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Journal of Function Spaces
Volume 2014, Article ID 787840, 11 pages
http://dx.doi.org/10.1155/2014/787840
Research Article

An Odd Rearrangement of ()

1Department of Mathematics, Zhejiang University, Hangzhou 310027, China
2Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China

Received 13 November 2013; Accepted 31 December 2013; Published 26 February 2014

Academic Editor: Dumitru Motreanu

Copyright © 2014 Zheng Wang and Jiecheng Chen. 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

We introduce an odd rearrangement defined by , , where is a decreasing rearrangement of the measurable function . With the help of this odd rearrangement, we show that for each , there exists a such that , where is an distribution function of . Moreover, we study the boundedness of singular integral operators when they are restricted to odd rearrangement of , and we give some results on Hilbert transform.