Pointwise multipliers on weak Orlicz spaces

Abstract

It is well known that the set of all functions $g$ such that “$f \in L^{p_1}\Rightarrow fg \in L^{p_2}$” is $L^{p_3}$, if $1/p_2 = 1/p_1 + 1/p_3$ with $p_i \in (0,\infty], i = 1, 2, 3$. In this paper we characterize the set of all functions $g$ such that “$f \in \mathrm w L^{\phi_1} \Rightarrow fg \in \mathrm w L^{\phi_2}$”, where $\mathrm w L^{\phi_i}, i = 1, 2$, are weak Orlicz spaces.