\(L^ q\)-estimates of spherical functions and an invariant mean-value property
type: Item
instance of: scholarly article
title: \(L^ q\)-estimates of spherical functions and an invariant mean-value property
zbMATH Open document ID: 0837.31003
DOI: 10.1007/BF01197533
author: Genkai Zhang
author: Jonathan Arazy
published in: Integral Equations and Operator Theory
publication date: 1996-05-21T00:00:00Z
review text: 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 [\textit{P. Ahern}, \textit{M. Flores} and \textit{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 \(1\leq q< \infty\).
Mathematics Subject Classification ID: 31B10
Mathematics Subject Classification ID: 32M15
Mathematics Subject Classification ID: 53C35
zbMATH DE Number: 811609
zbMATH Keywords: symmetric Riemann space
zbMATH Keywords: spherical function
zbMATH Keywords: Cartan domain
zbMATH Keywords: biholomorphic automorphisms
zbMATH Keywords: harmonic function
zbMATH Keywords: invariant mean value property
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 811609