Weak type and restricted weak type \((p,p)\) operators in Orlicz spaces. (Q595881)

Let \((\Omega,\mu)\) be a finite measure space and \({\mathcal M}(\Omega)\) be the space of measurable functions from \(\Omega\) into \(\mathbb{R}\cup\{\infty\}\). Let \(\psi\) be a nondecreasing continuous function such that \(\psi(0)= 0\) and \(\lim_{t\to\infty}\psi(t)= \infty\). Then the family of functions \[ L^\psi(\Omega)= \biggl \{f\in{\mathcal M}(\Omega)\,\biggl|\, \int\psi(\varepsilon|f|)\,d\mu< \infty\text{ for some }\varepsilon> 0\biggr\} \] is called an Orlicz space. Consider \(L^\varphi(\Omega)\) and \(L^\psi(\Omega)\), where \[ \varphi(t)= \int^t_0 a(s)\,ds,\quad \psi(t)= \int^t_0 b(s)\,ds \] with \(a\) and \(b\) being positive continuous functions defined on \([0,\infty)\). The main goal of the paper under review is to find a relationship between \(a\) and \(b\) to provide that \(K\), a sublinear and positive homogeneous operator of restricted weak type \((p,p)\) and of type \((\infty,\infty)\), maps \(L^\psi(\Omega)\) into \(L^\varphi(\Omega)\). If the above Orlicz spaces are normable, the authors' result implies the continuity of \(K\).