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On the Green Function and Poisson Integrals of the Dunkl Laplacian

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We prove the existence and study properties of the Green function of the unit ball for the Dunkl Laplacian △ k in \(\mathbb {R}^{d}\). As applications we derive the Poisson-Jensen formula for △ k -subharmonic functions and Hardy-Stein identities for the Poisson integrals of △ k . We also obtain sharp estimates of the Newton potential kernel, Green function and Poisson kernel in the rank one case in \(\mathbb {R}^{d}\). These estimates contrast sharply with the well-known results in the potential theory of the classical Laplacian.

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References

  1. Ancona, A.: First eigenvalues and comparison of Green’s functions for elliptic operators on manifolds or domains. J. Anal, Math. 72, 45–92 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  2. Aronson, D.G.: On the Green’s function for second order parabolic differential equations with discontinuous coefficients. Bull. Amer. Math. Soc. 69(6), 841–847 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bogdan, K.: Sharp estimates for the Green function in Lipschitz domains. J. Math. Anal. Appl. 243, 326–337 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bogdan, K., Dyda, B., Luks, T.: On Hardy spaces of local and nonlocal operators. Hiroshima Math. J. 44(2), 193–215 (2014)

    MathSciNet  MATH  Google Scholar 

  5. Bogdan, K., Jakubowski, T.: Estimates of the Green function for the fractional Laplacian perturbed by gradient. Potential Anal. 36(3), 455–481 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  6. Chen, Z.-Q., Song, R.: Estimates on Green functions and Poisson kernels for symmetric stable processes. Math. Ann. 312(3), 465–501 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  7. Ben Chrouda, M.: On the Dirichlet problem associated with the Dunkl Laplacian. Ann. Polon. Math. 117(1), 79–87 (2016)

    MathSciNet  MATH  Google Scholar 

  8. Cranston, M., Zhao, Z.: Conditional transformation of drift formula and potential theory for \(\frac 12{\Delta }+b(\cdot )\cdot \nabla \). Comm. Math. Phys. 112, 613–625 (1987)

  9. van Diejen, J.F., Vinet, L.: Calogero-Sutherland-Moser Models. Springer-Verlag, CRM Series in Mathematical Physics (2000)

  10. Doob, J.L.: Classical Potential Theory and Its Probabilistic Counterpart. Springer-Verlag, New York (1984)

    Book  MATH  Google Scholar 

  11. Dunkl, C.F.: Differential-difference operators associated to reflection groups. Trans. Amer. Math. Soc. 311, 167–183 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  12. Dunkl, C.F.: Integral kernels with reflection group invariance. Canad. J. Math. 43, 1213–1227 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  13. Dunkl, C.F., Xu, Y.: Orthogonal Polynomials of Several Variables Encyclopedia of Mathematics and Its Applications, vol. 81. Cambridge University Press, Cambridge (2001)

  14. El Kamel, J., Yacoub, C.H.: Poisson integrals and kelvin transform associated to Dunkl-Laplacian operator. Global J. Pure Appl. Math. 3(3), 351 (2007)

    Google Scholar 

  15. Gallardo, L., Rejeb, C.: A new mean value property for harmonic functions relative to the Dunkl-Laplacian operator and applications. Trans. Amer. Math. Soc. 368, 3727–3753 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  16. Gallardo, L., Rejeb, C.: Newtonian potentials and subharmonic functions associated to root systems, preprint (2016), available at https://hal.archives-ouvertes.fr/hal-01368871

  17. Gallardo, L., Yor, M.: Some new examples of Markov processes which enjoy the time-inversion property. Probab. Theory Relat. Fields 132, 150–162 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  18. Grüter, M., Widman, K.-O.: The Green function for uniformly elliptic equations. Manuscripta Math. 37, 303–342 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  19. Grzywny, T., Ryznar, M.: Estimates of Green functions for some perturbations of fractional Laplacian. Ill. J. Math. 51(4), 1409–1438 (2007)

    MathSciNet  MATH  Google Scholar 

  20. Hassine, K.: Mean value property of d k -harmonic functions on W-invariant open sets. Afr. Mat. 27(7), 1275–1286 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  21. Hayman, W.K., Kennedy, P.B.: Subharmonic functions, vol. 1. Academic Press, London (1976)

  22. Jakubowski, T.: The estimates for the Green function in Lipschitz domains for the symmetric stable processes. Probab. Math. Stat. 22, 419–441 (2002)

    MathSciNet  MATH  Google Scholar 

  23. Kim, P., Song, R.: Estimates on Green functions and schrödinger-type equations for non-symmetric diffusions with measure-valued drifts. J. Math. Anal. Appl. 332, 57–80 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  24. Kulczycki, T.: Properties of Green function of symmetric stable processes. Probab. Math. Stat. 17, 339–364 (1997)

    MathSciNet  MATH  Google Scholar 

  25. Littman, W., Stampacchia, G., Weinberger, H.F.: Regular points for elliptic equations with discontinuous coefficients. Ann. Scuola Norm. Sup. Pisa (III) 17, 43–77 (1963)

    MathSciNet  MATH  Google Scholar 

  26. Maslouhi, M., Youssfi, E.H.: Harmonic functions associated to Dunkl Laplacian. Monatsh. Math. 152, 337–345 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  27. Rejeb, C.: Harmonic and subharmonic functions associated to root systems, Ph.D. thesis, University of Tours, University of Tunis El Manar. available at https://tel.archives-ouvertes.fr/tel-01291741/ (2015)

  28. Rösler, M.: Positivity of Dunkl’s intertwining operator. Duke Math. J. 98, 445–463 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  29. Rösler, M.: A positive radial product formula for the Dunkl kernel. Trans. Amer. Math. Soc. 355, 2413–2438 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  30. Rösler, M.: Dunkl operators: Theory and applications Orthogonal polynomials and special functions, Leuven 2002, Springer Lecture Notes in Math, vol. 1817, pp. 93–135 (2003)

  31. Rösler, M., Voit, M.: Dunkl Theory, Convolution Algebras, and Related Markov Processes. In: Graczyk, P., Rösler, M., Yor, M. (eds.) Harmonic and Stochastic Analysis of Dunkl Processes. Travaux en cours 71, pp. 1–112. Hermann, Paris (2008)

    Google Scholar 

  32. Stein, E.M.: Boundary Behavior of Holomorphic Functions of Several Complex Variables, Princeton University Press and University of Tokyo Press, Princeton, New Jersey (1972)

  33. Trimèche, K.: Paley-wiener Theorems for the Dunkl transform and Dunkl translation operators. Integral Transform. Spec. Funct. 13, 17–38 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  34. Widman, K.-O.: Inequalities for the Green function and boundary continuity of the gradient of solutions of elliptic differential equations. Math. Scand. 21, 17–37 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  35. Zhao, Z.: Uniform boundedness of conditional gauge and schrödinger equations. Comm. Math. Phys. 93, 19–31 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  36. Zhao, Z.: Green function for schrödinger operator and conditioned Feynman-Kac gauge. J. Math. Anal. Appl. 116, 309–334 (1986)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. LAREMA, Université d’Angers, 2 Bd Lavoisier, 49045, Angers Cedex 1, France

    Piotr Graczyk

  2. Institut für Mathematik, Universität Paderborn, Warburger Strasse 100, D-33098, Paderborn, Germany

    Tomasz Luks & Margit Rösler

Authors
  1. Piotr Graczyk
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  2. Tomasz Luks
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  3. Margit Rösler
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Correspondence to Tomasz Luks.

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Graczyk, P., Luks, T. & Rösler, M. On the Green Function and Poisson Integrals of the Dunkl Laplacian. Potential Anal 48, 337–360 (2018). https://doi.org/10.1007/s11118-017-9638-6

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