Summary: Invariant Mean Value and Harmonicity in Cartan and Siegel Domains
J. ARAZY Department of Mathematics, University of Haifa, Haifa 31905, Israel
G. ZHANG 1 School of Mathematics, University of New South Wales, Kensington, N.S.W.
2033, Australia
Abstract
Let D be a Cartan domain of rank r in C n . Let K be the maximal connected com­
pact subgroup of holomorphic automorphisms of D, and let ¯ be a K­invariant, absolutely
continuous probability measure on D, satisfying some mild growth conditions near the
boundary @D. We show that for every 1 Ÿ q ! 1 there is an r \Gamma 1 dimensional complex
submanifold X ¯;q of C r so that the spherical functions fOE – ; – 2 X ¯;q g are non­harmonic,
belong to L q (D; ¯), and satisfy the invariant mean value property OE – \Lambda ¯ = OE – . Conse­
quently, if r ? 1 then this invariant mean value property for L q (D; ¯)­functions does not
imply harmonicity.
0 INTRODUCTION
This paper continues our previous work [3], in which we showed that the invariant
mean value property in the context of Cartan domains D of rank r ? 1 with respect
to certain weighted volume probability measures ¯ Ÿ does not imply harmonicity. In this
paper we obtain the same results for a much larger class of K­invariant, absolutely con­
tinuous probability measures ¯ on D. We show that the set of spherical functions OE – in a
neighborhood of the function OE \Gammaae (z) j 1 which satisfy the invariant mean value property

Source: Arazy, Jonathan - Department of Mathematics, University of Haifa

Collections: Mathematics