A maximal Function Approach to Two-Measure Poincaré Inequalities

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Researchers

Research units

  • University of Jyväskylä

Abstract

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.

Details

Original languageEnglish
JournalJOURNAL OF GEOMETRIC ANALYSIS
Publication statusE-pub ahead of print - 2018
MoE publication typeA1 Journal article-refereed

    Research areas

  • Geodesic two-measure space, Poincaré inequality, Self-improvement

ID: 31918033