Endpoint estimates for commutators of singular integrals on spaces of homogeneous type
type: Item
instance of: scholarly article
title: Endpoint estimates for commutators of singular integrals on spaces of homogeneous type
zbMATH Open document ID: 1019.43005
DOI: 10.1016/S0022-247X(03)00172-0
published in: Journal of Mathematical Analysis and Applications
publication date: 2003-07-30T00:00:00Z
review text: Let \((X,d,\mu)\) be a space of homogeneous type (\(d\) is a quasimetric on \(X\) and \(\mu\) is a non negative Borel measure on \(X\)). The BMO space of \(X\) is \(BMO(X,\mu)=\{f\in L_{loc}^{1}(X): \|f\|_{*}<+\infty\}\) where \(\|f\|_{*}=Sup_{B}\frac{1}{\mu(B)}\int_{B}\mid f(y)-f_{B} \mid d\mu(y)\) and \(f_{B}=\frac{1}{\mu(B)}\int_{B}f(x)d\mu(x)\) for a ball \(B\) in \(X\). The main result of the paper under review is stated as follows: let \(T\) be a singular integral operator which is bounded on \(L^{2}(X,\mu)\) and \(b\in BMO(X,\mu)\). Then there exists a positive constant \(C\) such that for each bounded function \(f\) with bounded support and for all \(\lambda >0\) \[ \mu\big(\{y\in X: \mid[b,T]f(y)\mid>\lambda\}\big)\leq C\|b\|_{*}\int_{X}\frac{\mid f(y)\mid}{y}(1+log^{+}(\frac{\mid f(y)\mid}{y})) d\mu(y) \] where \([b,T]\) is the commutator of \(b\) and \(T\).
reviewed by: Salah Mehdi
Mathematics Subject Classification ID: 43A85
zbMATH DE Number: 1957233
zbMATH Keywords: singular integrals
zbMATH Keywords: estimates
zbMATH Keywords: spaces of homogeneous type
author: Wengu Chen
author: Eric T. Sawyer
MaRDI profile type: Publication
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full work available at URL: s0022-247x(03)00172-0
OpenAlex ID: W2091247227
description: scientific article; zbMATH DE number 1957233