A variant of the notion of a space of homogeneous type
type: Item
instance of: scholarly article
title: A variant of the notion of a space of homogeneous type
zbMATH Open document ID: 0912.42011
author: Anthony Carbery
author: James Vance
author: Stephen Wainger
author: James Wrigth
published in: Journal of Functional Analysis
publication date: 1995-08-27T00:00:00Z
review text: In studying maximal functions and singular integral operators associated to certain curves or surfaces, one is led to the study of maximal functions and singular integral operators for which there is a natural family of balls, but such that these balls are not the balls of the space of homogeneous type. In order to overcome this difficulty, the authors give a double infinite sequence of nonnegative continuous functions \(\{\rho_j(x,y)\}\) on \(X\times X\), where \(X\) is a locally compact Hausdorff space equipped with a regular Borel measure \(\mu\), and corresponding balls \(B_j(x,r)= \{y\in X: \rho_j(y,x)< r\}\). Then the authors prove that if \(\{\rho_j\}\) and \(\{B_j\}\) satisfy certain conditions then the Hardy-Littlewood maximal function, associated to \(\{B_j\}\) is of strong type \((p,p)\), \(p>1\), and of weak type \((1,1)\). To the singular operator \[ Tf(x)= \int_X K(x,y) f(y) d\mu(y), \] the authors prove that if \(\{\rho_j\}\) and \(\{B_j\}\) satisfy certain conditions, \(Tf\) is bounded on \(L^2(X)\), and the kernel \(K(x,y)\) satisfies a Hörmander condition related to \(\{\rho_j\}\), then \(Tf\) is of weak type \((1,1)\) and of strong type \((p,p)\), \(1<p\leq 2\). As applications, the authors obtain weak type \((1,1)\) bounds and strong type \((p,p)\) bounds for some singular integral operators on a homogeneous Lie group. The authors also get similar results for maximal functions and singular integral operators on \(\mathbb{R}^n\) related to a continuous nonnegative pseudo-metric on \(\mathbb{R}^n\).
Mathematics Subject Classification ID: 42B25
Mathematics Subject Classification ID: 42B20
Mathematics Subject Classification ID: 43A85
zbMATH DE Number: 790376
zbMATH Keywords: singular integral operators
zbMATH Keywords: space of homogeneous type
zbMATH Keywords: Hardy-Littlewood maximal function
zbMATH Keywords: strong type \((p,p)\)
zbMATH Keywords: weak type \((1,1)\)
reviewed by: Dashan Fan
MaRDI profile type: Publication
full work available at URL: jfan.1995.1102
OpenAlex ID: W2012109212
DOI: 10.1006/JFAN.1995.1102
description: scientific article; zbMATH DE number 790376