Notes on some classical series associated with discrete subgroups of \(U(1,n;\mathbb{C}{})\) on \(\partial{}B^ n\times{}\partial{}B^ n\times{}\partial{}B^ n\) (Q1201498)

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 \(\partial B^ n\).