\(L^2\) boundedness for commutator of rough singular integral with variable kernel
type: Item
instance of: scholarly article
title: \(L^2\) boundedness for commutator of rough singular integral with variable kernel
zbMATH Open document ID: 1165.42004
DOI: 10.4171/RMI/545
published in: Revista Matemática Iberoamericana
publication date: 2008-11-05T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.rmi/1218475352
full work available at URL: https://eudml.org/doc/42383
review text: The authors obtain an improvement on a result of \textit{F. Chiarenza, M. Frasca} and \textit{P. Longo} [Ric. Mat. 40, No. 1, 149--168 (1991; Zbl 0772.35017)] which, in turn, was preceded by a theorem of \textit{A. P. Calderón} and \textit{A. Zygmund} on the \(L^{2}\) boundedness of a singular integral operator [Trans. Am. Math. Soc. 78, 209--224 (1955; Zbl 0065.04104)]. The operator, \(T\), is defined by \( Tf(x)=p.v.\int_{R^{n}}\frac{\Omega (x,x-y)}{| x-y| ^{n}}f(y)dy\), when \(f\in L^{2}(\mathbb{R}^{n})\) and \(\Omega \) is a function that satisfies the conditions (\(\mathbb{S}^{n-1}\) is the unit sphere in \(\mathbb{R}^{n}\), \(d\sigma \) is normalized Lebesgue measure): 1) for any \(x,z\in \mathbb{R}^{n},\;\lambda >0,\) \(\Omega (x,\lambda z)=\Omega (x,z).\) 2) \(\sup \{x\in \mathbb{R}^{n},( \int_{S^{n-1}}| \Omega (x,z^{\prime })| ^{q}d\sigma (z^{\prime })) ^{1/q}\}<\infty ,\) \(z^{\prime }=\frac{z}{| z| },\) \(z\neq 0.\) 3) \(\int_{S^{n-1}}\Omega (x,z^{\prime })d\sigma (z^{\prime })=0.\;\) for all \(x\in \mathbb{R}^{n}.\) The theorem proved in this paper is Theorem 1: If \(\Omega (x,z^{\prime })\) satisfies the above conditions with \(q>\frac{2(n-1)}{n}\), then for \(k\in \mathbb{N}\), there is a constant \(C>0\) such that \(\| T_{b,k}f\| _{L^{2}}\leq C\| b\| _{\ast }^{k}\cdot \| f\| _{L^{2}}\). \(T_{b,k}f(x)=p.v.\int_{R^{n}}\frac{\Omega (x,x-y)}{| x-y| ^{n}}(b(x)-b(y))^{k}f(y)dy\), with \(b\in BMO(\mathbb{R}^{n})\), i.e. \(\| b\| _{\ast }=\sup \{Q\) any cube in \( \mathbb{R}^{n},\frac{1}{| Q| }\int_{Q}| b(x)-b_{Q}| dx\}<\infty ,\) \(b_{Q}=\frac{1}{| Q| }\int_{Q}b(x)dx\). After briefly discussing previous work, which, in addition to the work mentioned above, includes \textit{L. Tang} and \textit{D. Yang} [J. Beijing Norm. Univ., Nat. Sci. 36, No. 6, 741--745 (2000; Zbl 1070.42010)] and \textit{G. Di Fazio} and \textit{M. A. Ragusa} [J. Funct. Anal. 112, No. 2, 241--256 (1993; Zbl 0822.35036)], the authors state four technical lemmas. Two of these lemmas involv…
reviewed by: Caroline Sweezy
Mathematics Subject Classification ID: 42B20
Mathematics Subject Classification ID: 42B25
zbMATH DE Number: 5361872
zbMATH Keywords: commutator
zbMATH Keywords: singular integral
zbMATH Keywords: variable kernel
zbMATH Keywords: bounded mean oscillation
zbMATH Keywords: spherical harmonic function
author: Yanping Chen
author: Yong Ding
MaRDI profile type: Publication
cites work: On a Problem of Mihlin
cites work: On singular integrals with variable kernels
cites work: Interior \(W^{2,p}\) estimates for non-divergence elliptic equations with discontinuous coefficients
cites work: Factorization theorems for Hardy spaces in several variables
cites work: Maximal and singular integral operators via Fourier transform estimates
cites work: Interior estimates in Morrey spaces for strong solutions to nondivergence form equations with discontinuous coefficients
cites work: Lp(Rn) boundedness for the commutator of a homogeneous singular integral operator
cites work: Q5640160
cites work: Q5463711
cites work: Q5562752
OpenAlex ID: W2079213736
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description: scientific article; zbMATH DE number 5361872