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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 \(1\leq q< \infty\).

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
Full Text: DOI
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