Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 4 (2008), 076, 6 pages      arXiv:0811.0962      https://doi.org/10.3842/SIGMA.2008.076
Contribution to the Special Issue on Dunkl Operators and Related Topics

Liouville Theorem for Dunkl Polyharmonic Functions

Guangbin Ren a, b and Liang Liu a
a) Department of Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, P.R. China
b) Departamento de Matemática, Universidade de Aveiro, P-3810-193, Aveiro, Portugal

Received July 03, 2008, in final form October 30, 2008; Published online November 06, 2008

Abstract
Assume that f is Dunkl polyharmonic in Rn (i.e. (Δh)p f = 0 for some integer p, where Δh is the Dunkl Laplacian associated to a root system R and to a multiplicity function κ, defined on R and invariant with respect to the finite Coxeter group). Necessary and successful condition that f is a polynomial of degree ≤ s for s ≥ 2p – 2 is proved. As a direct corollary, a Dunkl harmonic function bounded above or below is constant.

Key words: Liouville theorem; Dunkl Laplacian; polyharmonic functions.

pdf (187 kb)   ps (145 kb)   tex (9 kb)

References

  1. Axler S., Bourdon P., Ramey W., Harmonic function theory, 2nd ed., Graduate Texts in Mathematics, Vol. 137. Springer-Verlag, New York, 2001.
  2. Armitage D.H., A Liouville theorem for polyharmonic functions, Hiroshima Math. J. 31 (2001), 367-370.
  3. Dunkl C.F., Reflection groups and orthogonal polynomials on the sphere, Math. Z. 197 (1988), 33-60.
  4. Dunkl C.F., de Jeu M.F.E., Opdam E.M., Singular polymomials for finite groups, Trans. Amer. Math. Soc. 346 (1994), 237-256.
  5. Dunkl C.F., Xu Y., Orthogonal polynomials of several variables, Encyclopedia of Mathematics and its Applications, Vol. 81, Cambridge University Press, Cambridge, 2001.
  6. Futamura T., Kishi K., Mizuta Y., A generalization of the Liouville theorem to polyharmonic functions, J. Math. Soc. Japan 53 (2001), 113-118.
  7. Gallardo L., Godefroy L., Propriété de Liouville et équation de Poisson pour le Laplacien généralisé de Dunkl, C. R. Math. Acad. Sci. Paris 337 (2003), 639-644.
  8. González Vieli F.J., A new proof of a Liouville-type theorem for polyharmonic functions, Real Anal. Exchange 30 (2004/05), 319-322.
  9. Kuran Ü., Generalizations of a theorem on harmonic functions, J. London Math. Soc. 41 (1966), 145-152.
  10. Li A., Li Y.Y., A fully nonlinear version of the Yamabe problem and a Harnack type inequality, C. R. Math. Acad. Sci. Paris 336 (2003), 319-324, math.AP/0212031.
  11. Li A., Li Y.Y., On some conformally invariant fully nonlinear equations, C. R. Math. Acad. Sci. Paris 337 (2003), 639-644.
  12. Maslouhi M., Youssfi E.H., Harmonic functions associated to Dunkl operators, Monatsh. Math. 152 (2007), 337-345.
  13. Mejjaoli H., Trimèche K., On a mean value property associated with the Dunkl Laplacian operator and applications, Integral Transform. Spec. Funct. 12 (2001), 279-302.
  14. Nicolesco M., Sur les fonctions de n variables, harmoniques d'order p, Bull. Soc. Math. France 60 (1932), 129-151.
  15. Ren G.B., Almansi decomposition for Dunkl operators, Sci. China Ser. A 48 (2005), 333-342.
  16. Ren G.B., Howe duality in Dunkl superspace, Preprint.
  17. Rösler M., Dunkl operators: theory and applications, in Orthogonal Polynomials and Special Functions (Leuven, 2002), Editors E. Koelink et al., Lecture Notes in Math., Vol. 1817, Springer, Berlin, 2003, 93-135, math.CA/0210366.

Previous article   Next article   Contents of Volume 4 (2008)