Advertisement

The Journal of Geometric Analysis

, Volume 29, Issue 2, pp 1193–1205 | Cite as

On Harmonic Measure and Rectifiability in Uniform Domains

  • Mihalis MourgoglouEmail author
Article
  • 83 Downloads

Abstract

Let \(\Omega \subset \mathbb {R}^{d+1}\), \(d \ge 1\), be a uniform domain with lower d-Ahlfors–David regular and d-rectifiable boundary. We show that if the d-Hausdorff measure \(\mathcal {H}^d|_{\partial \Omega }\) is locally finite, then \(\mathcal {H}^d|_{\partial \Omega }\) is absolutely continuous with respect to harmonic measure for \(\Omega \).

Mathematics Subject Classification

31A15 28A75 28A78 

Notes

Acknowledgements

We warmly thank J. Azzam and X. Tolsa for their encouragement and several discussions pertaining to this work and rectifiability. We are particularly grateful to J. Azzam for explaining the techniques developed in his earlier work on the same topic. We would also like to thank the anonymous referees for their valuable comments that helped us improve the paper. The current manuscript was finished and uploaded on ArXiv in mid-2015 when the author was a post-doc of X. Tolsa at Universitat Autònoma de Barcelona supported by the ERC Grant 320501 of the European Research Council (FP7/2007–2013).

References

  1. 1.
    Akman, M., Badger, M., Hofmann, S., Martell, J.M.: Rectifiability and elliptic measures on \(1\)-sided NTA domains with Ahlfors–David regular boundaries. Trans. Am. Math. Soc. 369, 5711–5745 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Azzam, J.: Sets of absolute continuity for harmonic measure in NTA domains. Potential Anal. 45(3), 403–433 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Azzam, J., Hofmann, S., Martell, J.M., Nyström, K., Toro, T.: A new characterization of chord-arc domains. J. Eur. Math. Soc. 19(4), 967–981 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Azzam, J., Mourgoglou, M., Tolsa, X.: Singular sets for harmonic measure on locally flat domains with locally finite surface measure. Int. Math. Res. Not. 2017(12), 3751–3773 (2017)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Badger, M.: Null sets of harmonic measure on NTA domains: Lipschitz approximation revisited. Math. Z. 270(1–2), 241–262 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bishop, C.J., Jones, P.W.: Harmonic measure and arclength. Ann. of Math. 132(3), 511–547 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bortz, S., Hofmann, S.: Harmonic measure and approximation of uniformly rectifiable sets. Rev. Mat. Iberoam. 33(1), 351–373 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Christ, M.: A \(T(b)\) theorem with remarks on analytic capacity and the Cauchy integral. Colloq. Math. 60(2), 601–628 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Dahlberg, B.E.J.: Estimates of harmonic measure. Arch. Ration. Mech. Anal. 65(3), 275–288 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    David, G.: Morceaux de graphes lipschitziens et intégrales singulières sur une surface. Rev. Mat. Iberoam. 4(1), 73–114 (1988)CrossRefzbMATHGoogle Scholar
  11. 11.
    David, G., Jerison, D.: Lipschitz approximation to hypersurfaces, harmonic measure, and singular integrals. Indiana Univ. Math. J. 39(3), 831–845 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    David, G., Semmes, S.W.: Singular integrals and rectifiable sets in \({ R}^n\): beyond Lipschitz graphs. Astérisque 193, 152 (1991)Google Scholar
  13. 13.
    David, G., Semmes, S.W.: Analysis of and on Uniformly Rectifiable Sets. Mathematical Surveys and Monographs, vol. 38. American Mathematical Society, Providence, RI (1993)zbMATHGoogle Scholar
  14. 14.
    Hofmann, S., Martell, J.M.: Uniform rectifiability and harmonic measure I: uniform rectifiability implies poisson kernels in \(L^{p}\). Ann. Sci. Ecol. Norm. Sup. 47(3), 577–654 (2014)CrossRefzbMATHGoogle Scholar
  15. 15.
    Hofmann, S., Martell, J.M.: Uniform rectifiability and harmonic measure, IV: Ahlfors regularity plus Poisson kernels in \(L^p\) impies uniform rectifiability. arXiv:1505.06499 (2015)
  16. 16.
    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(3), 653–694 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Hofmann, S., Martell, J.M., Uriarte-Tuero, I.: Uniform rectifiability and harmonic measure, II: Poisson kernels in \(L^p\) imply uniform rectifiability. Duke Math. J. 8, 1601–1654 (2014)CrossRefzbMATHGoogle Scholar
  18. 18.
    Hytönen, T., Martikainen, H.: Non-homogeneous \(Tb\) theorem and random dyadic cubes on metric measure spaces. J. Geom. Anal. 22(4), 1071–1107 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Jerison, D.S., Kenig, C.E.: Boundary behavior of harmonic functions in nontangentially accessible domains. Adv. Math. 46(1), 80–147 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Lavrentiev, M.: Boundary problems in the theory of univalent functions (Russian), Math Sb. 43, 815–846 (1936). AMS Transl. Series 32, 1–35 (1936)Google Scholar
  21. 21.
    Mattila, P.: Geometry of sets and measures in Euclidean spaces. In: Fractals and Rectifiability. Cambridge Studies in Advanced Mathematics, vol. 44. Cambridge University Press, Cambridge (1995)Google Scholar
  22. 22.
    Mourgoglou, M., Tolsa, X.: Harmonic measure and Riesz transform in uniform and general domains, to appear in J. Reine Angew. Math. arXiv:1509.08386 (2015)
  23. 23.
    Riesz, F., Riesz, M.: Über Randwerte einer analytischen Functionen. In: Compte rendu du quatrième Congrès des Mathématiciens Scandinaves: tenu à Stockholm du 30 août au 2 Septembre 1916, pp. 27–44. Malmö (1955)Google Scholar
  24. 24.
    Wolff, T.H.: Counter Examples with Harmonic Gradients in \({ R}^3\). Essays on Fourier Analysis in Honor of Elias M. Stein. Mathematics Series, vol. 42, pp. 321–384. Princeton University Press, Princeton (1991)Google Scholar
  25. 25.
    Wu, J.-M.: On singularity of harmonic measure in space. Pac. J. Math 121(no. 2), 485–496 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Ziemer, W.: Some remarks on harmonic measure in space. Pac. J. Math. 55(2), 629–637 (1974)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Mathematica Josephina, Inc. 2018

Authors and Affiliations

  1. 1.Departament de MatemàtiquesUniversitat Autònoma de Barcelona and Centre de Reserca Matemàtica, Edifici C Facultat de CiènciesBellaterraCatalonia
  2. 2.Departamento de MatemáticasUniversidad del País VascoLeioaSpain
  3. 3.Ikerbasque, Basque Foundation for ScienceBilbaoSpain

Personalised recommendations