On the Poisson transform on symmetric spaces of real rank one
A characterization of the image of \(L^2(S^n)\) under the Poisson transform on hyperbolic spaces has been conjectured by \textit{R. S. Strichartz} in [J. Funct. Anal. 87, 51-148 (1989; Zbl 0694.43008) and J. Funct. Anal. 109, 457-460 (1992; Zbl 0789.43004)]. To complete the proposed proof one needed to establish certain average estimates for generalized spherical functions. In this paper the author works in the more general context of real rank-one Riemannian symmetric spaces \(G/K\) and proves an \(L^2\)-type characterization of the image under the Poisson transform of the space \(L^2(K/M)\) of \(L^2\)-functions on the Furstenberg boundary \(K/M\). This provides, in particular, a proof for Strichartz' conjecture. The key step is to establish the average estimates. Here they are given in a version which is slightly weaker than the one conjectured for the hyperbolic spaces, but enough to yield the required characterization. A related Plancherel-type theorem is then deduced.
scientific article; zbMATH DE number 1496050