Dichotomies for Lorentz spaces

Assume that L ^p,q , $L^{p_1 ,q_1 } ,...,L^{p_n ,q_n } $ are Lorentz spaces. This article studies the question: what is the size of the set $E = \{ (f_1 ,...,f_n ) \in L^{p_{1,} q_1 } \times \cdots \times L^{p_n ,q_n } :f_1 \cdots f_n \in L^{p,q} \} $ . We prove the following dichotomy: either $E = L^{p_1 ,q_1 } \times \cdots \times L^{p_n ,q_n } $ or E is σ-porous in $L^{p_1 ,q_1 } \times \cdots \times L^{p_n ,q_n } $ , provided 1/p ≠ 1/p ₁ + … + 1/p _n . In general case we obtain that either $E = L^{p_1 ,q_1 } \times \cdots \times L^{p_n ,q_n } $ or E is meager. This is a generalization of the results for classical L ^p spaces.