Isomorphic copies in the lattice E and its symmetrization E(∗) with applications to Orlicz–Lorentz spaces

The paper is devoted to the isomorphic structure of symmetrizations of quasi-Banach ideal function or sequence lattices. The symmetrization E(∗) of a quasi-Banach ideal lattice E of measurable functions on I = (0, a), 0 < a ∞, or I = N, consists of all functions with decreasing rearrangement belonging to E. For an order continuous E we show that every subsymmetric basic sequence in E(∗) which converges to zero in measure is equivalent to another one in the cone of positive decreasing elements in E, and conversely. Among several consequences we show that, provided E is order continuous with Fatou property, E(∗) contains an order isomorphic copy of p if and only if either E contains a normalized p-basic sequence which converges to zero in measure, or E(∗) contains the function t−1/p. We apply these results to the family of two-weighted Orlicz–Lorentz spaces Λϕ,w,v(I ) defined on I = N or I = (0, a), 0 < a ∞. This family contains usual Orlicz–Lorentz spaces Λϕ,w(I ) when v ≡ 1 and Orlicz–Marcinkiewicz spaces Mϕ,w(I ) when v = 1/w. We show that for a large class of weights w,v, it is equivalent for the space Λϕ,w,v(0, 1), and for the non-weighted Orlicz space Lϕ(0, 1) to contain a given sequential Orlicz space hψ isomorphically as a sublattice in their respective order continuous parts. We provide a complete characterization of order isomorphic copies of p in these spaces over (0, 1) or N exclusively in terms of the indices of ϕ. If I = (0,∞) we show that the set of exponents p for which p lattice embeds in the order continuous part of Λϕ,w,v(I ) is the union of three intervals determined respectively by the indices of ϕ and by the condition that the function t−1/p belongs to the space.