Non-local Gehring Lemmas in Spaces of Homogeneous Type and Applications

  • Pascal Auscher
  • Simon Bortz
  • Moritz Egert
  • Olli SaariEmail author


We prove a self-improving property for reverse Hölder inequalities with non-local right-hand side. We attempt to cover all the most important situations that one encounters when studying elliptic and parabolic partial differential equations. We present applications to non-local extensions of \(A_{\infty }\) weights and fractional elliptic divergence form equations. We write our results in spaces of homogeneous type.


Gehring’s lemma (non-local) Reverse Hölder inequalities Spaces of homogeneous type (very weak) \(A_\infty \) weights \(C_p\) weights Fractional elliptic equations Self-improvement properties 

Mathematics Subject Classification

Primary: 30L99 Secondary: 34A08 42B25 



We thank Tuomas Hytönen for an enlightening discussion on the topics of this work that led to the results extending the \(A_{\infty }\) class and Carlos Pérez for pointing out the connection to the \(C_p\) class. We also thank an anonymous referee for suggesting that our results should apply to the fractional divergence form equation of Shieh–Spector [25] rather than the toy model investigated in an earlier version of our manuscript.


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Authors and Affiliations

  1. 1.Laboratoire de Mathématiques d’Orsay, Univ. Paris-Sud, CNRSUniversité Paris-SaclayOrsayFrance
  2. 2.Laboratoire Amiénois de Mathématique Fondamentale et Appliquée, UMR 7352 du CNRSUniversité de Picardie-Jules VerneAmiensFrance
  3. 3.Department of MathematicsUniversity of WashingtonSeattleUSA
  4. 4.Department of Mathematics and Systems AnalysisAalto UniversityAaltoFinland
  5. 5.Mathematical InstituteUniversity of BonnBonnGermany

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