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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Spherical functions and conformal densities on spherically symmetric $CAT(-1)$-spaces
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by Michel Coornaert and Athanase Papadopoulos PDF
Trans. Amer. Math. Soc. 351 (1999), 2745-2762 Request permission

Abstract:

Let $X$ be a $CAT(-1)$-space which is spherically symmetric around some point $x_{0}\in X$ and whose boundary has finite positive $s-$dimensional Hausdorff measure. Let $\mu =(\mu _{x})_{x\in X}$ be a conformal density of dimension $d>s/2$ on $\partial X$. We prove that $\mu _{x_{0}}$ is a weak limit of measures supported on spheres centered at $x_{0}$. These measures are expressed in terms of the total mass function of $\mu$ and of the $d-$dimensional spherical function on $X$. In particular, this result proves that $\mu$ 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.
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Additional Information
  • Michel Coornaert
  • Affiliation: Institut de Recherche Mathématique Avancée, Université Louis Pasteur et CNRS, 7, rue René Descartes, 67084 Strasbourg Cedex France
  • Email: coornaert@math.u-strasbg.fr
  • Athanase Papadopoulos
  • Affiliation: Institut de Recherche Mathématique Avancée, Université Louis Pasteur et CNRS, 7, rue René Descartes, 67084 Strasbourg Cedex France
  • MR Author ID: 135835
  • Email: papadopoulos@math.u-strasbg.fr
  • Received by editor(s): January 30, 1996
  • Received by editor(s) in revised form: June 12, 1997
  • Published electronically: February 5, 1999
  • Additional Notes: The second author was also supported by the Max-Planck-Institut für Mathematik (Bonn)
  • © Copyright 1999 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 351 (1999), 2745-2762
  • MSC (1991): Primary 53C35
  • DOI: https://doi.org/10.1090/S0002-9947-99-02155-8
  • MathSciNet review: 1466945