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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
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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Additional Information

Miroslav Englis
Affiliation: MÚ AV ČR, Žitná 25, 11567 Prague 1, Czech Republic

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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