Predual of Weak Orlicz Spaces and Its Applications to Fefferman-Stein Vector-valued Maximal Inequality

In this paper the predual spaces of weak Orlicz spaces are shown to exist. As an application, the Fefferman-Stein vector-valued maximal inequality is obtained for weak Orlicz spaces. In order to prove this inequality, Orlicz-Lorentz spaces are introduced and it is shown that the Hardy-Littlewood maximal operator is bounded on these spaces. As is proved in this paper, weak Orlicz spaces are not reflexive. This makes it difficult to use the duality argument for the proof of the boundedness of operators. To avoid this problem, predual spaces are used.