A mean value theorem on bounded symmetric domains

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_{...

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Bibliographic Details
Published in:Proceedings of the American Mathematical Society 1999-11, Vol.127 (11), p.3259-3268
Main Author: Englis, Miroslav
Format: Article
Language:English
Subjects:
Algebra
Differential operators
Harmonic functions
Mathematical functions
Mathematical rings
Mean value theorems
Polynomials
Radius of a sphere
Symmetry
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Summary: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 .
ISSN:0002-9939
1088-6826
DOI:10.1090/S0002-9939-99-05052-2