\(L^2\) boundedness for commutator of rough singular integral with variable kernel
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 involve spherical harmonics, one is quoted from Stein and Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Univ. Press, 1971; in the other lemma, the authors establish estimates they use to prove Theorem 1. In the third lemma, they establish an \(L^{2}\) estimate on a sum of k\(^{th}\) order commutators, and also use a result from \textit{G. Hu} [Stud. Math. 154, No. 1, 13--27 (2003; Zbl 1011.42009)], to prove their theorem.
scientific article; zbMATH DE number 5361872
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