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Zbl 1109.43006
Kellil, Ferdaous; Rousseau, Guy
Poisson transform on a locally finite tree. (Transformation de Poisson sur un arbre localement fini.) (French)
[J] Ann. Math. Blaise Pascal 12, No. 1, 91-116 (2005). ISSN 1259-1734

Let $X$ be a locally finite tree and let $R$ be a complex-valued kernel on $X.$ Assume either that $R(s,t)=0$ whenever $d(s,t)>1$ or $R(s,t)=0$ whenever $d(s,t)\neq 0$ and $d(s,t)\neq 2.$ Under some further natural conditions, $R$ defines a bounded operator acting on $\ell^r(X)$ for $r\geq 1.$ The authors give an explicit formula for the associated Green function $(z-R)^{-1}$. Under invariance assumptions on $R,$ they recover some results by {\it K. Aomoto} [Proc. Japan Acad., Ser. A 64, 123-125 (1988; Zbl 0699.05025)] as well as {\it A. Figà-Talamanca} and {\it T. Steger} [Mem. Am. Math. Soc. 531 (1994; Zbl 0836.43019)]. They also study a Poisson transform associated to $R$ and extend results by {\it A. Korányi, M. A. Picardello} and {\it M. H. Taibleson} [Symp. Math. 29, 205-254 (1987; Zbl 0637.31004)] and {\it M. A. Picardello, M.H. Taibleson} and {\it W. Woess} [J. Funct. Anal. 102, 379-400 (1991; Zbl 0746.05044)] as well as Figà-Talamanca and Steger ({\it loc. cit.}).
[Bachir Bekka (Rennes)]

MSC 2000:: *43A85 Analysis on homogeneous spaces
20E08 Groups acting on trees
47A30 Operator norms and inequalities
47B38 Operators on function spaces

Keywords: locally finite tree; Green operator; Poisson transform

Citations: Zbl 0699.05025; Zbl 0836.43019; Zbl 0637.31004; Zbl 0746.05044

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