A maximal function approach to two-measure Poincar\'e inequalities

TL;DR

This paper extends self-improvement results for two-measure Poincaré inequalities, showing that under certain conditions, these inequalities improve to stronger forms, with applications to maximal functions and characterizations via harmonic analysis and PDE techniques.

Contribution

It introduces a new approach using maximal functions to analyze two-measure Poincaré inequalities and establishes conditions for their self-improvement.

Findings

Two-measure $(p,p)$-Poincaré inequalities improve to $(p,p- ext{small})$-inequalities under a balance condition.

Maximal Poincaré inequalities hold without the balance condition and characterize the self-improvement.

Examples illustrate the necessity of assumptions for the inequalities' validity.

Abstract

This paper extends the self-improvement result of Keith and Zhong in [16] to the two-measure case. Our main result shows that a two-measure $(p,p)$-Poincar\'e inequality for $1<p<\infty$ improves to a $(p,p-\varepsilon)$-Poincar\'e inequality for some $\varepsilon>0$ under a balance condition on the measures. The corresponding result for a maximal Poincar\'e inequality is also considered. In this case the left-hand side in the Poincar\'e inequality is replaced with an integral of a sharp maximal function and the results hold without a balance condition. Moreover, validity of maximal Poincar\'e inequalities is used to characterize the self-improvement of two-measure Poincar\'e inequalities. Examples are constructed to illustrate the role of the assumptions. Harmonic analysis and PDE techniques are used extensively in the arguments.