Relative Fatou's theorem for (-Δ)α/2-harmonic functions in bounded κ-fat open sets

Journal of Functional Analysis

Volumn 234, Issue 1, 2006, Pages 70-105

  Kim, Panki ^a  

^a UNIVERSITY OF ILLINOIS AT URBANA CHAMPAIGN   (United States)

Author keywords

Feynman Kac transforms; Green function; Harmonic function; Martin boundary; Martin kernel; Martin representation; Relative Fatou's theorem; Symmetric stable process

Indexed keywords

EID: 33645841631     PISSN: 00221236     EISSN: 10960783     Source Type: Journal    
DOI: 10.1016/j.jfa.2005.12.001     Document Type: Article

References (33)

1. H. Aikawa Boundary Harnack principle and Martin boundary for a uniform domain J. Math. Soc. Japan 53 1 2001 119-145
2. R.F. Bass Probabilistic Techniques in Analysis 1995 Springer-Verlag
3. R.F. Bass D. You A Fatou theorem for α-harmonic functions Bull. Sci. Math. 127 7 2003 635-648
4. K. Bogdan The boundary Harnack principle for the fractional Laplacian Studia Math. 123 1 1997 43-80
5. K. Bogdan T. Byczkowski Potential theory for the α-stable Schrödinger operator on bounded Lipschitz domains Studia Math. 133 1 1999 53-92
6. K. Bogdan B. Dyda Relative Fatou theorem for harmonic functions of rotation invariant stable processes in smooth domain Studia Math. 157 1 2003 83-96
7. K. Bogdan K. Burdzy Z.-Q. Chen Censored stable processes Probab. Theory Related Fields 127 2003 89-152
8. Z.-Q. Chen Multidimensional symmetric stable processes Korean J. Comput. Appl. Math. 6 2 1999 227-266
9. Z.-Q. Chen Gaugeability and conditional gaugeability Trans. Amer. Math. Soc. 354 2002 4639-4679
10. Z.-Q. Chen P. Kim Stability of Martin boundary under nonlocal Feynman-Kac perturbations Probab. Theory Related Fields 128 2004 525-564
11. Z.-Q. Chen R. Song Estimates on Green functions and Poisson kernels of symmetric stable processes Math. Ann. 312 1998 465-501
12. Z.-Q. Chen R. Song Martin boundary and integral representation for harmonic functions of symmetric stable processes J. Funct. Anal. 159 1998 267-294
13. Z.-Q. Chen R. Song General gauge and conditional gauge theorems Ann. Probab. 30 2002 1313-1339
14. Z.-Q. Chen R. Song Conditional gauge theorem for nonlocal Feynman-Kac transforms Probab. Theory Related Fields 125 2003 45-72
15. Z.-Q. Chen R. Song Drift transforms and Green function estimates for discontinuous processes J. Funct. Anal. 201 1 2003 262-281
16. Z.-Q. Chen R. Durrett G. Ma Holomorphic diffusions and boundary behavior of harmonic functions Ann. Probab. 25 1997 1103-1134
17. K.L. Chung Z. Zhao From Brownian Motion to Schrödinger's Equation 1995 Springer
18. J.L. Doob A non-probabilistic proof of the relative Fatou theorem Ann. Inst. Fourier (Grenoble) 9 1959 293-300
19. J.L. Doob A relativized Fatou theorem Proc. Natl. Acad. Sci. USA 45 1959 215-222
20. J.L. Doob Classical Potential Theory and Its Probabilistic Counterpart 1984 Springer
21. R. Durrett Brownian Motion and Martingales in Analysis 1984 Wadsworth
22. P. Fatou Séries trigonométriques et series de Taylor Acta Math. 30 1906 335-400
23. R.A. Hunt R.L. Wheeden On the boundary value of harmonic functions Trans. Amer. Math. Soc. 132 2 1968 307-322
24. R.A. Hunt R.L. Wheeden Positive harmonic functions on Lipschitz domains Trans. Amer. Math. Soc. 147 2 1968 507-527
25. D.S. Jerison C.E. Kenig Boundary behavior of harmonic functions in nontangentially accessible domains Adv. Math. 46 1982 80-147
26. P. Kim Fatou's theorem for censored stable processes Stochastic Process. Appl. 108 1 2003 63-92
27. H. Kunita T. Watanabe Markov processes and Martin boundaries Illinois J. Math. 9 1965 485-526
28. J.E. Littlewood On a theorem of Fatou Proc. London Math. Soc. 2 1927 172-176
29. O. Martio M. Vuorinen Whitney cubes, p-capacity, and Minkowski content Expos. Math. 5 1 1987 17-40
30. K. Michalik M. Ryznar Relative Fatou theorem for α-harmonic functions in Lipschitz domains Illinois J. Math. 48 3 2004 977-998
31. M. Sharpe General Theory of Markov Processes 1988 Academic Press
32. R. Song J. Wu Boundary Harnack principle for symmetric stable processes J. Funct. Anal. 168 2 1999 403-427
33. J.G. Wu Comparisons of kernel functions, boundary Harnack principle and relative Fatou theorem on Lipschitz domains Ann. Inst. Fourier (Grenoble) 28 4 1978 147-167