An Odd Rearrangement of L1(Rn). (26th February 2014) - Wang, Zheng; Chen, Jiecheng
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.