Notes on some classical series associated with discrete subgroups of \(U(1,n;\mathbb{C}{})\) on \(\partial{}B^ n\times{}\partial{}B^ n\times{}\partial{}B^ n\)
type: Item
instance of: scholarly article
title: Notes on some classical series associated with discrete subgroups of \(U(1,n;\mathbb{C}{})\) on \(\partial{}B^ n\times{}\partial{}B^ n\times{}\partial{}B^ n\)
zbMATH Open document ID: 0764.30034
author: Shigeyasu Kamiya
published in: Proceedings of the Japan Academy. Series A
publication date: 1993-01-17T00:00:00Z
review text: Let \(U(1,n;\mathbb{C})\) be the group of unitary transformations. Our purpose is to show two theorems on some classical series associated with discrete subgroups of \(U(1,n;\mathbb{C})\) acting on \(\partial B^ n\times\partial B^ n\times\partial B^ n\). Denote by \(G\) a discrete subgroup of \(U(1,n;\mathbb{C})\). Let \(\{g_ 1,g_ 2,\dots\}\) be a complete list of elements of \(G\). If \(g_ k\) is an element of \(G\), then \(g_ k\) is represented by a matrix \((a_{aj}^{(k)})_{1\leq i,j\leq n+1}\). Let \(x=(x_ 1,\dots,x_ n)\), \(y=(y_ 1,\dots,y_ n)\) and \(z=(z_ 1,\dots,z_ n)\) be points in \(\partial B^ n\). Theorem 1. The series \[ \sum_{g_ k\in G} \left(\left| a_{11}^{(k)} +\sum_{j=2}^{n+1} a_{1j}^{(k)} x_{j- 1}\right| \left| a_{11}^{(k)}+ \sum_{j=2}^{n+1} a_{1j}^{(k)} y_{j-1}\right| \left| a_{11}^{(k)}+ \sum_{j=2}^{n+1} a_{1j}^{(k)} z_{j-1}\right|\right)^{- 2n} \] converges for almost every triple \((x,y,z)\) in \(\partial B^ n\times\partial B^ n\times\partial B^ n\). Theorem 2. If \(\sum_{g_ k\in G} | a_{11}^{(k)}|^{-m}\) converges for \(m>0\), then the series \[ \sum_{g_ k\in G} \left(\left| a_{11}^{(k)}+ \sum_{j=2}^{n+1} a_{1j}^{(k)} x_{j-1}\right| \left| a_{11}^{(k)}+ \sum_{j=2}^{n+1} a_{1j}^{(k)} y_{j-1}\right| \left| a_{11}^{(k)}+ \sum_{j=2}^{n+1} a_{1j}^{(k)} z_{j-1}\right|\right)^{- m} \] converges for every distinct points \(x\), \(y\) and \(z\) in \(\partial B^ n\).
reviewed by: Shigeyasu Kamiya
Mathematics Subject Classification ID: 30F40
zbMATH DE Number: 97947
zbMATH Keywords: unitary transformations
MaRDI profile type: Publication
full work available at URL: pjaa.68.137
OpenAlex ID: W1983742018
cites work: Q3686976
cites work: DISCRETE SUBGROUPS OF U(1, n; C) ON THE PRODUCT SPACE ∂Bn × ∂Bn × … × ∂Bn
cites work: Discrete subgroups of convergence type of U(1,n;C)
cites work: Q3993094
DOI: 10.3792/PJAA.68.137
description: scientific article; zbMATH DE number 97947