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$$L^ q$$-estimates of spherical functions and an invariant mean-value property. (English) Zbl 0837.31003
Let $$D$$ be a Cartan domain of rank $$r$$ in $$\mathbb{C}^n$$ and $$G$$ be the connected component of the identity in the group of biholomorphic automorphisms of $$D$$. A function $$f\in C^\infty (D)$$ is called harmonic on $$D$$ if $$Tf=0$$ for all $$G$$-invariant differential operators $$T$$ such that $$T1=0$$. It is known that any harmonic function $$f$$ satisfies the invariant mean value property $\int_D f(g (w)) dm(w)= f(g (0)), \qquad \forall g\in G, \tag{1}$ $$m$$ being the normalized volume measure on $$D$$. The converse is also true for functions $$f\in L^\infty (D)$$.
On the other hand, it was shown in [P. Ahern, M. Flores and W. Rudin, J. Funct. Anal. 111, 380-397 (1993; Zbl 0771.32006)] that for the unit ball in $$\mathbb{C}^n$$ (the case of the rank $$r=1$$) condition (1) for $$f\in L^1$$ implies its harmonicity if and only if $$n< 12$$.
In the paper under review it is proved that for Cartan domains of rank $$r>1$$ harmonicity of a function $$f\in L^q (D)$$ does not follow from the invariant mean value property (1) for all $$1\leq q< \infty$$.

##### MSC:
 31B10 Integral representations, integral operators, integral equations methods in higher dimensions 32M15 Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects) 53C35 Differential geometry of symmetric spaces
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##### References:
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