Integral Equations and Operator Theory

, Volume 23, Issue 2, pp 123–144

Lq-Estimates of spherical functions and an invariant mean-value property

  • Jonathan Arazy
  • Genkai Zhang


We find someLq-estimates for the spherical functions on Cartan domains. As an application we prove that if the rank of the Cartan domainD is greater than one, then for any 1<-q<∞, the invariant mean-value property forLq-function onD does not imply harmonicity (the converse is known to be true even in the context of general non-compact Riemannian symmetric spacesG/K).

AMS subject classification

31B10 32M15 


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  1. [AFR] P. Ahern, M. Flores and W. Rudin,An invariant volume-mean-value property, J. Funct. Anal.111 (1993), 380–397.Google Scholar
  2. [B] F. A. Berezin,General concept of quantization, Commun. Math. Phys.40 (1975), 153–174.Google Scholar
  3. [E] A. Erdelyi et al,Higher transdental functions, vol. 1, McGraw-Hill, New York-Toronto-London, 1953.Google Scholar
  4. [FK1] J. Faraut and A. Koranyi,Function spaces and reproducing kernels on bounded symmetric domains, J. Func. Anal.89 (1990), 64–89.Google Scholar
  5. [FK2] J. Faraut and A. Koranyi,Analysis on symmetric cones, to appear.Google Scholar
  6. [Fu] H. Furstenberg,Poisson formula for semisimple Lie groups, Ann. Math.77 (1963), 335–386.Google Scholar
  7. [G] R. Godement.,Une généralisation des représentations de la moyenne pour les fonctions harmoniques, C. R. Acad. Sci. Paris234 (1952), 2137–2139.Google Scholar
  8. [H] S. Helgason,Groups and geometric analysis, Academic Press, London, 1984.Google Scholar
  9. [L] O. Loos,Bounded Symmetric Domains and Jordan Pairs University of California, Irvine, 1977.Google Scholar
  10. [P] J. Peetre,Berezin transform and Ha-plitz opertors, J. Oper. Theory24 (1990), 165–168.Google Scholar
  11. [R] W. Rudin,Function Theory in the Unit Ball ofn, Springer-Verlag, 1980.Google Scholar
  12. [UU] A. Unterberger and H. Upmeier,Berezin transform and invariant differential operators, preprint (1993).Google Scholar
  13. [Up] H. Upmeier,Jordan Algebras in Analysis, Operator Theory, and Quantum mechanics, Regional Conference in Mathematics No. 67, Amer. Math.Soc., 1987.Google Scholar

Copyright information

© Birkhäuser Verlag 1995

Authors and Affiliations

  • Jonathan Arazy
    • 1
  • Genkai Zhang
    • 2
  1. 1.Department of MathematicsUniversity of HaifaHaifaIsrael
  2. 2.Institut for Matematik og DatalogiOdense UniversitetOdense MDenmark
  3. 3.School of MathematicsUniversity of New South WalesKensingtonAustralia

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