Journal of Applied Mathematics
Volume 2013 (2013), Article ID 178961, 8 pages
http://dx.doi.org/10.1155/2013/178961
Research Article

Commutators with Lipschitz Functions and Nonintegral Operators

Peizhu Xie1,2 and Ruming Gong1,2

1School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, China
2Key Laboratory of Mathematics and Interdisciplinary Sciences of Guangdong Higher Education Institutes, Guangzhou University, Guangzhou 510006, China

Received 29 March 2013; Accepted 30 May 2013

Academic Editor: Alberto Cabada

Copyright © 2013 Peizhu Xie and Ruming Gong. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let be a singular nonintegral operator; that is, it does not have an integral representation by a kernel with size estimates, even rough. In this paper, we consider the boundedness of commutators with and Lipschitz functions. Applications include spectral multipliers of self-adjoint, positive operators, Riesz transforms of second-order divergence form operators, and fractional power of elliptic operators.