A maximal Function Approach to Two-Measure Poincaré Inequalities
Kinnunen, J., Korte, R., Lehrbäck, J., & Vähäkangas, A. (2019). A maximal Function Approach to Two-Measure Poincaré Inequalities. Journal of Geometric Analysis, 29(2), 1763-1810. https://doi.org/10.1007/s12220-018-0061-z
Published inJournal of Geometric Analysis
© Mathematica Josephina, Inc. 2018
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
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.
PublisherSpringer New York
ISSN Search the Publication Forum1050-6926
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