Spherical functions and conformal densities on spherically symmetric 𝐶𝐴𝑇(-1)-spaces

Abstract

Let X X be a C A T ( − 1 ) CAT(-1) -space which is spherically symmetric around some point x 0 ∈ X x_{0}\in X and whose boundary has finite positive s − s- dimensional Hausdorff measure. Let μ = ( μ x ) x ∈ X \mu =(\mu _{x})_{x\in X} be a conformal density of dimension d > s / 2 d>s/2 on ∂ X \partial X . We prove that μ x 0 \mu _{x_{0}} is a weak limit of measures supported on spheres centered at x 0 x_{0} . These measures are expressed in terms of the total mass function of μ \mu and of the d − d- dimensional spherical function on X X . In particular, this result proves that μ \mu is entirely determined by its dimension and its total mass function. The results of this paper apply in particular for symmetric spaces of rank one and semi-homogeneous trees.