Dichotomies for Lp spaces

Assume that (X, Sigma, mu) is a measure space and p(1), . . . , p(n), r > 0. We prove that {(f(1), . . . , f(n)) is an element of L-p1 x . . . x L-pn; f(1) . . . f(n) is an element of L-r} is either L-p1 x . . . x L-pn or a sigma-porous subset of L-p1 x . . . x L-pn. This dichotomy depends on properties of mu and the sign of the number 1/r - 1/p(1) - . . . - 1/p(n). (C) 2010 Elsevier Inc. All rights reserved.