Poincar\'e inequalities for the maximal function

TL;DR

This paper establishes that functions satisfying certain Poincaré inequalities have their maximal functions obey similar bounds, providing a unified framework for maximal operator boundedness across various function spaces.

Contribution

It introduces a generalized approach linking Poincaré inequalities to maximal function estimates, unifying boundedness results for Sobolev, Lipschitz, and BMO spaces.

Findings

Maximal functions inherit Poincaré-type inequalities from original functions.

Unified proof of boundedness of maximal operator on multiple function spaces.

Provides a new perspective on the relationship between inequalities and maximal function estimates.

Abstract

We study generalized Poincar\'e inequalities. We prove that if a function satisfies a suitable inequality of Poincar\'e type, then the Hardy-Littlewood maximal function also obeys a meaningful estimate of similar form. As a by-product, we get a unified approach to proving that the maximal operator is bounded on Sobolev, Lipschitz and BMO spaces.