Dichotomies for Orlicz spaces (Q2810997)

Let \(L^{\phi},L^{\phi_1},\dots ,L^{\phi_n}\) be Orlicz spaces. The author proves that, under some mild condition on \(\phi ,\phi_1,\dots ,\phi_n\), the set \(\bigl \{(f_1,\dots ,f_n)\in L^{\phi_1}\times \dots \times L^{\phi_n}:f_1\cdots f_n\in L^{\phi}\bigr \}\) either equals \(L^{\phi_1}\times \dots \times L^{\phi_n}\) or it is \(\sigma \)-porous in \(L^{\phi_1}\times \dots \times L^{\phi_n}\). In the general situation, the set \(\bigl \{(f_1,\dots ,f_n)\in L^{\phi_1}\times \dots \times L^{\phi_n}:f_1\dots f_n\in L^{\phi}\bigr \}\) either equals \(L^{\phi_1}\times \dots \times L^{\phi_n}\) or is a meager subset of \(L^{\phi_1}\times \dots \times L^{\phi_n}\).