Dichotomies for Lorentz spaces

Abstract

Assume that L-p,L-q; L-p1,L-q1 , ..., L-pn,L-qn are Lorentz spaces. This article studies the question: what is the size of the set E = {(f(1),..., f(n)) is an element of L-p1,L-q1 x ... x L-pn,L-qn : f(1) ... f(n) is an element of L-p,L-q}. We prove the following dichotomy: either E = L-p1,L-q1 x ... x L-pn,L-qn or E is sigma-porous in L-p1,L-q1 x ... x L-pn,L-qn , provided 1/p not equal 1/p(1) + ... + 1/p(n). In general case we obtain that either E = L-p1,L-q1 x ... x L-pn,L-qn or E is meager. This is a generalization of the results for classical L-p spaces.