, Volume 23, Issue 2, pp 123-144

L q -Estimates of spherical functions and an invariant mean-value property

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Abstract

We find someL q -estimates for the spherical functions on Cartan domains. As an application we prove that if the rank of the Cartan domainD is greater than one, then for any 1<-q<∞, the invariant mean-value property forL q -function onD does not imply harmonicity (the converse is known to be true even in the context of general non-compact Riemannian symmetric spacesG/K).