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A mean value theorem
on bounded symmetric domains


Author: Miroslav Englis
Journal: Proc. Amer. Math. Soc. 127 (1999), 3259-3268
MSC (1991): Primary 31C05, 32M15; Secondary 31B10
DOI: https://doi.org/10.1090/S0002-9939-99-05052-2
Published electronically: May 4, 1999
MathSciNet review: 1625737
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Abstract: Let $\Omega $ be a Cartan domain of rank $r$ and genus $p$ and $B_\nu$, $\nu >p-1$, the Berezin transform on $\Omega $; the number $B_{\nu }f(z)$ can be interpreted as a certain invariant-mean-value of a function $f$ around $z$. We show that a Lebesgue integrable function satisfying $f=B_\nu f=B_{\nu +1}f=\dots =B_{\nu +r}f$, $\nu \ge p$, must be $\mathcal{M}$-harmonic. In a sense, this result is reminiscent of Delsarte's two-radius mean-value theorem for ordinary harmonic functions on the complex $n$-space $\mathbf{C}^{n}$, but with the role of radius $r$ played by the quantity $1/\nu $.


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  • [AFR] P. Ahern, M. Flores, W. Rudin, An invariant volume-mean-value property, J. Funct. Anal. 111 (1993), 380-397. MR 94b:31002
  • [AZ] J. Arazy, G. Zhang, $L^{q}$-estimates of spherical functions and an invariant mean-value property, Integr. Eq. Oper. Theory 23 (1995), 123-144. MR 96g:22015
  • [AC] S. Axler, Z. \v{C}u\v{c}kovi\'{c}, Commuting Toeplitz operators with harmonic symbols, Integr. Eq. Oper. Theory 14 (1991), 1-12. MR 92f:47018
  • [Dix] J. Dixmier, Les algèbres d'opérateurs dans l'espace hilbertien, $2^{\text{nd}}\,$edition, Gauthier-Villars, Paris, 1969. MR 50:5482
  • [E1] M. Engli\v{s}, Functions invariant under the Berezin transform, J. Funct. Anal. 121 (1994), 233-254. MR 95h:31001
  • [E2] M. Engli\v{s}, Invariant operators and the Berezin transform on Cartan domains, to appear, Math. Nachrichten.
  • [FK1] J. Faraut, A. Korányi, Function spaces and reproducing kernels on bounded symmetric domains, J. Funct. Anal. 88 (1990), 64-89. MR 90m:32049
  • [FK2] J. Faraut, A. Korányi, Analysis on symmetric cones, Clarendon Press, Oxford, 1994. MR 98g:17031
  • [F] M.V. Fedoryuk, Asymptotics, integrals, series, Nauka, Moscow, 1987 (in Russian). MR 89j:41045
  • [Fu] H. Fürstenberg, Poisson formula for semisimple Lie groups, Ann. Math. 77 (1963), 335-386. MR 26:3820
  • [He] S. Helgason, Lie groups and symmetric spaces, ``Battele Rencontres'', Benjamin, New York, 1968, pp. 1-71. MR 38:4622
  • [Ka] R.M. Kauffman, Eigenfunction Expansions, Operator Algebras and Riemannian Symmetric Spaces, Pitman Research Notes in Mathematics, vol. 355, Addison Wesley Longman, Harlow, 1996. MR 97j:47031
  • [Lo] O. Loos, Bounded symmetric domains and Jordan pairs, University of California, Irvine, 1977.
  • [NV] I. Netuka, J. Veselý, Mean value property and harmonic functions, Classical and modern potential theory and applications (Chateau de Bonas, 1993), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 430, Kluwer Acad. Publ., Dordrecht, 1994, pp. 359-398. MR 96c:31001
  • [R] W. Rudin, Function theory in the unit ball of ${\bold C}^{n}$, Springer-Verlag, Berlin-Heidelberg-New York, 1980. MR 82i:32002
  • [UU] A. Unterberger, H. Upmeier, Berezin transform and invariant differential operators, Comm. Math. Phys. 164 (1994), 563-598. MR 96h:58170
  • [Up] H. Upmeier, Toeplitz Operators and Index Theory in Several Complex Variables, Operator Theory: Advances and Applications, vol. 81, Birkhäuser, Basel, 1996. MR 97f:47022
  • [Wa] G. Warner, Harmonic analysis on semi-simple Lie groups I, II, Grundlehren der Math. Wiss., vol. 188, 189, Springer-Verlag, Berlin-Heidelberg, 1972. MR 58:16979; MR 58:16980
  • [Za] L. Zalcman, Offbeat integral geometry, Amer. Math. Monthly 87 (1980), 161-175. MR 81b:53046

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Additional Information

Miroslav Englis
Affiliation: MÚ AV ČR, Žitná 25, 11567 Prague 1, Czech Republic
Email: englis@math.cas.cz

DOI: https://doi.org/10.1090/S0002-9939-99-05052-2
Keywords: Berezin transform, bounded symmetric domains, invariant mean-value property
Received by editor(s): January 28, 1998
Published electronically: May 4, 1999
Additional Notes: The author’s research was supported by GA AV ČR grant A1019701 and by GA ČR grant 201/96/0411.
Communicated by: Christopher D. Sogge
Article copyright: © Copyright 1999 American Mathematical Society

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