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Compactness of the commutators of homogeneous singular integral operators

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Abstract

Let T Ω be the singular integral operator with kernel \(\frac{{\Omega (x)}} {{\left| x \right|^n }}\), where Ω is homogeneous of degree zero, integrable and has mean value zero on the unit sphere S n−1. In this paper, by Fourier transform estimates, Littlewood-Paley theory and approximation, the authors prove that if Ω ∈ L(lnL)2(S n−1), then the commutator generated by T Ω and CMO(ℝn) function, and the corresponding discrete maximal operator, are compact on \(L^p \left( {\mathbb{R}^n ,\left| x \right|^{\gamma _p } } \right)\) for p ∈ (1, ∞) and γ p ∈ (−1, p − 1).

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Guo, X., Hu, G. Compactness of the commutators of homogeneous singular integral operators. Sci. China Math. 58, 2347–2362 (2015). https://doi.org/10.1007/s11425-015-5017-1

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