paper establishes new endpoint bounds for the lacunary spherical maximal operator, leading to almost everywhere convergence of lacunary spherical means for functions with very weak integrability conditions.

Contribution

It provides improved endpoint bounds for the lacunary spherical maximal operator and demonstrates convergence results for functions in a nearly critical function space.

Findings

New endpoint bounds for the lacunary spherical maximal operator

Almost everywhere convergence for functions in $L ext{log} ext{log} ext{log}L( ext{log} ext{log} ext{log} ext{log}L)^{1+ ext{epsilon}}$

Extension of convergence results to functions with minimal regularity

Abstract

We prove new endpoint bounds for the lacunary spherical maximal operator and as a consequence obtain almost everywhere pointwise convergence of lacunary spherical means for functions locally in $L\log\log\log L(\log\log\log\log L)^{1+\epsilon}$ for any $\epsilon>0$.