Potential Analysis

, Volume 48, Issue 3, pp 337–360 | Cite as

On the Green Function and Poisson Integrals of the Dunkl Laplacian

  • Piotr Graczyk
  • Tomasz Luks
  • Margit Rösler


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.


Dunkl Laplacian Green function Newton kernel Poisson kernel Hardy-Stein identity 

Mathematics Subject Classification (2010)

Primary 31B05 31B25 60J50 Secondary 42B30 51F15 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bogdan, K.: Sharp estimates for the Green function in Lipschitz domains. J. Math. Anal. Appl. 243, 326–337 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bogdan, K., Dyda, B., Luks, T.: On Hardy spaces of local and nonlocal operators. Hiroshima Math. J. 44(2), 193–215 (2014)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Bogdan, K., Jakubowski, T.: Estimates of the Green function for the fractional Laplacian perturbed by gradient. Potential Anal. 36(3), 455–481 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Chen, Z.-Q., Song, R.: Estimates on Green functions and Poisson kernels for symmetric stable processes. Math. Ann. 312(3), 465–501 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Ben Chrouda, M.: On the Dirichlet problem associated with the Dunkl Laplacian. Ann. Polon. Math. 117(1), 79–87 (2016)MathSciNetzbMATHGoogle Scholar
  8. 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)Google Scholar
  9. 9.
    van Diejen, J.F., Vinet, L.: Calogero-Sutherland-Moser Models. Springer-Verlag, CRM Series in Mathematical Physics (2000)Google Scholar
  10. 10.
    Doob, J.L.: Classical Potential Theory and Its Probabilistic Counterpart. Springer-Verlag, New York (1984)CrossRefzbMATHGoogle Scholar
  11. 11.
    Dunkl, C.F.: Differential-difference operators associated to reflection groups. Trans. Amer. Math. Soc. 311, 167–183 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Dunkl, C.F.: Integral kernels with reflection group invariance. Canad. J. Math. 43, 1213–1227 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Dunkl, C.F., Xu, Y.: Orthogonal Polynomials of Several Variables Encyclopedia of Mathematics and Its Applications, vol. 81. Cambridge University Press, Cambridge (2001)Google Scholar
  14. 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. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Gallardo, L., Rejeb, C.: Newtonian potentials and subharmonic functions associated to root systems, preprint (2016), available at
  17. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Grüter, M., Widman, K.-O.: The Green function for uniformly elliptic equations. Manuscripta Math. 37, 303–342 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Grzywny, T., Ryznar, M.: Estimates of Green functions for some perturbations of fractional Laplacian. Ill. J. Math. 51(4), 1409–1438 (2007)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Hassine, K.: Mean value property of d k-harmonic functions on W-invariant open sets. Afr. Mat. 27(7), 1275–1286 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Hayman, W.K., Kennedy, P.B.: Subharmonic functions, vol. 1. Academic Press, London (1976)Google Scholar
  22. 22.
    Jakubowski, T.: The estimates for the Green function in Lipschitz domains for the symmetric stable processes. Probab. Math. Stat. 22, 419–441 (2002)MathSciNetzbMATHGoogle Scholar
  23. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Kulczycki, T.: Properties of Green function of symmetric stable processes. Probab. Math. Stat. 17, 339–364 (1997)MathSciNetzbMATHGoogle Scholar
  25. 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)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Maslouhi, M., Youssfi, E.H.: Harmonic functions associated to Dunkl Laplacian. Monatsh. Math. 152, 337–345 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Rejeb, C.: Harmonic and subharmonic functions associated to root systems, Ph.D. thesis, University of Tours, University of Tunis El Manar. available at (2015)
  28. 28.
    Rösler, M.: Positivity of Dunkl’s intertwining operator. Duke Math. J. 98, 445–463 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Rösler, M.: A positive radial product formula for the Dunkl kernel. Trans. Amer. Math. Soc. 355, 2413–2438 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 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)Google Scholar
  31. 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. 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)Google Scholar
  33. 33.
    Trimèche, K.: Paley-wiener Theorems for the Dunkl transform and Dunkl translation operators. Integral Transform. Spec. Funct. 13, 17–38 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Zhao, Z.: Uniform boundedness of conditional gauge and schrödinger equations. Comm. Math. Phys. 93, 19–31 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Zhao, Z.: Green function for schrödinger operator and conditioned Feynman-Kac gauge. J. Math. Anal. Appl. 116, 309–334 (1986)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2017

Authors and Affiliations

  1. 1.LAREMAUniversité d’AngersAngers Cedex 1France
  2. 2.Institut für MathematikUniversität PaderbornPaderbornGermany

Personalised recommendations