Weak estimates for commutators of fractional integral operators
type: Item
instance of: scholarly article
title: Weak estimates for commutators of fractional integral operators
zbMATH Open document ID: 1014.42012
DOI: 10.1007/BF02880137
author: Pu Zhang
author: Yong Ding
author: Shanzhen Lu
published in: Science in China. Series A
publication date: 2002-08-15T00:00:00Z
review text: The authors consider the commutators \([b,I_\alpha]f(x) = b(x)I_\alpha f(x)-I_\alpha(bf)(x)\) generated by the fractional integral operator \(I_\alpha\) defined by \[ I_\alpha f(x)\int_{\mathbb{R}^n}\frac{f(y)}{|x-y|^{n-\alpha}} dy\quad (0<\alpha<n), \] and a BMO function \(b(\mathbb{R}^n)\). It is proved that for \(q = n/(n-\alpha)\), \([b,I_\alpha]\) satisfies the following weak type \(L\log L\) estimates: Let \(b\in \text{BMO}(\mathbb{R}^n)\), \(0 <\alpha < n\), \(\Phi(t) = t(1+\log^+t)\). Then there exists a positive constant \(C\) such that for any \(\lambda > 0\) \[ |\{x\in\mathbb{R}^n:|[b,I_\alpha]f(x)|>\lambda\}|^{1/q}\leq C\Phi(\Phi(\|b\|_*))\|\phi(|f(\cdot)|/\lambda)\|_1\{1+\tfrac{\alpha}{n} \log^+\|\Phi(|f(\cdot)|/\lambda\|_1\}; \] where \(\|b\|_*\) is the BMO norm of \(b\). In addition, for the higher order commutators the weak type \(L \log L\) estimates are shown by using the technique of sharp functions.
reviewed by: Sunggeum Hong
Mathematics Subject Classification ID: 42B20
Mathematics Subject Classification ID: 26A33
Mathematics Subject Classification ID: 47B38
Mathematics Subject Classification ID: 47B47
Mathematics Subject Classification ID: 47G10
zbMATH DE Number: 1782133
zbMATH Keywords: fractional integral operator
zbMATH Keywords: BMO
zbMATH Keywords: higher order commutators
zbMATH Keywords: weak type
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1782133