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

Chen, Yanping; Ding, Yong; Hong, Guixiang

doi:10.2140/apde.2016.9.1497

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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

DOI: 10.2140/apde.2016.9.1497

Abstract

For $b \in L_{loc}^{1} (ℝ^{n})$ and $α \in (0, 1)$ , let $D^{α}$ 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 $[b, D^{α} T]$ is a bounded operator on a function space such as $L^{p} (ℝ^{n})$ and $L^{p, λ} (ℝ^{n})$ for any $1 < p < \infty$ . Furthermore, we establish a necessary and sufficient condition on the function $b$ to guarantee that $[b, D^{α} T]$ is a bounded operator from $L^{\infty} (ℝ^{n})$ to $BMO (ℝ^{n})$ and from $L^{1} (ℝ^{n})$ to $L^{1, \infty} (ℝ^{n})$ . 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

Vol. 9, No. 6, 2016