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

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 $1<p<\mathrm{\infty }$ improves to a $(p,p-\epsilon )$-Poincar'e inequality for some $\epsilon >0$ 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.