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

Abstract

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 ℓp if and only if either E contains a normalized ℓp-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 ℓp 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 ℓp 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. © 2009 Elsevier Inc.

Publication Title

Journal of Functional Analysis

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