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Symmetrization, symmetric stable processes, and Riesz capacities


Author: Dimitrios Betsakos
Journal: Trans. Amer. Math. Soc. 356 (2004), 735-755
MSC (2000): Primary 31B15, 60J45
DOI: https://doi.org/10.1090/S0002-9947-03-03298-7
Published electronically: September 22, 2003
Addendum: Trans. Amer. Math. Soc. (recently posted)
MathSciNet review: 2022718
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Abstract: Let $\texttt{X}_t$ be a symmetric $\alpha$-stable process killed on exiting an open subset $D$ of $\mathbb R^n$. We prove a theorem that describes the behavior of its transition probabilities under polarization. We show that this result implies that the probability of hitting a given set $B$ in the complement of $D$ in the first exit moment from $D$ increases when $D$ and $B$ are polarized. It can also lead to symmetrization theorems for hitting probabilities, Green functions, and Riesz capacities. One such theorem is the following: Among all compact sets $K$ in $\mathbb R^n$with given volume, the balls have the least $\alpha$-capacity ( $0<\alpha<2$).


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  • 1. A. Baernstein II, A unified approach to symmetrization, in Partial Differential Equations of Elliptic type (Cortona 1992) pp.47-91, Cambridge Univ. Press, 1994. MR 96e:26019
  • 2. R. Bañuelos, R. Latala, and P. J. Méndez-Hernández, A Brascamp-Lieb-Luttinger-type inequality and applications to symmetric stable processes. Proc. Amer. Math. Soc. 129 (2001), 2997-3008. MR 2002c:60125
  • 3. D. Betsakos, Polarization, conformal invariants, and Brownian motion. Ann. Acad. Sci. Fenn. Ser. A I Math. 23 (1998), 59-82. MR 99g:31004
  • 4. R. M. Blumenthal and R. K. Getoor, Some theorems on stable processes. Trans. Amer. Math. Soc. 95 (1960), 263-273. MR 22:10013
  • 5. R. M. Blumenthal and R. K. Getoor, Markov Processes and Potential Theory. Academic Press, 1968. MR 41:9348
  • 6. K. Bogdan, The boundary Harnack principle for the fractional Laplacian. Studia Math. 123 (1997), 43-80. MR 98g:31005
  • 7. K. Bogdan and T. Byczkowski, Potential theory for the $\alpha$-stable Schrödinger operator on bounded Lipschitz domains. Studia Math. 133 (1999), 53-92. MR 99m:31010
  • 8. F. Brock and A. Yu. Solynin, An approach to symmetrization via polarization. Trans. Amer. Math. Soc. 352 (2000), 1759-1796. MR 2001a:26014
  • 9. Z. Q. Chen and R. Song, Intrinsic ultracontractivity and conditional gauge for symmetric stable processes. J. Funct. Anal. 150 (1997), 204-239. MR 98j:60103
  • 10. 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
  • 11. Z. Q. Chen and R. Song, Martin boundary and integral representation for harmonic functions of symmetric stable processes. J. Funct. Anal. 159 (1998), 267-294. MR 2000d:60128
  • 12. K. L. Chung, Lectures from Markov Processes to Brownian Motion. Springer-Verlag, 1982. MR 84c:60091
  • 13. K. L. Chung and M. Rao, Equilibrium and energy. Probab. Math. Statist. 1 (1980), 99-108. MR 83f:60102
  • 14. V. N. Dubinin, Capacities and geometric transformations of subsets in n-space. Geometric and Functional Anal. 3 (1993), 342-369. MR 94f:31008
  • 15. V. N. Dubinin, Symmetrization in the geometric theory of functions of a complex variable. Russian Math. Surveys 49 (1994), no. 1, 1-79. MR 96b:30054
  • 16. E. B. Dynkin, Markov Processes, Volume I. Springer 1965. MR 33:1887
  • 17. E. B. Dynkin, Markov Processes, Volume II. Springer 1965. MR 33:1887
  • 18. N. S. Landkof, Foundations of Modern Potential Theory. Springer-Verlag, 1972. MR 50:2520
  • 19. P. Mattila, Orthogonal projections, Riesz capacities, and Minkowski content. Indiana Univ. Math. J. 39 (1990), 185-198. MR 91d:28018
  • 20. G. Pólya and G. Szegö, Isoperimetric Inequalities in Mathematical Physics. Princeton Univ. Press, 1951. MR 13:270d
  • 21. S. C. Port, On hitting places for stable processes. Ann. Math. Statist. 38 (1967), 1021-1026. MR 35:5002
  • 22. S. C. Port, A remark on hitting places for transient stable process. Ann. Math. Statist. 39 (1968), 365-371. MR 37:980
  • 23. S. C. Port and C. J. Stone, Brownian Motion and Classical Potential Theory. Academic Press, 1978. MR 58:11459
  • 24. J. Sarvas, Symmetrization of condensers in $n$-space. Ann. Acad. Sci. Fenn. Ser. A I No. 522 (1972), 1-44. MR 50:606
  • 25. A. Yu. Solynin, Functional inequalities via polarization. Algebra i Analiz 8 (1996), 148-185 (in Russian); English transl. in St. Petersburg Math. J. 8 (1997), 1015-1038. MR 98e:30001a
  • 26. R. Song and J.-M. Wu, Boundary Harnack principle for symmetric stable processes. J. Funct. Anal. 168 (1999), 403-427. MR 2001b:60092
  • 27. G. Szegö, Über einige Extremalaufgaben der Potentialtheorie, Math. Zeitschrift 31 (1930), 583-593.
  • 28. V. Wolontis, Properties of conformal invariants. Amer. J. of Math. 74 (1952), 587-606. MR 14:36c

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Additional Information

Dimitrios Betsakos
Affiliation: Department of Mathematics, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece
Email: betsakos@auth.gr

DOI: https://doi.org/10.1090/S0002-9947-03-03298-7
Keywords: Symmetrization, symmetric stable process, polarization, transition function, $\alpha$-harmonic measure, Green function, Riesz capacity
Received by editor(s): July 14, 2002
Received by editor(s) in revised form: January 23, 2003
Published electronically: September 22, 2003
Dedicated: Dedicated to Albert Baernstein on the occasion of the thirty years of his star-function
Article copyright: © Copyright 2003 American Mathematical Society

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