Vol. 9, No. 6, 2016

Download this article
Download this article For screen
For printing
Recent Issues

Volume 17
Issue 2, 379–756
Issue 1, 1–377

Volume 16, 10 issues

Volume 15, 8 issues

Volume 14, 8 issues

Volume 13, 8 issues

Volume 12, 8 issues

Volume 11, 8 issues

Volume 10, 8 issues

Volume 9, 8 issues

Volume 8, 8 issues

Volume 7, 8 issues

Volume 6, 8 issues

Volume 5, 5 issues

Volume 4, 5 issues

Volume 3, 4 issues

Volume 2, 3 issues

Volume 1, 3 issues

The Journal
About the Journal
Editorial Board
Editors’ Interests
Submission Guidelines
Submission Form
Policies for Authors
Ethics Statement
ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
Author Index
To Appear
Other MSP Journals
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