On geometry of Orlicz spaces

Abstract

In terms of the function Φ it is established when Orlicz space L_Φ does not contain l ⁿ_∞ uniformly and when it has some type or cotype.

We study some geometrical properties of Orlicz spaces. Namely using the results of [1] we establish when the given Orlicz space does not contain l ⁿ_∞ uniformly and when it has some type or cotype.

Let X be a real Banach space, X* the dual space and let (ɛ_k)_k ∈ N be the sequence of independent random variables with P[ɛ_k=1]=P[ɛ_k=−1]=1/2 (the Bernoulli, or the Rademacher sequence). A Banach space X is said to be a space of type p, 0<p⩽2, if there exists a constant c>0 such that for each finite collection x₁,...,x_n of elements of X there holds the inequality

$$E\parallel \sum\limits_{k = 1}^n {x_k \varepsilon _k } \parallel _X^p \leqslant c\sum\limits_{k = 1}^n {\parallel x_k \parallel _X^p }$$

Here and below E denotes the mathematical expectation,

It is clear that every Banach space has type p,0<p⩽1.

A Banach space X is said to be a space of cotype q 2<q<∞, if there exists a constant c′>0 such that for each finite collection x₁,...,x_n of elements of X there holds the inequality

$$E\parallel \sum\limits_{k = 1}^n {x_k \varepsilon _k } \parallel _X^q \geqslant c'\sum\limits_{k = 1}^n {\parallel x_k \parallel _X^q .}$$

If X is of type p, 1<p⩽2, then X* is of cotype p′=p/p−1. Đenote by l ⁿ_∞ the Rⁿ with the maximum-norm. We shall say that a Banach space X contains l ⁿ_∞ uniformly if for each ɛ>0 and any integer n there exists an injective linear operator J: l ⁿ_∞ → X such that ‖J‖‖J⁻¹‖<1+ɛ. X does not contain l ⁿ_∞ uniformly if and only if it has certain cotype q ([2]).

Let (T, Σ, ϑ) be a positive σ-finite measure space and Φ : R⁺ → R⁺ denotes a convex continious non-decreasing and vanishing at zero function. For measurable function x: T → R define

$$\rho _\Phi (x) = \int\limits_T {\Phi (\left| {x(t)} \right|)d \vartheta (t)}$$

and denote L_Φ=L_Φ (T, Σ, ϑ) the collection of all measurable functions x with ρ_Φ(λ x)<∞ for some λ>0. L_Φ is a vector space. Moreover, L_Φ is Banach space under the norm

$$\left\| x \right\|_\Phi = \inf \{ \lambda > 0:\rho _\Phi (x/\lambda ) \leqslant 1\}$$

and this space is said to be Orlicz space.

By Δ₂ we denote the family of functions Φ that satisfy the so-called Δ₂ condition (i.e. Φ (2u)⩽c Φ (u) for some c>0 and every u ∈ R⁺). If Φ ∈ Δ₂, then x ∈ L_Φ if and only if ρ_Φ(x)<∞.