Vol. 9, No. 6, 2016

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ISSN: 1948-206X (e-only)
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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

For b Lloc1(n) and α (0,1), let Dα 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 [b,DαT] is a bounded operator on a function space such as Lp(n) and Lp,λ(n) for any 1 < p < . Furthermore, we establish a necessary and sufficient condition on the function b to guarantee that [b,DαT] is a bounded operator from L(n) to BMO(n) and from L1(n) to L1,(n). This is a new theory. Finally, we apply our general theory to the Hilbert and Riesz transforms.

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