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The Journal of Geometric Analysis

, Volume 29, Issue 2, pp 1763–1810 | Cite as

A maximal Function Approach to Two-Measure Poincaré Inequalities

  • Juha Kinnunen
  • Riikka Korte
  • Juha Lehrbäck
  • Antti V. VähäkangasEmail author
Article
  • 63 Downloads

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 (pp)-Poincaré inequality for \(1<p<\infty \) improves to a \((p,p-{\varepsilon })\)-Poincaré inequality for some \({\varepsilon }>0\) 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.

Keywords

Poincaré inequality Self-improvement Geodesic two-measure space 

Mathematics Subject Classification

31E05 35A23 46E35 

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Copyright information

© Mathematica Josephina, Inc. 2018

Authors and Affiliations

  1. 1.Department of MathematicsAalto UniversityEspooFinland
  2. 2.Department of Mathematics and StatisticsUniversity of JyvaskylaJyvaskylaFinland

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