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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.

In this paper, we formulate necessary conditions for decay rates of L operator norms of weighted oscillatory integral operators R and give sharp L estimates and nearly sharp L estimates.