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


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.


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

Mathematics Subject Classification

42B99 42B25 35J25 42B20 



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


  1. 1.
    Bennewitz, B., Lewis, J.L.: On weak reverse Hölder inequalities for nondoubling harmonic measures. Complex Var. Theory Appl. 49(7–9), 571–582 (2004)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Bishop, C., Jones, P.: Harmonic measure and arclength. Ann. Math. 2(132), 511–547 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bourgain, J.: On the Hausdorff dimension of harmonic measure in higher dimensions. Invent. Math. 87, 477–483 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    David, G., Semmes, S.: Singular integrals and rectifiable sets in \({\mathbb{R}}^n\): beyond Lipschitz graphs. Asterisque 193 (1991)Google Scholar
  5. 5.
    David, G., Semmes, S.: Analysis of and on Uniformly Rectifiable Sets, Mathematical Monographs and Surveys. AMS, Providence (1993)CrossRefzbMATHGoogle Scholar
  6. 6.
    Dindos, M., Kenig, C.E., Pipher, J.: BMO solvability and the \(A_\infty \) condition for elliptic operators. J. Geom. Anal. 21, 78–95 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Fabes, E.B., Neri, U.: Dirichlet problem in Lipschitz domains with BMO data. Proc. Am. Math. Soc. 78, 33–39 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Garcia-Cuerva, J., Rubio de Francia, J.L.: Weighted Norm Inequalities and Related Topics, North-Holland Mathematics Studies. Elsevier, Amsterdam (1985)zbMATHGoogle Scholar
  9. 9.
    Garnett, J., Mourgoglou, M., Tolsa, X.: Uniform rectifiability from Carleson measure estimates and \({\varepsilon }\)-approximability of bounded harmonic functions, preprint. arXiv:1611.00264
  10. 10.
    Heinonen, J., Kilpeläinen, T., Martio, O.: Nonlinear Potential Theory of Degenerate Elliptic Equations, Rev edn. Dover, New York (2006)zbMATHGoogle Scholar
  11. 11.
    Hofmann, S., Le, P., Martell, J.M., Nyström, K.: The weak-\(A_\infty \) property of harmonic and \(p\)-harmonic measures implies uniform rectifiability. Anal. PDE 10, 513–558 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Hofmann, S., Martell, J.M.: Uniform Rectifiability and harmonic measure IV: Ahlfors regularity plus Poisson kernels in \(L^p\) implies uniform rectifiability, unpublished manuscript. arXiv:1505.06499
  13. 13.
    Hofmann, S., Martell, J.M., Mayboroda, S.: Uniform rectifiability, carleson measure estimates, and approximation of harmonic functions. Duke Math. J. 165, 2331–2389 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Hofmann, S., Martell, J.M., Mayboroda, S.: Transference of scale-invariant estimates from Lipschitz to Non-tangentially accessible to Uniformly rectifiable domains (in preparation)Google Scholar
  15. 15.
    Hofmann, S., Martell, J.M., Toro, T.: General Divergence Form Elliptic Operators on Domains with ADR Boundaries, and on 1-sided NTA Domains (in preparation)Google Scholar
  16. 16.
    Jerison, D., Kenig, C.: Boundary behavior of harmonic functions in nontangentially accessible domains. Adv. Math. 46(1), 80–147 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Kenig, C.E.: Harmonic analysis techniques for second order elliptic boundary value problems, CBMS Regional Conference Series in Mathematics, 83. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI (1994)Google Scholar
  18. 18.
    Kenig, C., Kirchheim, B., Pipher, J., Toro, T.: Square functions and the \(A_\infty \) property of elliptic measures. J. Geom. Anal. 26, 2383–2410 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Mattila, P., Melnikov, M., Verdera, J.: The Cauchy integral, analytic capacity, and uniform rectifiability. Ann. Math. 144(1), 127–136 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Mourgoglou, M., Tolsa, X.: Harmonic measure and Riesz transform in uniform and general domains, preprint. arXiv:1509.08386
  21. 21.
    Nazarov, F., Tolsa, X., Volberg, A.: On the uniform rectifiability of ad-regular measures with bounded Riesz transform operator: the case of codimension 1. Acta Math. 213, 237–321 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Zhao, Z..: BMO solvability and the \(A_\infty \) condition of the elliptic measure in uniform domains. J. Geom. Anal. (to appear)Google Scholar

Copyright information

© Mathematica Josephina, Inc. 2017

Authors and Affiliations

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

Personalised recommendations