  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.
