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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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References:
 [1] E.P. Van Den Ban - H. Schlichtkrull , Asymptotic expansions and boundary values of eigenfunctions on Riemannian symmetric spaces . J. reine angew. Math. 380 ( 1987 ), pp. 108 - 165 . Article | MR 916202 | Zbl 0631.58028 · Zbl 0631.58028 · crelle:GDZPPN002204878 · eudml:152961 [2] A. Figà-Talamanca - M.A. Picardello , Harmonic analysis on free groups . Marcel Dekker , 1983 . MR 710827 | Zbl 0536.43001 · Zbl 0536.43001 [3] G.B. Folland - E.M. Stein , Hardy spaces on homogeneous groups . Princeton University Press , 1982 . MR 657581 | Zbl 0508.42025 · Zbl 0508.42025 [4] S. Helgason , A duality for symmetric spaces with applications to group representations II. Differential equations and eigenspace representations . Adv. in Math. 22 ( 1976 ), pp. 187 - 219 . MR 430162 | Zbl 0351.53037 · Zbl 0351.53037 · doi:10.1016/0001-8708(76)90153-5 [5] S. Helgason , Differential geometry, Lie groups, and symmetric spaces . Academic Press , 1978 . MR 514561 | Zbl 0451.53038 · Zbl 0451.53038 [6] S. Helgason , Groups and geometric analysis: Integral geometry, invariant differential operators, and spherical functions . Academic Press , 1984 . MR 754767 | Zbl 0543.58001 · Zbl 0543.58001 [7] A.W. Knapp - E.M. Stein , Intertwining operators for semisimple groups, II . Invent. Math. 60 ( 1980 ), pp. 9 - 84 . MR 582703 | Zbl 0454.22010 · Zbl 0454.22010 · doi:10.1007/BF01389898 · eudml:142742 [8] A. Korányi - M.A. Picardello , Boundary behaviour of eigenfunctions of the Laplace operator on trees . Ann. Scuola Norm. Sup. Pisa , 13 ( 1986 ), pp. 389 - 399 . Numdam | MR 881098 | Zbl 0621.31005 · Zbl 0621.31005 · numdam:ASNSP_1986_4_13_3_389_0 · eudml:83984 [9] P. Sjögren , Convergence for the square root of the Poisson kernel . Pacific J. Math. 131 ( 1988 ), pp. 361 - 391 . Article | MR 922224 | Zbl 0601.31001 · Zbl 0601.31001 · doi:10.2140/pjm.1988.131.361 · minidml.mathdoc.fr
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