This paper is devoted to investigating the boundedness, continuity and compactness for variation operators of singular integrals and their commutators on Morrey spaces and Besov spaces. More precisely, we establish the boundedness for the variation operators of singular integrals with rough kernels Ω ∈ L q (S n −1 ) ( q > 1) and their commutators on Morrey spaces as well as the compactness for the above commutators on Lebesgue spaces and Morrey spaces. In addition, we present a criterion on the boundedness and continuity for a class of variation operators of singular integrals and their commutators on Besov spaces. As applications, we obtain the boundedness and continuity for the variation operators of Hilbert transform, Hermit Riesz transform, Riesz transforms and rough singular integrals as well as their commutators on Besov spaces.

中文翻译：

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Morrey空间和Besov空间上粗糙奇异积分及其对易子的变分不等式

本文致力于研究奇异积分的变分算子及其对易子在莫雷空间和贝索夫空间上的有界性、连续性和紧性。更准确地说，我们建立了具有粗糙核 Ω ∈ L q (S n −1 ) ( q > 1) 的奇异积分的变分算子及其在 Morrey 空间上的交换子的有界性，以及上述交换子在 Lebesgue 空间上的紧致性和莫雷空间。此外，我们提出了一类奇异积分变分算子及其对易子在 Besov 空间上的有界性和连续性判据。作为应用，我们获得了Hilbert变换、Hermit Riesz变换、Riesz变换和粗糙奇异积分的变分算子及其在Besov空间上的对易子的有界性和连续性。