L q ESTIMATES OF SPHERICAL FUNCTIONS
AND AN INVARIANT MEANVALUE PROPERTY
Jonathan Arazy and Genkai Zhang
We find some L q estimates for the spherical functions on Cartan domains. As an ap
plication we prove that if the rank of the Cartan domain D is greater than one, then for
any 1 ź q ! 1, the invariant meanvalue property for L q function on D does not imply
harmonicity (the converse is known to be true even in the context of general noncompact
Riemannian symmetric spaces G=K).
It is wellunderstood that the harmonicity of a function u 2 C 2 on a domain in the
complex plane is characterized in following equivalent ways
1) u is in the kernel of the Laplace operator,
2) u satisfies the meanvalue property.
Much work has been done on the generalization of this equivalence to domains in higher
dimensions. In this paper we will consider the case of a general Cartan domain.
Let D be a Cartan domain of rank r in C n , i.e. D is an irreducible bounded symmetric
domain in the HarishChandra form (namely, D circular and convex). Let Aut(D) be the
group of biholomorphic automorphisms of D. Denote by G = Aut(D) 0 , the connected