A maximal function approach to two-measure Poincar\'e inequalities
Abstract
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 -Poincar\'e inequality for improves to a -Poincar\'e inequality for some 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.
Cite
@article{arxiv.1801.06978,
title = {A maximal function approach to two-measure Poincar\'e inequalities},
author = {Juha Kinnunen and Riikka Korte and Juha Lehrbäck and Antti V. Vähäkangas},
journal= {arXiv preprint arXiv:1801.06978},
year = {2018}
}