Orlicz Space LΦ Research Articles

Orlicz Spaces Without Strongly Extreme Points and Without H-Points

AbstractW. Kurc [5] has proved that in the unit sphere of Orlicz space LΦ(μ) generated by an Orlicz function Φ satisfying the suitable Δ2-condition and equipped with the Luxemburg norm every extreme point is strongly extreme. In this paper it is proved in the case of a nonatomic measure μ that the unit sphere of the Orlicz space LΦ(μ) generated by an Orlicz function Φ which does not satisfy the suitable Δ2-condition and equipped with the Luxemburg norm has no strongly extreme point and no H-point.

Closedness of the Set of Extreme Points in Orlicz Spaces

AbstractThe aim of the paper is to give necessary and sufficient conditions under which the set of extreme points of the unit ball of an Orlicz space Lϕ(μ), equipped with the Luxemburg norm, is closed. Using that description a theorem is given saying when the notions “extremal” and “nice”, for a linear bounded compact operator from Lϕ(μ) into C(Z), coincide.

Finite representability ofl p(X) in Orlicz function spaces

We show that iflp(X),p ≠ 2, is finitely crudely representable in an Orlicz spaceLϕ (which does not containc0) then the Banach spaceX is isomorphic to a subspace ofLp. The same remains true forp = 2 whenLϕ is 2-concave or 2-convex, or ifX has local unconditional structure. We extend a theorem of Guerre and Levy to Orlicz function spaces.

Estimates for solutions of elliptic equations in a limit case

Let u be a week solution of homogeneous Dirichlet problem for a second order elliptic equation of divergence form, in a bounded open subset of ℝn. We prove, that if the right hand side of the equation is an element of H−1, n(Ω), then u belongs to the Orlicz space LΦ where Φ(t) = exp(|t|n/(n−1)) − 1. We employ the properties of the Schwartz symmetrization thus obtaining the “best” constant of the estimate.

A characterization of best Φ-approximants

Let T T be an operator from an Orlicz space L Φ {L_\Phi } into itself. It is shown in this paper that four algebraic conditions and one integration condition assure that T T is the best Φ \Phi -approximator, given a suitable σ \sigma -lattice.

Best ?- and N ?-approximants in Orlicz spaces of vector valued functions

Let (Ω, A, P) be a probability space and E be a Banach space. We study the approximation of an E-valued random variable X, which is an element of the Orlicz space LΦ(Ω, A, P; E), by a function Y∃LΦ, which is measurable with respect to a sub-σ-field of A and takes values in a closed convex subset of E. Two types of approximation are considered: ∫Φ(∥X − Y∥) dP=inf, and NΦ(X−Y)=inf with the Orlicz space norm NΦ. We give conditions for the existence of best approximants. If E is reflexive, we obtain martingale type convergence theorems for best approximants and discuss the continuity of the operator X → best approximant of X.

Orlicz and Sobolev spaces on unbounded domains

This paper deals with embedding theorems involving the Sobolev spaces H m,p (Ω) and H 0 m , p ( Ω ) on an unbounded domain Ω in R n ( n > 1 ) . It is shown, for example, that the Sobolev space H 0 1 , n ( Ω ) is continuously embedded in the Orlicz space L Φ* (Ω), where Φ ( t ) = | t | n exp ⁡ ( | t | n / ( n − 1 ) ) ; and that multiplication by suitable functions acts as a compact map of H 0 1 , n ( Ω ) to L Ψ ∗ ( Ω ) for any Orlicz function Ψ subordinate to Φ in a certain sense. These results extend the earlier work of Trudinger, who dealt with the case in which Ω is bounded. Examples are given of unbounded domains Ω for which the natural embedding of in H 1 , p ( Ω ) in L p ( Ω ) ( 1 < p < ∞ ) is a k -set contraction for some k < 1: the case k = 0 corresponds to a compact embedding. Applications are made to the Dirichlet problem in an unbounded domain for elliptic equations with violent non-linearities.

Extensions of the Hausdorff-Young theorem

In this paper the clasical Hausdorff-Young theorem, which states that iff ∈Lp, 1≦p≦2, on the line and\(\hat f\) is its Fourier transform, then\(\left\| {\hat f} \right\|_q \leqq \left\| f \right\|_p \) whereq−1+p−1=1, is extended in two ways for certain Orlicz spacesLΦ. IfLΦ is based on (G, μ), (1) an arbitrary compact topological group with Haar measure, and (2) a locally compact abelian topological group andμ is again the Haar measure, then the above inequality is extended to these cases. Various other related results and remarks are also included.