Isomorphic copies in the lattice E and its symmetrization [formula omitted] 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 ℓ p if and only if either E contains a normalized ℓ 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 Λ φ , w , v ( I ) defined on I = N or I = ( 0 , a ) , 0 < a ⩽ ∞ . This family contains usual Orlicz–Lorentz spaces Λ φ , w ( I ) when v ≡ 1 and Orlicz–Marcinkiewicz spaces M φ , w ( I ) when v = 1 / w . We show that for a large class of weights w , v , it is equivalent for the space Λ φ , w , v ( 0 , 1 ) , and for the non-weighted Orlicz space L φ ( 0 , 1 ) to contain a given sequential Orlicz space h ψ isomorphically as a sublattice in their respective order continuous parts. We provide a complete characterization of order isomorphic copies of ℓ 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 ℓ p lattice embeds in the order continuous part of Λ φ , 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.