Potential Analysis

, Volume 31, Issue 1, pp 57–77 | Cite as

Intrinsic Ultracontractivity for Stable Semigroups on Unbounded Open Sets



We study the intrinsic ultracontractivity of the semigroup of the isotropic stable process on an unbounded open set. A two-sided estimate of the first eigenfunction of the semigroup is derived. We obtain general geometric conditions for intrinsic ultracontractivity which yield a necessary and sufficient condition for non-degenerate horn-shaped sets.


Intrinsic ultracontractivity Isotropic stable process First eigenfunction 

Mathematics Subject Classifications (2000)

Primary 47D07 60G52 Secondary 60J45 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bañuelos, R.: Intrinsic ultracontractivity and eigenvalue estimates for Schrödinger operators. J. Funct. Anal. 100, 181–206 (1991)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Bañuelos, R.: Lifetime and heat kernel estimates in nonsmooth domains. In: Proc. University of Chicago Conference in Partial Differential Equations with Minimal Smoothness, vol. 42. IMA, Minneapolis (1990)Google Scholar
  3. 3.
    Bañuelos, R.: Sharp estimates for Dirichlet eigenfunctions in simply connected domains. J. Differential Equations 125(1), 282–298 (1996)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Bañuelos, R., van den Berg, M.: Dirichlet eigenfunctions for horn-shaped regions and Laplacians on cross sections. J. London Math. Soc. 53, 503–511 (1996)MATHMathSciNetGoogle Scholar
  5. 5.
    Bañuelos, R., Davis, B.: Geometrical characterization of intrinsic ultracontractivity for planar domains with boundaries given by graphs of functions. Indiana Univ. Math J. 41, 885–913 (1992)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Bañuelos, R., Davis, B.: Sharp estimates for Dirichlet eigenfunctions in horn-shaped regions. Comm. Math. Phys. 150(1), 209–215 (1992)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Bañuelos, R., Kulczycki, T.: The Cauchy process and the Steklov problem. J. Funct. Anal. 211(2), 355–423 (2004)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Bass, R.F., Burdzy, K.: Lifetimes of conditioned diffusions. Probab. Theory Related Fields 91, 405–443 (1992)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Bliedtner, J., Hansen, W.: Potential Theory. An Analytic and Probabilistic Approach to Balayage. Springer, Berlin (1986)MATHGoogle Scholar
  10. 10.
    Blumenthal, R.M., Getoor, R.K., Ray, D.B.: On the distribution of first hits for the symmetric stable processes. Trans. Amer. Math. Soc. 99, 540–554 (1961)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Bogdan, K.: The boundary Harnack principle for the fractional Laplacian. Stud. Math. 123, 43–80 (1997)MATHMathSciNetGoogle Scholar
  12. 12.
    Bogdan, K., Byczkowski, T.: Potential theory for the α-stable Schrödinger operator on bounded Lipschitz domains. Stud. Math. 133(1), 53–92 (1999)MATHMathSciNetGoogle Scholar
  13. 13.
    Bogdan, K., Byczkowski, T.: Probabilistic proof of boundary Harnack principle for α-harmonic functions. Potential Anal. 11(2), 135–156 (1999)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Bogdan, K., Kulczycki, T., Kwaśnicki, M.: Estimates and structure of α-harmonic functions. Probab. Theory Related Fields 140(3–4), 345–381 (2008)MATHMathSciNetGoogle Scholar
  15. 15.
    Bogdan, K., Żak, T.: On Kelvin transformation. J. Theoret. Probab. 19(1), 89–120 (2006)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Chen, Z.Q.: Multidimensional symmetric stable processes. Korean J. Comput. Appl. Math. 6(2), 227–266 (1999)MATHMathSciNetGoogle Scholar
  17. 17.
    Chen, Z.Q., Song, R.: Intrinsic ultracontractivity and conditional gauge for symmetric stable processes. J. Funct. Anal. 150(1), 204–239 (1997)MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Chen, Z.Q., Song, R.: Intrinsic ultracontractivity, conditional lifetimes and conditional gauge for symmetric stable processes on rough domains. Illinois J. Math. 44(1), 138–160 (2000)MATHMathSciNetGoogle Scholar
  19. 19.
    Davies, E.B.: Criteria for ultracontractivity. Ann. Inst. Henri Poincare 43A, 181–194 (1985)Google Scholar
  20. 20.
    Davies, E.B., Simon, B.: Ultracontractivity and the heat kernel for Schrödinger operators and Dirichlet Laplacians. J. Funct. Anal. 59, 335–395 (1984)MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Davis, B.: Intrinsic ultracontractivity and the Dirichlet Laplacian. J. Funct. Anal. 100(1), 162–180 (1991)MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    DeBlassie, D.: The lifetime of conditioned Brownian motion in certain Lipschitz domains. Probab. Theory Related Fields 75, 431–458 (1987)CrossRefMathSciNetGoogle Scholar
  23. 23.
    Dziubański, J.: Asymptotic behaviour of densities of stable semigroups of measures. Probab. Theory Related Fields 87(4), 459–467 (1992)CrossRefGoogle Scholar
  24. 24.
    Fabes, E.B., Garofalo, N., Salsa, S.: A backward Harnack inequality and Fatou theorem for nonnegative solutions of parabolic equations. Illinois J. Math. 30, 536–565 (1986)MATHMathSciNetGoogle Scholar
  25. 25.
    Getoor, R.K.: Markov operators and their associated semi-groups. Pacific J. Math. 9, 449–472 (1959)MATHMathSciNetGoogle Scholar
  26. 26.
    Grzywny, T.: Intrinsic ultracontractivity for Lévy processes. Probab. Math. Statist. 28(1), 91–106 (2008)MATHMathSciNetGoogle Scholar
  27. 27.
    Kenig, C.E., Pipher, J.: The h-path distribution of the lifetime of conditioned Brownian motion for nonsmooth domains. Probab. Theory Related Fields 82(4), 615–623 (1989)MATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Kim, P., Song, R.: Intrinsic ultracontractivity for non-symmetric Levy processes. Forum Math. 21(1), 43–66 (2009)MATHCrossRefGoogle Scholar
  29. 29.
    Kulczycki, T.: Intrinsic ultracontractivity for symmetric stable processes. Bull. Polish Acad. Sci. Math. 46(3), 325–334 (1998)MATHMathSciNetGoogle Scholar
  30. 30.
    Landkof, N.S.: Foundations of Modern Potential theory. Springer, New York (1972)MATHGoogle Scholar
  31. 31.
    Lindeman, A., Pang, M.H., Zhao, Z.: Sharp bounds for ground state eigenfunctions on domains with horns and cusps. J. Math. Anal. Appl. 212(2), 381–416 (1997)MATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    Méndez-Hernández, P.J.: Toward a geometrical characterization of intrinsic ultracontractivity of the Dirichlet Laplacian. Michigan Math. J. 47, 79–99 (2000)MATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    Ouhabaz, E.M., Wang, F.-Y.: Sharp estimates for intrinsic ultracontractivity on C 1,α-domains. Manuscripta Math. 112, 229–244 (2007)CrossRefMathSciNetGoogle Scholar
  34. 34.
    Port, S.: Hitting times and potentials for recurrent stable processes. J. Anal. Math. 20, 371–395 (1967)MATHCrossRefMathSciNetGoogle Scholar
  35. 35.
    Song, R., Vondracek, Z.: Potential theory of subordinate killed Brownian motion in a domain. Probab. Theory Related Fields 125(4), 578–592 (2003)MATHCrossRefMathSciNetGoogle Scholar
  36. 36.
    Song, R., Vondracek, Z.: Potential theory of special subordinators and subordinate killed stable processes. J. Theoret. Probab. 19(4), 817–847 (2006)MATHCrossRefMathSciNetGoogle Scholar
  37. 37.
    Song, R., Vondracek, Z.: Parabolic Harnack inequality for the mixture of Brownian motion and stable process. Tohoku Math. J. 59(1), 1–19 (2007)MATHCrossRefMathSciNetGoogle Scholar
  38. 38.
    Song, R., Wu, J.M.: Boundary Harnack principle for symmetric stable processes. J. Funct. Anal. 168, 403–427 (1999)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  1. 1.Institute of Mathematics and Computer ScienceWrocław University of TechnologyWrocławPoland

Personalised recommendations