An Odd Rearrangement of L1(Rn) - Zheng Wang; Jiecheng Chen (Journal of Function Spaces)
We introduce an odd rearrangement f* defined by π(f)(x)=f*(x)=sgn(x1)f*(νn|x|n), x∈Rn, where f* is a decreasing rearrangement of the measurable function f. With the help of this odd rearrangement, we show that for each f∈L1(Rn), there exists a g∈H1(Rn) such that df=dg, where df is an distribution function of f. Moreover, we study the boundedness of singular integral operators when they are restricted to odd rearrangement of L1(Rn), and we give some results on Hilbert transform.