Some remarks on the characterization of the Poisson kernels for the hyperbolic spaces (Q1335291)

Let \(G/K\) be the classical hyperbolic space, i.e. \(G\) is one of the groups \(\text{SO}_0(1, n)\), \(\text{SU}(1, n)\) and \(\text{SP}(1, n)\) corresponding to the fields \(\mathbb{F}= \mathbb{R}\), \(\mathbb{C}\) and \(\mathbb{H}\), respectively. Let \(G= \text{KAN}\) be an Iwasawa decomposition of \(G\) and \(M\) the centralizer of \(A\) in \(K\). The Poisson kernel \(P: G/K\times K/M\to \mathbb{R}\) is given by \[ P(gK, kM)= \Biggl({1- |x|^2\over |1-{^t \overline x}\cdot b|^2}\Biggr)^\rho, \] where \(x= gK\), \(b= kM\) and \(\rho= d- 1+ ((n- 1)/2)d\), \(d= \dim_{\mathbb{R}}\mathbb{F}\). Let \(D\) be the Laplace-Beltrami operator on \(G/K\), then one can pose the following problem: Suppose that a real-valued \(C^2\)-class function \(F\neq 0\) on \(G/K\) satisfies the conditions \(DF= 0\) and \(DF^2= 8\rho^2 F^2\). Then do there exist \(c\in \mathbb{R}\) and \(kM\in K/M\) such that \(F(gK)= c\cdot P(gK, kM)\)? This problem has been solved in the affirmative for \(\mathbb{F}= \mathbb{R}\) by \textit{M. Chipot}, \textit{P. Eymard} and \textit{T. Tahani} [Harmonic analysis, symmetric spaces and probability theory, Cortona/Italy 1984, Symp. Math. 29, 111-129 (1987; Zbl 0646.58028)] and for \(\mathbb{F}= \mathbb{C}\) by \textit{T. Kawazoe} and \textit{T. Tahani} [Tokyo J. Math. 11, 37-55 (1988; Zbl 0691.35031)]. This paper gives a quick proof in the real case and also some preliminary results about this problem, which might be useful to solve the problem in other cases.