A maximal Function Approach to Two-Measure Poincaré Inequalities

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 )-Poincaré inequality for 1 < p < ∞ improves to a ( p , p - ε ) -Poincaré inequality for some ε > 0 under a balance condition on the measures. The corresponding result for a maximal Poincaré inequality is also considered. In this case the left-hand side in the Poincaré inequality is replaced with an integral of a sharp maximal function and the results hold without a balance condition. Moreover, validity of maximal Poincaré inequalities is used to characterize the self-improvement of two-measure Poincaré inequalities. Examples are constructed to illustrate the role of the assumptions. Harmonic analysis and PDE techniques are used extensively in the arguments.