Relative Fatou theorem for α-harmonic functions in lipschitz domains

Illinois Journal of Mathematics

Volumn 48, Issue 3, 2004, Pages 977-998

  Michalik, Krzysztof ^a   Ryznar, Michal ^a  

^a WROCLAW UNIVERSITY OF TECHNOLOGY   (Poland)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 17044423812     PISSN: 00192082     EISSN: None     Source Type: Journal    
DOI: 10.1215/ijm/1258131065     Document Type: Article

References (30)

1. R. Bañuelos and K. Bogdan, Symmetric stable processes in cones, preprint, 2003.
2. R. Bañuelos and C. N. Moore, Probabilistic behavior of harmonic functions, Progress in Mathematics, vol. 175, Birkhäuser Verlag, Basel, 1999. MR 2001j:31003
3. R. F. Bass, Probabilistic techniques in analysis, Probability and its Applications (New York), Springer-Verlag, New York, 1995. MR 96e:60001
4. R. F. Bass and K. Burdzy, A probabilistic proof of the boundary Harnack principle, Seminar on Stochastic Processes, 1989 (San Diego, CA, 1989), Progr. Probab., vol. 18, Birkhäuser Boston, Boston, MA, 1990, pp. 1-16. MR 92c:60106
5. A. S. Besicovitch, A general form of the covering principle and relative differentiation of additive functions. II, Proc. Cambridge Philos. Soc. 42 (1946), 1-10. MR 7,281e
6. R. M. Blumenthal and R. K. Getoor, Markov processes and potential theory, Pure and Applied Mathematics, Vol. 29, Academic Press, New York, 1968. MR 41 #9348
7. R. M. Blumenthal, R. K. Getoor, and D. B. Ray, On the distribution of first hits for the symmetric stable processes., Trans. Amer. Math. Soc. 99 (1961), 540-554. MR 23 #A4179
8. K. Bogdan, The boundary Harnack principle for the fractional Laplacian, Studia Math. 123 (1997), 43-80. MR 98g:31005
9. _, Representation of α-harmonic functions in Lipschitz domains, Hiroshima Math. J. 29 (1999), 227-243. MR 2001a:31011
10. K. Bogdan and T. Byczkowski, Potential theory for the α-stable Schrödinger operator on bounded Lipschitz domains, Studia Math. 133 (1999), 53-92. MR 99m:31010
11. _, Probabilistic proof of boundary Harnack principle for α-harmonic functions, Potential Anal. 11 (1999), 135-156. MR 2001a:60089
12. K. Bogdan and B. Dyda, Relative Fatou theorem for harmonic functions of rotation invariant stable processes in smooth domains, Studia Math. 157 (2003), 83-96. MR 1 980 119
13. K. Bogdan, T. Kulczycki, and A. Nowak, Gradient estimates for harmonic and q-harmonic functions of symmetric stable processes, Illinois J. Math. 46 (2002), 541-556. MR 2004b:60188
14. K. Bogdan, A. Stós, and P. Sztonyk, Potential theory for Lévy stable processes, Bull. Polish Acad. Sci. Math. 50 (2002), 361-372. MR 2003m:60125
15. L. Carleson, Selected problems on exceptional sets, Van Nostrand Mathematical Studies, No. 13, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1967. MR 37 #1576
16. Z.-Q. Chen and R. Song, Estimates on Green functions and Poisson kernels for symmetric stable processes, Math. Ann. 312 (1998), 465-501. MR 2000b:60179
17. J. L. Doob, A relativized Fatou theorem, Proc. Nat. Acad. Sci. U.S.A. 45 (1959), 215-222. MR 21 #5822
18. _, Classical potential theory and its probabilistic counterpart, Grundlehren der Mathematischen Wissenschaften, vol. 262, Springer-Verlag, New York, 1984. MR 85k:31001
19. T. Jakubowski, The estimates for the Green function in Lipschitz domains for the symmetric stable processes, Probab. Math. Statist. 22 (2002), 419-441. MR 2004g:60069
20. _, Problème de Dirichlet pour les fonctions α-harmoniquea sur les domaines coniques, preprint, 2003.
21. α/2)-/ iarmomc function in κ-fat open set, preprint, 2004.
22. T. Kulczycki, Properties of Green function of symmetric stable processes, Probab. Math. Statist. 17 (1997), 339-364. MR 98m:60119
23. N. S. Landkof, Foundations of modern potential theory, Springer-Verlag, New York, 1972. MR 50 #2520
24. n of symmetric stable processes, Illinois J. Math. 46 (2002), 155-163. MR 20031:60070
25. K. Michalik, Sharp estimates of the Green function, Poisson kernel and Martin kernel of cones for symmetric stable processes, preprint, 2003.
26. R. Song and J.-M. Wu, Boundary Harnack principle for symmetric stable processes, J. Fund. Anal. 168 (1999), 403-427. MR 2001b:60092
27. E. M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 44 #7280
28. P. Sztonyk, On harmonic measure for Lévy processes, Probab. Math. Statist. 20 (2000), 383-390. MR 2002c:60126
29. J. M. Wu, Comparisons of kernel functions, boundary Harnack principle and relative Fatou theorem on Lipschitz domains, Ann. Inst. Fourier (Grenoble) 28 (1978), 147-167, vi. MR 80g:31005
30. _, Harmonic measures for symmetric stable processes, Studia Math. 149 (2002), 281-293. MR 2003f:60139