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Generalized Bessel and Riesz Potentials on Metric Measure Spaces

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Abstract

We introduce generalized Bessel and Riesz potentials on metric measure spaces and the corresponding potential spaces. Estimates of the Bessel and Riesz kernels are given which reflect the intrinsic structure of the spaces. Finally, we state the relationship between Bessel (or Riesz) operators and subordinate semigroups.

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References

  1. Adams, D.R., Hedberg, L.I.: Function Spaces and Potential Theory. Springer, Berlin (1996)

    Google Scholar 

  2. Barlow, M.: Diffusions on fractals. In: Lect. Notes Math. vol. 1690, pp. 1–121. Springer (1998)

  3. Barlow, M., Bass, R.F.: Brownian motion and harmonic analysis on Sierpínski carpets. Canad. J. Math. 51, 673–744 (1999)

    MATH  MathSciNet  Google Scholar 

  4. Barlow, M., Perkins, E.A.: Brownian motion on the Sierpínski gasket. Probab. Theory Related Fields 79, 543–623 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  5. Bernstein, S.: Sur les fonctions absolument monotones (french). Acta Math. 52, 1–66 (1929)

    Article  Google Scholar 

  6. Bertoin, J.: Lévy Processes. Cambridge University Press, Cambridge (1996)

    MATH  Google Scholar 

  7. Bochner, S.: Diffusion equation and stochastic processes. Proc. Nat. Acad. Sci. U. S. A. 35, 368–370 (1949)

    Article  MATH  MathSciNet  Google Scholar 

  8. Falconer, K.J.: Fractal Geometry—Mathematical Foundations and Applications. Wiley, Chichester (1990)

    MATH  Google Scholar 

  9. Fukushima, M., Oshima, Y., Takeda, M.: Dirichlet Forms and Symmetric Markov Processes. Walter de Gruyter, Berlin (1994)

    MATH  Google Scholar 

  10. Grigor’yan, A., Hu, J., Lau, K.-S.: Heat kernels on metric-measure spaces and an application to semilinear elliptic equations. Trans. Amer. Math. Soc. 355, 2065–2095 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  11. Hambly, B.M., Kumagai, T.: Transition density estimates for diffusion processes on post critically finite self-similar fractals. Proc. London Math. Soc. 79, 431–458 (1997)

    MathSciNet  Google Scholar 

  12. Han, Y., Yang, D.: New characterizations and applications of inhomogeneous Besov spaces and Triebel-Lizorkin spaces on homogeneous type spaces and fractals. Diss. Math. 403, 1–102 (2002)

    Article  MathSciNet  Google Scholar 

  13. Hausdorff, F.: Summationsmethoden und Momentfolgen I. Math. Zeit. 9, 74–109 (1921)

    Article  MathSciNet  Google Scholar 

  14. Heinonen, J.: Lectures on Analysis on Metric Spaces. Springer, New York (2001)

    MATH  Google Scholar 

  15. Hu, J., Zähle, M.: Jump processes and nonlinear fractional heat equations on fractals. Math. Nachr. 279, 150–163 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  16. Jacob, N.: Pseudo-Differential Operators and Markov Processes, vol. 1: Fourier Analysis and Semigroups. Imperial College Press, London (2001)

    Google Scholar 

  17. Kigami, J.: Analysis on Fractals. Cambridge University Press, Cambridge (2001)

    MATH  Google Scholar 

  18. Phillips, R.S.: On the generation of semigroups of linear operators. Pacific J. Math. 2, 343–369 (1952)

    MATH  MathSciNet  Google Scholar 

  19. Rubin, B.: Fractional Integrals and Potentials. Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 82. Longman, Harlow (1996)

    Google Scholar 

  20. Samko, S.G., Kilbas, A.A., Marichev, O.L.: Fractional Integrals and Derivatives. Gordon and Breach Science, Amsterdam (1993)

    MATH  Google Scholar 

  21. Schilling, R.L.: On the domain of the generator of a subordinate semigroup. In: Kral, J., et al. (eds.) Potential Theory-ICPT94. Proceedings Internat. Conf. Potential Theory, 1994, pp. 449–462. De Gruyter, Berlin (1996)

    MathSciNet  Google Scholar 

  22. Schilling, R.L.: Subordination in the sense of Bochner and a related functional calculus. J. Austral. Math. Soc. Ser. A 64, 368–396 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  23. Stein, E.M.: Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton (1970)

    MATH  Google Scholar 

  24. Strichartz, R.S.: Differential Equations on Fractals: A Tutorial. Princeton University Press, Princeton (2006)

    MATH  Google Scholar 

  25. Triebel, H.: Theory of Function Spaces III. Birkhäuser-Verlag, Basel (2007)

    Google Scholar 

  26. Triebel, H., Yang, D.: Spectral theory of Riesz potentials on quasi-metric spaces. Math. Nachr. 238, 160–184 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  27. Widder, D.V.: The Laplace Transform. Princeton University Press, Princeton (1941)

    MATH  Google Scholar 

  28. Zähle, M.: Riesz potentials and Liouville operators on fractals. Potential Anal. 21, 193–208 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  29. Zähle, M.: Potential spaces and traces of Lévy processes on h-sets. J. Contemp. Math. Anal. 2(44) (2009, to appear)

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Authors and Affiliations

  1. Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, China

    J. Hu

  2. Mathematical Institute, University of Jena, 07737, Jena, Germany

    M. Zähle

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  1. J. Hu
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  2. M. Zähle
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Correspondence to J. Hu.

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Hu, J., Zähle, M. Generalized Bessel and Riesz Potentials on Metric Measure Spaces. Potential Anal 30, 315–340 (2009). https://doi.org/10.1007/s11118-009-9117-9

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  • DOI: https://doi.org/10.1007/s11118-009-9117-9

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