Dichotomies for Orlicz spaces

Abstract

Abstract Assume that L ψ , L ψ 1 , …, L ψ n are Orlicz spaces. We consider the size of the set E ψ ( ψ 1 , … , ψ n ) = { ( f 1 , … , f n ) ∈ L ψ 1 × … × L ψ n :   f 1 ⋯ f n ∈ L ψ } $E_\psi ^{({\psi _1}, \ldots, {\psi _n})} = \{ ({f_1}, \ldots, {f_n}) \in {{\mathbf{L}}^{{\psi _1}}} \times \ldots \times {{\mathbf{L}}^{{\psi _n}}}:\;{f_1} \cdots {f_n} \in {{\mathbf{L}}^\psi }\} $ . We show that either it is very small (meager or σ-porous), or it is equal to L ψ 1 × … × L ψ n . This work is a continuation of our previous papers, where we considered a similar problem for L p spaces and Lorentz spaces.