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Asymptotic behaviour of generalized Poisson integrals in rank one symmetric spaces and in trees. (English) Zbl 0696.43008
Let G/K be a noncompact rank-one symmetric space, let $${\mathfrak A}_{{\mathbb{C}}}$$ be the complexified Cartan subalgebra of the Lie algebra of G, and $$\lambda\in {\mathfrak A}'_{{\mathbb{C}}}$$. The $$\lambda$$-Poisson transform of $$f\in L^ 1(K/M)$$ is defined as $P_{\lambda}f(g\cdot \circ)=\int_{K/M}f(kM)\exp <-(\rho +\lambda),H(g^{-1}k)>d(kM)$ where $$\rho$$ is the Weyl shift and H($$\cdot)\in {\mathfrak A}$$ is the appropriate factor in the Iwasawa decomposition. If Re $$\lambda$$ $$>0$$, denote by $${\mathcal P}_{\lambda}$$ the normalized Poisson transform: $${\mathcal P}_{\lambda}f=P_{\lambda}f/P_{\lambda}1$$. It is known that $${\mathcal P}_{\lambda}f$$ converges to f radially, and, more generally, admissibly. The same is true for $$\lambda =0$$. On the other hand, if $$\lambda$$ is purely imaginary (“principal series”), then $${\mathcal P}_{\lambda}$$ cannot be defined as above because the spherical function $$P_{\lambda}1$$ has an unbounded sequence of zeroes. Asymptotically, however, $$P_{\lambda}1\approx 2c(\lambda)e^{<\lambda -\rho,H>}\neq 0$$, where c is the Harish Chandra c-function. Therefore $${\mathcal P}_{\lambda}$$ can be defined as $${\mathcal P}_{\lambda}f=e^{<\rho,H>}P_{\lambda}f$$. The main result shows that, for admissible convergence to the boundary K/M, $${\mathcal P}_{\lambda}f$$ behaves asymptotically as a linear combination of two oscillating terms $$\exp <\pm \lambda,H>$$. The author determines the coefficients and estimates the remainder. This extends results of S. Helgason [Adv. Math. 22, 187-219 (1976; Zbl 0351.53037)], which hold for K-finite functions. Finally an analogous result is proved for homogeneous trees equipped with the nearest-neighbour isotropic Laplacian. In this context, the asymptotic-behaviour of $${\mathcal P}_{\lambda}f$$ for Re $$\lambda$$ $$>0$$ (or $$\lambda =0)$$ has been determined by A. Korányi and the reviewer [Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 13, 389-399 (1986; Zbl 0621.31005)], and, for $$\lambda$$ positive, by P. Cartier [Symp. Math. 9, 203-270 (1972; Zbl 0283.31005)].
Reviewer: M.Picardello

##### MSC:
 43A90 Harmonic analysis and spherical functions 43A85 Harmonic analysis on homogeneous spaces 53C35 Differential geometry of symmetric spaces 43A32 Other transforms and operators of Fourier type 31C05 Harmonic, subharmonic, superharmonic functions on other spaces
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