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A Fatou theorem for α-harmonic functions in Lipschitz domains
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  • Published: 15 March 2005

A Fatou theorem for α-harmonic functions in Lipschitz domains

  • Richard F. Bass1 &
  • Dahae You1 

Probability Theory and Related Fields volume 133, pages 391–408 (2005)Cite this article

  • 141 Accesses

  • 4 Citations

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Abstract.

We study α-harmonic functions in Lipschitz domains. We prove a Fatou theorem when the boundary function is bounded and Lp-Hölder continuous of order β with βp > 1.

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References

  1. Bass, R.F.: Probabilistic Techniques in Analysis. Springer Verlag, Berlin, 1995

  2. Bass, R.F., Levin, D.A.: Harnack inequalities for jump processes. Potential Anal. 17, 375–388 (2002)

    Article  Google Scholar 

  3. Bass, R.F., You, D.: A Fatou theorem for α-harmonic functions. Bull. Sci. Math. 127, 635–648 (2003)

    Article  Google Scholar 

  4. Blumenthal, R.M., Getoor, R.K.: Markov Processes and Their Potential Theory. Academic Press, New York, 1968

  5. Bogdan, K.: The boundary Harnack principle for the fractional Laplacian. Studia Math. 123, 43–80 (1997)

    Google Scholar 

  6. Bogdan, K.: Representation of α-harmonic functions in Lipschitz domains. Hiroshima Math. J. 29, 227–243 (1999)

    Google Scholar 

  7. Bogdan, K., Byczkowski, T.: On the Schrödinger operator based on the fractional Laplacian. Bull. Polish Acad. Sci. Math. 49, 291–301 (2001)

    Google Scholar 

  8. Bogdan, K., Byczkowski, T.: Potential theory of Schrödinger operator based on fractional Laplacian. Probab. Math. Statist. 20, 293–335 (2000)

    Google Scholar 

  9. Bogdan, K., Byczkowski, T.: Probabilistic proof of boundary Harnack principle for α-harmonic functions. Potential Anal. 11, 135–156 (1999)

    Article  Google Scholar 

  10. Bogdan, K., Byczkowski, T.: Potential theory for the α-stable Schrödinger operator on bounded Lipschitz domains. Studia Math. 133, 53–92 (1999)

    Google Scholar 

  11. Bogdan, K., Kulczycki, T., Nowak, A.: Gradient estimates for harmonic and q-harmonic functions of symmetric stable processes. Illinois J. Math. 46, 541–556 (2002)

    Google Scholar 

  12. Bogdan, K., Stós, A., Sztonyk, P.: Potential theory for Lévy stable processes. Bull. Polish Acad. Sci. Math. 50, 361–372 (2002)

    Google Scholar 

  13. Chen, Z.-Q.: Multidimensional symmetric stable processes. Korean J. Comput. Appl. Math. 6, 227–266 (1999)

    Google Scholar 

  14. Chen, Z.-Q., Song, R.: Martin boundary and integral representation for harmonic functions of symmetric stable processes. J. Funct. Anal. 159, 267–294 (1998)

    Article  Google Scholar 

  15. Chen, Z.-Q., Song, R.: Estimates on Green functions and Poisson kernels for symmetric stable processes. Math. Ann. 312, 465–501 (1998)

    Article  Google Scholar 

  16. Hunt, R.A., Wheeden, R.L.: On the boundary values of harmonic functions. Trans. Am. Math. Soc. 132, 307–322 (1968)

    Google Scholar 

  17. Ikeda, N., Watanabe, S.: On some relations between the harmonic measure and the Lévy measure for a certain class of Markov processes. J. Math. Kyoto Univ. 2, 79–95 (1962)

    Google Scholar 

  18. Jakubowski, T.: The estimates for the Green function in Lipschitz domains for the symmetric stable processes. Probab. Math. Statist. 22, 419–441 (2002)

    Google Scholar 

  19. Landkof, N.S.: Foundations of Modern Potential Theory. Springer, New York, 1972

  20. Song, R., Wu, J.-M.: Boundary Harnack principle for symmetric stable processes. J. Funct. Anal. 168, 403–427 (1999)

    Article  Google Scholar 

  21. Stein, E.M.: Singular Integrals and Differentiability Properties of Functions. Princeton Univ. Press, Princeton, 1970

  22. Jonsson, A., Wallin, H.: Function spaces on subsets of Rn. Math. Rep. 2 (1), 1984

    Google Scholar 

  23. Wu, J.-M.: Harmonic measures for symmetric stable processes. Studia Math. 149, 281–293 (2002)

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Connecticut, Storrs, CT06269, USA

    Richard F. Bass & Dahae You

Authors
  1. Richard F. Bass
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  2. Dahae You
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Corresponding author

Correspondence to Richard F. Bass.

Additional information

Mathematics Subject Classifications (2000): Primary 31B25; Secondary 60J50

*Research supported by NSF Grant DMS0244737

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Cite this article

Bass, R., You, D. A Fatou theorem for α-harmonic functions in Lipschitz domains. Probab. Theory Relat. Fields 133, 391–408 (2005). https://doi.org/10.1007/s00440-005-0431-x

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  • Received: 16 July 2004

  • Revised: 05 January 2005

  • Published: 15 March 2005

  • Issue Date: November 2005

  • DOI: https://doi.org/10.1007/s00440-005-0431-x

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Keywords

  • Stable processes
  • Fatou theorem
  • α-harmonic
  • Nontangential convergence
  • Maximal function
  • Hitting probabilities
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