Baltasar Rodriguez-Salinas

Onl p-complemented copies in Orlicz spaces II

For anyp > 1, the existence is shown of Orlicz spacesLF andlF with indicesp containingsingular lp-complemented copies, extending a result of N. Kalton ([6]). Also the following is proved:Let 1 <α ≦β < ∞and H be an arbitrary closed subset of the interval [α, β].There exist Orlicz sequence spaces lF (resp. Orlicz function spaces LF)with indices α and β containing only singular lp-complemented copies and such that the set of values p > 1for which lp is complementably embedded into lF (resp. LF)is exactly the set H (resp. H ∪ {2}). An explicitly defined class of minimal Orlicz spaces is given.

Lattice-embedding scales ofLp spaces into orlicz spaces

We study the setPX of scalarsp such thatLp is lattice-isomorphically embedded into a given rearrangement invariant (r.i.) function spaceX[0, 1]. Given 0<α≤β<∞, we construct a family of Orlicz function spacesX=LF[0, 1], with Boyd indicesα andβ, whose associated setsPX are the closed intervals [γ, β], for everyγ withα≤γ≤β. In particular forα>2, this proves the existence of separable 2-convex r.i. function spaces on [0,1] containing isomorphically scales ofLp-spaces for different values ofp. We also show that, in general, the associated setPX is not closed. Similar questions in the setting of Banach spaces with uncountable symmetric basis are also considered. Thus, we construct a family of Orlicz spaces ℓF(I), with symmetric basis and indices fixed in advance, containing ℓp(Γ-subspaces for differentp’s and uncountable Λ⊂I. In contrast with the behavior in the countable case (Lindenstrauss and Tzafriri [L-T1]), we show that the set of scalarsp for which ℓp(Γ) is isomorphic to a subspace of a given Orlicz space ℓF(I) is not in general closed.

Lattice-embeddingLp into Orlicz spaces

Given 0<α≤p≤β<∞, we construct Orlicz function spacesLF[0, 1] with Boyd indicesα andβ such thatLp is lattice isomorphic to a sublattice ofLF[0, 1]. Forp>2 this shows the existence of (non-trivial) separable r.i. spaces on [0, 1] containing an isomorphic copy ofLp. The discrete case of Orlicz spaces ℓF (I) containing an isomorphic copy of ℓp(Γ) for uncountable sets Γ ⊂I is also considered.

Strictly localizable measures

It is proved that, under the Continuum Hypotesis (CH), every complete, locally determined, Radon measures of type (H), on a topological space with countable basis, is strictly localizable. This result is useful in the theory of invariant measures on a topological group and, in particular, in the theory of Hausdorff measures.

Lattice-universal Orlicz spaces on probability spaces

Lattice-universal Orlicz function spacesLFα,β[0, 1] with prefixed Boyd indices are constructed. Namely, given 0<α<β<∞ arbitrary there exists Orlicz function spacesLFα,β[0, 1] with indices α and β such that every Orlicz function spaceLG[0, 1] with indices between α and β is lattice-isomorphic to a sublattice ofLFα,β[0, 1]. The existence of classes of universal Orlicz spaceslFα,β(I) with uncountable symmetric basis and prefixed indices α and β is also proved in the uncountable discrete case.