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Estimates and structure of α-harmonic functions
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  • Published: 27 March 2007

Estimates and structure of α-harmonic functions

  • Krzysztof Bogdan1,2,
  • Tadeusz Kulczycki2 &
  • Mateusz Kwaśnicki2 

Probability Theory and Related Fields volume 140, pages 345–381 (2008)Cite this article

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  • 60 Citations

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Abstract

We prove a uniform boundary Harnack inequality for nonnegative harmonic functions of the fractional Laplacian on arbitrary open set D. This yields a unique representation of such functions as integrals against measures on D c∪ {∞} satisfying an integrability condition. The corresponding Martin boundary of D is a subset of the Euclidean boundary determined by an integral test.

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Authors and Affiliations

  1. Department of Statistics, Purdue University, West Lafayette, IN, 47906, USA

    Krzysztof Bogdan

  2. Institute of Mathematics and Computer Science, Wrocław University of Technology, ul. Wybrzeże Wyspiańskiego 27, 50-370, Wrocław, Poland

    Krzysztof Bogdan, Tadeusz Kulczycki & Mateusz Kwaśnicki

Authors
  1. Krzysztof Bogdan
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  2. Tadeusz Kulczycki
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  3. Mateusz Kwaśnicki
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Corresponding author

Correspondence to Krzysztof Bogdan.

Additional information

K. Bogdan was supported by KBN grant 1 P03A 026 29 and RTN contract HPRN-CT-2001-00273-HARP. T. Kulczycki was supported by KBN grant 1 P03A 020 28 and RTN contract HPRN-CT-2001-00273-HARP. M. Kwaśnicki was supported by KBN grant 1 P03A 020 28 and RTN contractHPRN-CT-2001-00273-HARP.

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Bogdan, K., Kulczycki, T. & Kwaśnicki, M. Estimates and structure of α-harmonic functions. Probab. Theory Relat. Fields 140, 345–381 (2008). https://doi.org/10.1007/s00440-007-0067-0

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  • Received: 21 July 2006

  • Revised: 21 February 2007

  • Published: 27 March 2007

  • Issue Date: March 2008

  • DOI: https://doi.org/10.1007/s00440-007-0067-0

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Keywords

  • Boundary Harnack inequality
  • Martin representation
  • Stable process

Mathematics Subject Classification (2000)ss

  • Primary: 31C35
  • 60J50
  • Secondary: 31B05
  • 60G51
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