Local ergodic theorems for \(K\)-spherical averages on the Heisenberg group
type: Item
instance of: scholarly article
title: Local ergodic theorems for \(K\)-spherical averages on the Heisenberg group
zbMATH Open document ID: 0957.22005
published in: Mathematische Zeitschrift
publication date: 2001-03-25T00:00:00Z
review text: Let \(H^n=\mathbb{C}^n \times\mathbb{R}\) denote the Heisenberg group. Let \((H^n,K)\) be a Gelfand pair, where \(K\) is a subgroup of the unitary group \(U(n)\). Given a point \(w\in\mathbb{C}^n\) there is a measure \(\mu_{K\cdot w}\) which is supported on the \(K\)-orbit \(K\cdot w\) through \(w\) such that \[ f*\mu_{K\cdot w}(z,t) =\int_K f\bigl((z,t) (k\cdot w,0)^{-1} \bigr)dk, \] where \(dk\) is the normalized Haar measure on \(K\). The aim of this paper is to prove local ergodic theorems for the sphere and ball averages of the measure \(\mu_{K\cdot x}\) supported on orbits through real points \(x\in\mathbb{R}^n\). We define the ball averages \(\nu_r\) and the averages \(\sigma_r\) of the measure \(\mu_{K\cdot x}\) by \(\nu_r= \int_{|x|\leq r}\mu_{K\cdot x}dx\) and \(\sigma_r= \int_{|x |=r} \mu_{K\cdot x}d \mu_r^{n-1}\), where \(\mu_r^{n-1}\) is the normalized surface measure on the sphere of radius \(r\) in \(\mathbb{R}^n\). Theorem 1: The ball averages \(\nu_r\) are a local ergodic family in \(L^p\) for all \(1<p< \infty\). Theorem 2: Let \(n\geq 3\) and \(p>{n \over n-1}\). Then the sphere averages \(\sigma_r\) are a local ergodic family in \(L^p\). The author also considers averages over annuli in the case of the reduced Heisenberg group and he shows that if the function has zero mean value then the maximal function associated to the annulus averages has a better behaviour than the spherical maximal function.
Mathematics Subject Classification ID: 22D40
Mathematics Subject Classification ID: 43A80
Mathematics Subject Classification ID: 42B25
zbMATH DE Number: 1488558
zbMATH Keywords: Heisenberg group
zbMATH Keywords: Gelfand pair
zbMATH Keywords: local ergodic theorems
zbMATH Keywords: maximal function
zbMATH Keywords: spherical maximal function
author: S. Thangavelu
reviewed by: Koichi Saka
MaRDI profile type: Publication
full work available at URL: s002099900108
OpenAlex ID: W2035676363
DOI: 10.1007/S002099900108
description: scientific article; zbMATH DE number 1488558