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This paper extends the self-improvement result of Keith and Zhong in [16] tothe 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 abalance condition on the measures. The corresponding result for a maximalPoincar\'e inequality is also considered. In this case the left-hand side inthe Poincar\'e inequality is replaced with an integral of a sharp maximalfunction and the results hold without a balance condition. Moreover, validityof maximal Poincar\'e inequalities is used to characterize the self-improvementof two-measure Poincar\'e inequalities. Examples are constructed to illustratethe role of the assumptions. Harmonic analysis and PDE techniques are usedextensively in the arguments.

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