An Odd Rearrangement of L1(Rn). (26th February 2014)

• Record Type: Journal Article
• Authors: Wang, Zheng; Chen, Jiecheng
• Other Names: Motreanu Dumitru Academic Editor.
• Is Part Of: Journal of function spaces. Volume 2014(2014)
• Publication Date: 2014-02-26
• Subjects: Function spaces -- Periodicals; 515.7305
• DOI: 10.1155/2014/787840
• Languages: English
• ISSNs: 2314-8896

Abstract

Abstract : We introduce an odd rearrangementf * defined byπ ( f ) ( x ) = f * ( x ) = s g n ( x 1 ) f * ( ν n | x | n ), x ∈ R n, wheref * is a decreasing rearrangement of the measurable functionf . With the help of this odd rearrangement, we show that for eachf ∈ L 1 ( R n ), there exists ag ∈ H 1 ( R n ) such thatd f = d g, whered f is an distribution function off . Moreover, we study the boundedness of singular integral operators when they are restricted to odd rearrangement ofL 1 ( R n ), and we give some results on Hilbert transform.