A maximal Function Approach to Two-Measure Poincaré Inequalities

Abstract

This paper extends the self-improvement result of Keith and Zhong in Keith and Zhong (Ann. Math. 167(2):575–599, 2008) to the two-measure case. Our main result shows that a two-measure (p, p)-Poincare inequality for $$1<p<\infty $$ improves to a $$(p,p-{\varepsilon })$$ -Poincare inequality for some $${\varepsilon }>0$$ under a balance condition on the measures. The corresponding result for a maximal Poincare inequality is also considered. In this case the left-hand side in the Poincare inequality is replaced with an integral of a sharp maximal function and the results hold without a balance condition. Moreover, validity of maximal Poincare inequalities is used to characterize the self-improvement of two-measure Poincare inequalities. Examples are constructed to illustrate the role of the assumptions. Harmonic analysis and PDE techniques are used extensively in the arguments.