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The Journal of Geometric Analysis

, Volume 28, Issue 4, pp 3278–3299 | Cite as

BMO Solvability and Absolute Continuity of Harmonic Measure

  • Steve Hofmann
  • Phi Le
Article
  • 73 Downloads

Abstract

We show that for a uniformly elliptic divergence form operator L, defined in an open set \(\Omega \) with Ahlfors–David regular boundary, BMO solvability implies scale-invariant quantitative absolute continuity (the weak-\(A_\infty \) property) of elliptic-harmonic measure with respect to surface measure on \(\partial \Omega \). We do not impose any connectivity hypothesis, qualitative, or quantitative; in particular, we do not assume the Harnack Chain condition, even within individual connected components of \(\Omega \). In this generality, our results are new even for the Laplacian. Moreover, we obtain a partial converse, assuming in addition that \(\Omega \) satisfies an interior Corkscrew condition, in the special case that L is the Laplacian.

Keywords

BMO Dirichlet problem Harmonic measure Divergence form elliptic equations Weak-\(A_\infty \) Ahlfors–David regularity Uniform rectifiability 

Mathematics Subject Classification

42B99 42B25 35J25 42B20 

Notes

Acknowledgements

We are grateful to Simon Bortz for a suggestion which has simplified one of our arguments in Sect. 4. We also thank the referee for a careful reading of the manuscript, for suggesting numerous improvements to the exposition, and especially for pointing out an error in the original proof of Lemma 3.3. Finally, we thank Chema Martell for suggesting the approach that we have used in Sect. 5 to treat the case of an unbounded domain with bounded boundary. The authors were supported by NSF Grant Number DMS-1361701. Steve Hofmann was also supported by NSF Grant Number DMS-1664047

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Copyright information

© Mathematica Josephina, Inc. 2017

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of MissouriColumbiaUSA
  2. 2.Department of MathematicsSyracuse UniversitySyracuseUSA

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