Some New Thoughts on Maximal Functions and Poisson Integrals

TL;DR

This paper explores the relationships between maximal functions, Poisson integrals, and convergence theorems in metric spaces, establishing their logical equivalence and providing useful estimates for the Poisson kernel.

Contribution

It demonstrates the equivalence of maximal functions, Poisson integrals, and convergence theorems in a general metric space setting, with new estimates for the Poisson kernel.

Findings

Maximal functions, Poisson integrals, and convergence theorems are essentially equivalent in general metric spaces.

New estimates for the Poisson kernel are provided.

The study extends classical results to more general metric space contexts.

Abstract

We study Wiener-type covering lemmas, Hardy-Littlewood-type maximal functions, and convergence theorems on metric spacs. Later we specialize down to a result for the Poisson integral. We show that, in a suitably general setting, these three phenomena are essentially logically equivalent. Along the way we discuss some useful estimates for the Poisson kernel.