Endpoint estimates for commutators of singular integral operators
type: Item
instance of: scholarly article
title: Endpoint estimates for commutators of singular integral operators
zbMATH Open document ID: 0831.42010
author: Carlos Pérez
published in: Journal of Functional Analysis
publication date: 1996-02-11T00:00:00Z
full work available at URL: https://idus.us.es/xmlui/handle/11441/48494
review text: Let \(b\) be a locally integrable function on \(\mathbb{R}^n\) and let \(T\) be a Calderón-Zygmund singular integral operator. Consider the commutator operator \([b, T]\) defined for smooth functions \(f\) by \([b, T] f= bT(f)- T(bf)\). As is now well-known, \([b, T]\) is a bounded operator on \(L^p(\mathbb{R}^n)\), \(1< p< \infty\), when \(b\) is a BMO function. However, it is not of weak type \((1,1)\), as the author shows by an example. In place of weak \((1, 1)\) inequality, he shows the following: Let \(T\) and \(b\) be as above. Then, there exists a \(C> 0\) such that for each smooth function with compact support \[ |\{ y\in \mathbb{R}^n; |[b, T] f(y)|> \lambda\}|\leq C|b|_{\text{BMO}} \int {|f(y)|\over \lambda} \Biggl(1+ \log^+\Biggl({|f(y)|\over \lambda}\Biggr)\Biggr) dy,\qquad \lambda> 0. \] By imposing a condition on \(b\) stronger than BMO, he sharpens this estimate, replacing \(L\log L\) by \(L(\log L)^\varepsilon\), \(0< \varepsilon< 1\). He also show that \([b, T]\) is bounded from a modified Hardy space \(H^1_b\) to \(L^1\). Higher order commutators are also treated.
reviewed by: Kôzô Yabuta
Mathematics Subject Classification ID: 42B20
Mathematics Subject Classification ID: 47B47
zbMATH DE Number: 764004
zbMATH Keywords: Calderón-Zygmund singular integral operator
zbMATH Keywords: commutator
zbMATH Keywords: BMO function
zbMATH Keywords: weak \((1, 1)\) inequality
zbMATH Keywords: \(L\log L\)
MaRDI profile type: Publication
OpenAlex ID: W2017550191
DOI: 10.1006/JFAN.1995.1027
description: scientific article; zbMATH DE number 764004