#### Vol. 9, No. 6, 2016

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Commutators with fractional differentiation and new characterizations of BMO-Sobolev spaces

### Yanping Chen, Yong Ding and Guixiang Hong

Vol. 9 (2016), No. 6, 1497–1522
##### Abstract

For $b\in {L}_{loc}^{1}\left({ℝ}^{n}\right)$ and $\alpha \in \left(0,1\right)$, let ${D}^{\alpha }$ be the fractional differential operator and $T$ be the singular integral operator. We obtain a necessary and sufficient condition on the function $b$ to guarantee that $\left[b,{D}^{\alpha }T\right]$ is a bounded operator on a function space such as ${L}^{p}\left({ℝ}^{n}\right)$ and ${L}^{p,\lambda }\left({ℝ}^{n}\right)$ for any $1. Furthermore, we establish a necessary and sufficient condition on the function $b$ to guarantee that $\left[b,{D}^{\alpha }T\right]$ is a bounded operator from ${L}^{\infty }\left({ℝ}^{n}\right)$ to $BMO\left({ℝ}^{n}\right)$ and from ${L}^{1}\left({ℝ}^{n}\right)$ to ${L}^{1,\infty }\left({ℝ}^{n}\right)$. This is a new theory. Finally, we apply our general theory to the Hilbert and Riesz transforms.

##### Keywords
commutator, fractional differentiation, BMO-Sobolev spaces, Littlewood–Paley theory
##### Mathematical Subject Classification 2010
Primary: 42B20, 42B25
##### Milestones
Received: 6 April 2016
Accepted: 12 May 2016
Published: 3 October 2016
##### Authors
 Yanping Chen Department of Applied Mathematics, School of Mathematics and Physics University of Science and Technology Beijing Beijing, 100083 China Yong Ding School of Mathematical Sciences, Laboratory of Mathematics and Complex Systems Beijing Normal University Beijing, 100875 China Guixiang Hong School of Mathematics and Statistics Wuhan University Wuhan, 430072 China