On rearrangements of vector-valued \(H_ 1\)-functions (Q908487)

Let X be a Banach space, \(T={\mathbb{R}}/{\mathbb{Z}}\), \(L_ 1(T,X)\) be the space of all Bochner integrable functions on T and \(H^ a_ 1(X)\) be the space of all \(f\in L_ 1(T,X)\) such that \(f=\sum_{n}c_ na_ n\), where each \(a_ n\) is an atom and \(\sum | c_ n| <\infty\). The function \(f\in L_ 1(T,X)\) is called to be a rearrangement of \(g\in L_ 1(T,X)\) if \(\mu_ f=\mu_ g\) where \(\mu_ f\) is the Borel measure on X defined by \(\mu_ f(B)=\lambda (f^{-1}(B))\) (B\(\subset X\) and \(\lambda\) is a normalized Haar measure on T). It is found the necessary and sufficient conditions that \(f\in L_ 1(T,X)\) has a rearrangement in \(H^ a_ 1(X)\).