Some New Thoughts on Maximal Functions and Poisson Integrals

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.

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