Acta Mathematica Sinica

, 22:1227 | Cite as

Towards an L p Potential Theory for Sub–Markovian Semigroups: Kernels and Capacities

  • Niels JacobEmail author
  • René L. Schilling


We study in fairly general measure spaces (X, μ) the (non–linear) potential theory of L p sub–Markovian semigroups which are given by kernels having a density with respect to the underlying measure. In terms of mapping properties of the operators we provide sufficient conditions for the existence (and regularity) of such densities. We give various (dual) representations for several associated capacities and, in the corresponding abstract Bessel potential spaces, we study the role of the truncation property. Examples are discussed in the case of ℝ n , where abstract Bessel potential spaces can be identified with concrete function spaces.


nonlinear potential theory (r, p)–capacity Bessel potential space gamma–transform sub–Markovian semigroup 

MR (2000) Subject Classification

31C15 31C45 31C25 35S05 47B34 47B65 47D07 60J35 


  1. 1.
    Hoh, W., Jacob, N.: Towards an L p potential theory for sub–Markovian semigroups: variational inequalities and balayage theory. J. Evol. Equ., 4, 297–312 (2004)zbMATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Malliavin, P.: Implicit functions in finite corank on the Wiener space, Itô, K. (ed.): Proceedings Taniguchi Symp. Stoch. Anal. Katata and Kyoto 1982, Math. Libr. 32, North–Holland, Amsterdam, 369–386, 1984Google Scholar
  3. 3.
    Fukushima, M.: Two topics related to Dirichlet forms: quasi–everywhere convergence and additive functionals, Dell’Antonio, G., Mosco, U. (eds.): Dirichlet Forms, Lect. Notes Math. 1563, Springer, Berlin, 21–53, 1993Google Scholar
  4. 4.
    Fukushima, M., Kaneko, H.: On (r, p)–capacities for general Markovian semigroups, Albeverio, S. (ed.): Infinite dimensional analysis and stochastic processes, Res. Notes Math. 124, Pitman, Boston (MA), 41–47, 1985Google Scholar
  5. 5.
    Fukushima, M., Oshima, Y., Takeda, M.: Dirichlet Forms and Symmetric Markov Processes, Stud. Math. 19, de Gruyter, Berlin, 1994Google Scholar
  6. 6.
    Kazumi, T., Shigekawa, I.: Measures of finite (r, p)–energy and potentials on a separable metric space, Azéma, J., Meyer, P.A., Yor, M. (eds.): Séminaire de Probabilités XXVI, Lect. Notes Math. 1526, Springer, Berlin, 415–444, 1992Google Scholar
  7. 7.
    Maz’ya, V. G., Havin, V. P.: Nonlinear potential theory. Russ. Math. Surv., 27, 71–148 (1972)CrossRefGoogle Scholar
  8. 8.
    Maz’ja (Maz’ya), V. G., Sobolev Spaces, Series in Soviet Math., Springer, Berlin, 1985Google Scholar
  9. 9.
    Mizuta, Y.: Potential theory in Euclidean spaces, GAKUTO Int. Ser., Math. Sci. Appl. 6, Gakkōtosho, Tokyo, 1996Google Scholar
  10. 10.
    Adams, D. R., Hedberg, L. I.: Function Spaces and Potential Theory, Grundlehren Math. Wiss. 314, Springer, Berlin 1999 (corr. 2nd printing)Google Scholar
  11. 11.
    Rao, M., Vondraček, Z.: Nonlinear Potentials in Function Spaces. Nagoya Math. J., 165, 91–116 (2002)zbMATHMathSciNetGoogle Scholar
  12. 12.
    Jacob, N.: Pseudo Differential Operators and Markov Processes. Vol. 3, Markov Processes and Applications, Imperial College Press, London, 2005Google Scholar
  13. 13.
    Farkas, W., Jacob, N., Schilling, R. L.: Function spaces related to continuous negative definite functions: ψ–Bessel potential spaces. Diss. Math., CCCXCIII, 1–62 (2001)CrossRefGoogle Scholar
  14. 14.
    Schilling, R. L.: Dirichlet operators and the positive maximum principle. Integral Equations Oper. Theory, 41, 74–92 (2001)zbMATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Jacob, N.: Pseudo–differential operators and Markov processes, Math. Res. 94, Akademie–Verlag, Berlin, 1996Google Scholar
  16. 16.
    Jacob, N.: Pseudo Differential Operators and Markov Processes, Vol. 2: Generators and their Potential Theory, Imperial College Press, London, 2002Google Scholar
  17. 17.
    Jacob, N., Schilling, R. L.: Lévy–type processes and pseudo–differential operators, Barndorff–Nielsen, O. E., et al. (eds.) Lévy processes: Theory and Applications, Birkhäuser, Boston, 139–168, 2001Google Scholar
  18. 18.
    Farkas, W., Jacob, N., Schilling, R. L.: Feller semigroups, L p–sub–Markovian semigroups, and applications to pseudo–differential operators with negative definite symbols. Forum Math., 13, 51–90 (2001)zbMATHMathSciNetCrossRefGoogle Scholar
  19. 19.
    Dunford, N., Pettis, B. J.: Linear operators on summable functions. Trans. Am. Math. Soc., 47, 323–349 (1940)zbMATHMathSciNetCrossRefGoogle Scholar
  20. 20.
    Robinson, D. W.: Elliptic operators and Lie groups, Oxf. Math. Monogr., Clarendon Press, Oxford, 1991Google Scholar
  21. 21.
    Schaefer, H. H.: Banach lattices and positive operators, Grundlehren Math. Wiss. 215, Springer, Berlin, 1974Google Scholar
  22. 22.
    Nagel, R. (ed.): One–parameter Semigroups of Positive Operators, Lect. Notes Math. 1184, Springer, Berlin, 1986Google Scholar
  23. 23.
    Bukhvalov, A. V.: Integral representations of linear operators. J. Soviet Math., 9, 129–137 (1978)zbMATHCrossRefGoogle Scholar
  24. 24.
    Bukhvalov, A. V.: Application of methods of the theory of order–bounded operators to the theory of operators in L p–spaces. Russ. Math. Surv., 38, 43–98 (1983)zbMATHMathSciNetCrossRefGoogle Scholar
  25. 25.
    Arendt, W., Bukhvalov, A. V.: Integral representations of resolvents and semigroups. Forum Math., 6, 111–135 (1994)zbMATHMathSciNetCrossRefGoogle Scholar
  26. 26.
    Doob, J. L.: One–parameter families of transformations. Duke Math. J., 4, 752–774 (1938)zbMATHMathSciNetCrossRefGoogle Scholar
  27. 27.
    Revuz, D.: Markov Chains (revised ed.), Elsevier, North–Holland Math. Libr., 11, Amsterdam, 1984Google Scholar
  28. 28.
    Jörgens, K.: Linear Integral Operators, Surv. Ref. Works Math., Pitman, Boston, 1982Google Scholar
  29. 29.
    Zaanen, A. C.: Linear Analysis, North Holland, Amsterdam, 1953Google Scholar
  30. 30.
    Blumenthal, R. M., Getoor, R. K.: Markov Processes and Potential Theory, Pure Appl. Math. 29, Academic Press, New York, 1968Google Scholar
  31. 31.
    Schilling, R. L.: Subordination in the sense of Bochner and a related functional calculus. J. Aust. Math. Soc. Ser. A, 64, 368–396 (1998)zbMATHMathSciNetCrossRefGoogle Scholar
  32. 32.
    Jacob, N.: Pseudo Differential Operators and Markov Processes, Vol. 1: Fourier Analysis and Semigroups, Imperial College Press, London, 2001Google Scholar
  33. 33.
    Kaneko, H.: On (r, p)–capacities for Markov processes. Osaka J. Math., 23, 325–336 (1986)zbMATHMathSciNetGoogle Scholar
  34. 34.
    Ouhabaz, E. M.: Propriétés d’ordre et de contractivité des semi–groupes avec applications aux opérateurs elliptiques, Thèse de l’Université de Franche–Comté, Besan¸con, 1992Google Scholar
  35. 35.
    Ma, Z. M., Overbeck, L., Röckner, M.: Markov processes associated with semi–Dirichlet forms. Osaka J. Math., 32, 97–119 (1995)zbMATHMathSciNetGoogle Scholar
  36. 36.
    Dellacherie, C., Meyer, P. A.: Probabilités et potentiel: chapitres I à IV, Hermann, Publ. Inst. Math. Univ. Strasbourg t., XV, Paris, 1975Google Scholar
  37. 37.
    Malliavin, P.: Stochastic Analysis, Grundlehren Math. Wiss. Bd., 313, Springer, Berlin, 1997Google Scholar
  38. 38.
    Triebel, H.: Truncation of functions. Forum Math., 12, 731–756 (2000)zbMATHMathSciNetCrossRefGoogle Scholar
  39. 39.
    Hirsch, F.: Lipschitz functions and fractional Sobolev spaces. Potential Anal., 11, 415–429 (1999)zbMATHMathSciNetCrossRefGoogle Scholar
  40. 40.
    Berg, C., Forst, G.: Potential Theory on Locally Compact Abelian Groups, Ergeb. Math. Grenzgeb. 87, Springer, Berlin, 1975Google Scholar
  41. 41.
    Berg, C., Forst, G.: Non–symmetric translation invariant Dirichlet forms. Invent. Math., 21, 199–212 (1973)zbMATHMathSciNetCrossRefGoogle Scholar
  42. 42.
    Hoh, W.: A symbolic calculus for pseudo differential operators generating Feller semigroups. Osaka J. Math., 35, 798–820 (1998)MathSciNetGoogle Scholar
  43. 43.
    Bakry, D.: Transformations de Riesz pour les semi–groupes symétriques. Seconde partie: Etudes sous la condition Γ2≥ 0, Azéma, J., Yor, M. (eds.): Sém. Probab. XIX, Lect. Notes Math. 1123, Springer Berlin, 145–174, 1985Google Scholar
  44. 44.
    Triebel, H.: Theory of Function Spaces, Monogr. Math. 78, Birkhäuser, Basel, 1983Google Scholar
  45. 45.
    Triebel, H.: Theory of Function Spaces II, Monogr. Math. 84, Birkhäuser, Basel, 1992Google Scholar
  46. 46.
    Choquet, G.: Lectures on Analysis, Volume 1: Integration and Topological Vector Spaces, W. A. Benjamin, New York, 1969Google Scholar
  47. 47.
    Dunford, N., Schwartz, J. T.: Linear Operators I, Pure Appl. Math. 7, Interscience, New York, 1957Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Wales SwanseaSwansea SA2 8PPUnited Kingdom
  2. 2.FB 12—MathematikPhilipps-Universität MarburgMarburgGermany

Personalised recommendations