Vol. 9, No. 6, 2016

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

Volume 10
Issue 2, 253–512
Issue 1, 1–252

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
Cover
About the Cover
Editorial Board
Editors’ Interests
About the Journal
Scientific Advantages
Submission Guidelines
Submission Form
Subscriptions
Editorial Login
Contacts
Author Index
To Appear
 
ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
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 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.

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