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