Chen, Yanping; Ding, Yong \(L^2\) boundedness for commutator of rough singular integral with variable kernel. (English) Zbl 1165.42004 Rev. Mat. Iberoam. 24, No. 2, 531-547 (2008). The authors obtain an improvement on a result of F. Chiarenza, M. Frasca and P. Longo [Ric. Mat. 40, No. 1, 149–168 (1991; Zbl 0772.35017)] which, in turn, was preceded by a theorem of A. P. Calderón and 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 isTheorem 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 L. Tang and D. Yang [J. Beijing Norm. Univ., Nat. Sci. 36, No. 6, 741–745 (2000; Zbl 1070.42010)] and G. Di Fazio and 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 G. Hu [Stud. Math. 154, No. 1, 13–27 (2003; Zbl 1011.42009)], to prove their theorem. Reviewer: Caroline Sweezy (Las Cruces, NM) Cited in 1 ReviewCited in 13 Documents MSC: 42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.) 42B25 Maximal functions, Littlewood-Paley theory Keywords:commutator; singular integral; variable kernel; bounded mean oscillation; spherical harmonic function Citations:Zbl 0772.35017; Zbl 0065.04104; Zbl 1070.42010; Zbl 0822.35036; Zbl 1011.42009 × Cite Format Result Cite Review PDF Full Text: DOI Euclid EuDML References: [1] Calderón, A. and Zygmund, A.: On a problem of Mihlin. Trans. Amer. Math. Soc. 78 (1955), 209-224. JSTOR: · Zbl 0065.04104 · doi:10.2307/1992955 [2] Calderón, A. and Zygmund, A.: On singular integrals with variable kernels. Applicable Anal. 7 (1977/78), no. 3, 221-238. · Zbl 0451.42012 · doi:10.1080/00036817808839193 [3] Chiarenza, F., Frasca, M. and Longo, P.: Interior \(W^2,p\) estimates for nondivergence elliptic equations with discontinuous coefficients. Ricerche Math. 40 (1991), no. 1, 149-168. · Zbl 0772.35017 [4] Coifman, R. Rocherberg, R., Weiss, G.: Factorization theorems for Hardy spaces in several variables. Ann. of Math. (2) 103 (1976), no. 3, 611-636. JSTOR: · Zbl 0326.32011 · doi:10.2307/1970954 [5] Duoandikoetxea, J. and Rubio de Francia, J. L.: Maximal and singular integral operators via Fourier transform estimates. Invent. Math. 84 (1986), no. 3, 541-561. · Zbl 0568.42012 · doi:10.1007/BF01388746 [6] Di Fazio, G. and Ragusa, M. A.: Interior estimates in Morrey spaces for strong solutions to nondivergence form equations with discontinuous coefficients. J. Funct. Anal. 112 (1993), no. 2, 241-256. · Zbl 0822.35036 · doi:10.1006/jfan.1993.1032 [7] Hu, G.: \(L^p(\mathbb R^n)\) boundedness for the commutator of a homogeneous singular integral operator. Studia Math. 154 (2003), no. 1, 13-27. · Zbl 1011.42009 · doi:10.4064/sm154-1-2 [8] Stein, E. and Weiss, G.: Introduction to Fourier Analysis on Euclidean Spaces. Princeton Mathematical Series 32 . Princeton University Press, Princeton, N.J, 1971. · Zbl 0232.42007 [9] Tang, L. and Yang, D.: \(L^2(\mathbbR^2)\)-boundedness of commutators of singular integrals of rough variable kernels. Beijing Shifan Daxue Xuebao 36 (2000), no. 6, 741-745. · Zbl 1070.42010 [10] Watson, G.: A Treatise on the Theory of Bessel Functions. Cambridge Mathematical Library. Cambridge University Press, Cambridge, 1966. · Zbl 0174.36202 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.