An odd rearrangement of \(L^1(\mathbf{R}^n)\) (Q2443815)

The odd rearrangement \(f_*\) of a measurable function \(f\) defined on \(\mathbb{R}^n\) is introduced and defined as \(f_{*}(x)=\text{sgn}(x_1)f^*(\nu_n|x|^n)\), where \(f^*\) denotes the classical decreasing rearrangement of \(f\). The definition allows to show that, for a given function \(f\in L^1(\mathbb{R}^n)\), there exists a function \(g\in H^1(\mathbb{R}^n)\) with the same distribution function as \(f\). As an application of this result, it is shown that, although it is not possible to give necessary conditions on singular integral operators to ensure that they map \(L^1(\mathbb{R}^n)\) into \(L^1(\mathbb{R}^n)\), some singular integral operators are of strong-type \((1,1)\), when they are restricted to odd rearrangements of \(L^1(\mathbb{R}^n)\). Also, making use of this rearrangement, the existence of an odd function in \(\displaystyle{\bigcap_{0<p\leq \infty} L^p(\mathbb{R})}\) but not in \(H^1(\mathbb{R})\) is proved and it is shown that for each \(f\in L^1(\mathbb{R})\) and \(c\in (0,\infty)\), there exists a function \(g\in H^1(\mathbb{R})\) equimeasurable with \(f\), such that \(\|Hg\|_1/\|g\|_1=c\), where \(Hg\) denotes the classical Hilbert transform of \(g\).