Pointwise ergodic theorems for radial averages on simple Lie groups. II
type: Item
instance of: scholarly article
title: Pointwise ergodic theorems for radial averages on simple Lie groups. II
zbMATH Open document ID: 0869.43005
DOI: 10.1215/S0012-7094-97-08607-5
author: Amos Nevo
published in: Duke Mathematical Journal
publication date: 1997-08-31T00:00:00Z
review text: The present paper is a continuation of Part I [ibid. 76, 113-140 (1994; Zbl 0838.43013)], and we begin by briefly recalling the setup and the notation: -- \(G=G_n= SO^0 (n,1)\) is the group of orientation-preserving isometries of \(n\)-dimensional real hyperbolic space \(H^n\), \(n\geq 2\). -- \(K=\) a fixed maximal compact subgroup. \(m_K=\) Haar probability measure. -- \(A= \{a_t|t\in\mathbb{R}\}=\) a one-parameter group of hyperbolic translations such that \(G=KA_+K\) is a Cartan decomposition. -- \(\sigma_t=\) the bi-\(K\)-invariant probability measure on \(G\) given by \(\sigma_t= m_K* \delta_{a_t}* m_K\), where \(*\) denotes convolution. Note that \(\sigma_0=m_K\). -- \(\mu_t= 1/t\int^t_0 \sigma_sds\), the uniform average of \(\sigma_s\), \(0\leq s\leq t\). We define \(\mu_0= m_K\). -- \(M(G,K)=\) the commutative convolution algebra (of bi-\(K\)-invariant complex bounded Borel measures on \(G)\) generated by \(\sigma_t\), \(t\geq 0\). -- \((X,{\mathcal B}, \lambda)=\) a standard Borel space with a Borel measurable \(G\)-action which preserves the probability measure \(\lambda\). -- \(\pi(\nu) f(x)= \int_Gf(g^{-1}x)d \nu(g)=\) the Markov operator on \(L^p(X)\) corresponding to a probability measure \(\nu\) on \(G\). -- \(M_\mu f(x)= \sup_{t \geq 0} |\pi (\mu_t) f(x)|\), and \(M_\sigma f(x) =\sup_{t\geq 0}|\pi(\sigma_t) f(x) |\), maximal functions associated with the action of \(\sigma_t\) and \(\mu_t\) in \(L^p(X)\), \(1\leq p\leq \infty\). Finally, recall also the following definition. Definition. Let \(\nu_t\), \(t\geq 0\), be a one-parameter family of probability measures on \(G\). Assume that \(t \mapsto\nu_t\in M(G)\) is continuous in the \(w^*\)-topology of \(M(G)\) as the dual of \(C_0(G)\). Let \((X, {\mathcal B}, \lambda) \) denote a \(G\)-space as above. (1) \(\nu_t\) is called a pointwise ergodic family in \(L^p\) if, for any \(f\in L^p(X)\), \(\lim_{t\to\infty} \pi(\nu_t) f(x)= E_1(f)(x)\), where the convergence is pointwise almost everywhere and in…
Mathematics Subject Classification ID: 43A90
Mathematics Subject Classification ID: 22D40
zbMATH DE Number: 997804
zbMATH Keywords: simple Lie groups
zbMATH Keywords: orientation-preserving isometries
zbMATH Keywords: real hyperbolic space
zbMATH Keywords: convolution
zbMATH Keywords: convolution algebra
zbMATH Keywords: probability measure
zbMATH Keywords: Markov operator
zbMATH Keywords: pointwise ergodic family
zbMATH Keywords: local ergodic theorem
MaRDI profile type: Publication
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full work available at URL: s0012-7094-97-08607-5
OpenAlex ID: W4255513410
description: scientific article; zbMATH DE number 997804