Analogs of Wiener's ergodic theorems for semisimple groups. I
type: Item
instance of: scholarly article
title: Analogs of Wiener's ergodic theorems for semisimple groups. I
zbMATH Open document ID: 0884.43004
DOI: 10.2307/2951845
author: Amos Nevo
author: Elias M. Stein
published in: Annals of Mathematics. Second Series
publication date: 1997-07-23T00:00:00Z
review text: The authors prove maximal and pointwise ergodic theorems for ball averages on simple real-rank one Lie groups which are analogs of N. Wiener's maximal and pointwise ergodic theorems for ball averages on \(\mathbb{R}^n\). More precisely, let \(G\) be a simple Lie group of real-rank one and let \(K\) be a maximal compact with normalized Haar measure \(m_K\). Let \(n(G)=dim_\mathbb{R}(G/K)\). If \(X\) is a standard Borel space with a Borel measurable \(G\)-action and a \(G\)-invariant probability measure \(\lambda\), let \(\pi:G\to Iso(L^p(X))\) be the induced isometric representation. If \(\nu_t\) is a family of probability measures on \(G\), define the corresponding Markov operators on \(L^p(X)\), \(\pi(\nu_t)f(x)=\int_Gf(g^{-1}x)d\nu_t(g)\) and the associated maximal function \(M_\nu f(x)=\sup_{t>0}|\pi(\nu_t)f(x)|\). Definition. Let \(t\to\nu_t\) be an assignment of probability measures on \(G\) which is continuous with respect to the weak-\(\ast\)-topology on \(C(G)^\ast\). (i) \(\nu_t\) is a pointwise ergodic family in \(L^p\) if for any Borel probability space \(X\) and any \(f\in L^p(X)\) one has \(\lim_{t\to\infty}\pi(\nu_t)f(x)=E_1f(x)\), for a.e. \(x\) and in the \(L^p\) norm, where \(E_1\) is the conditional expectation of \(f\) with respect to the \(\sigma\)-algebra of \(G\)-invariant sets. (ii) \(\nu_t\) satisfies the local ergodic theorem in \(L^p\) if for any Borel probability space \(X\) and any \(f\in L^p(X)\), one has \(\lim_{t\to0}\pi(\nu_t)f(x)=\pi(\nu_0)f(x)\) pointwise for a.e. \(x\) and in the \(L^p\) norm. The one-parameter families of probability measures considered in the paper are the following: (1) The convolution \(\sigma_t=m_K*\delta_{a_t}*m_K\), where \(A=\{a_t:t\in\mathbb{R}\}\) is a one-parameter subgroup of hyperbolic elements so that \(G=KA_+K\) is a Cartan decomposition. (2) The absolutely continuous measure \(\beta_t\) on \(G\) with density \(vol(B_t)^{-1}\chi_{B_t}\), where \(B_t=\{g\in G:d(gK,K)\leq t\}\) with \(d\) the \(G\)-inv…
Mathematics Subject Classification ID: 43A80
Mathematics Subject Classification ID: 28D15
Mathematics Subject Classification ID: 22D40
zbMATH DE Number: 1037287
zbMATH Keywords: maximal ergodic theorem
zbMATH Keywords: pointwise ergodic theorem
zbMATH Keywords: ball average
zbMATH Keywords: sphere average
zbMATH Keywords: simple real-rank one Lie groups
reviewed by: Alessandra Iozzi
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2307/2951845
OpenAlex ID: W2328750840
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description: scientific article; zbMATH DE number 1037287