Commutators of Marcinkiewicz integral with rough kernels on Sobolev spaces |
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Authors: | Yan Ping Chen Yong Ding Xin Xia Wang |
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Institution: | 1. Department of Mathematics and Mechanics, School of Applied Science, University of Science and Technology of Beijing, Beijing, 100083, P. R. China 2. School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems (BNU), Ministry of Education, Beijing, 100875, P. R. China 3. The College of Mathematics and Systems Science, Xinjiang University, Urumqi, 830046, P. R. China
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Abstract: | In this paper, the authors give the boundedness of the commutator b, ????,?? ] from the homogeneous Sobolev space $\dot L_\gamma ^p \left( {\mathbb{R}^n } \right)$ to the Lebesgue space L p (? n ) for 1 < p < ??, where ????,?? denotes the Marcinkiewicz integral with rough hypersingular kernel defined by $\mu _{\Omega ,\gamma } f\left( x \right) = \left( {\int_0^\infty {\left| {\int_{\left| {x - y} \right| \leqslant t} {\frac{{\Omega \left( {x - y} \right)}} {{\left| {x - y} \right|^{n - 1} }}f\left( y \right)dy} } \right|^2 \frac{{dt}} {{t^{3 + 2\gamma } }}} } \right)^{\frac{1} {2}} ,$ , with ?? ?? L 1(S n?1) for $0 < \gamma < min\left\{ {\frac{n} {2},\frac{n} {p}} \right\}$ or ?? ?? L(log+ L) ?? (S n?1) for $\left| {1 - \frac{2} {p}} \right| < \beta < 1\left( {0 < \gamma < \frac{n} {2}} \right)$ , respectively. |
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Keywords: | Marcinkiewicz integral commutator rough kernel Sobolev space Bony paraproduct |
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