Space LΦ Research Articles

Compactness of Bochner representable operators on Orlicz spaces

We study compactness properties of linear operators from an Orlicz space LΦ provided with a natural mixed topology \(\gamma_{L^{\Phi}}\) to a Banach space (X, || · ||X). We derive that every Bochner representable operator \(T : L^\Phi \rightarrow X\) is \((\gamma_{L^{\Phi}}, || \centerdot ||_X)\)-compact. In particular, it is shown that every Bochner representable operator \(T : L^\infty \rightarrow X\) is (τ(L∞, L1), || · ||X)-compact.

The Boundedness of Maximal Functions in Orlicz–Campanato Spaces of Homogeneous Type

Abstract Let (𝑋,𝑑,μ) be a normal space of homogeneous type, 𝑋+ be the upper half-space equipped with a Carleson measure β, and let Φ be an N-function and φ a suitable function satisfying the doubling property. We prove that the generalized Hardy–Littlewood maximal operator 𝑀 is bounded from the Orlicz–Campanato space 𝐿Φ,φ (𝑋,μ) to 𝐿Φ,φ (𝑋+,β).

On the latticity of projection and antiprojection sets in Orlicz-Musielak spaces

We discuss the latticial structure of projection sets and antiprojection sets in Orlicz-Musielak spaces LΦ, with the distance given by modular ρΦ and classical F-norms generated by ρΦ. In particular, we show that the case of Amemiya F-norm is essentially different from others. This extends results given for sets of best approximants by Landers and Rogge, Kilmer and Kozlowski, and Mazzone.

Schur property and lP isomorphic copies in Musielak–Orlicz sequence spaces

The author shows that if the dual of a Musielak–Orlicz sequence space lΦ is a stabilized asymptotic l∞, space with respect to the unit vector basis, then lΦ is saturated with complemented copies of l1 and has the Schur property. A sufficient condition is found for the isomorphic embedding of lp spaces into Musielak–Orlicz sequence spaces.

Approach regions for the square root of the Poisson kernel and boundary functions in certain Orlicz spaces

If the Poisson integral of the unit disc is replaced by its square root, it is known that normalized Poisson integrals of L p and weak L p boundary functions converge along approach regions wider than the ordinary nontangential cones, as proved by Ronning and the author, respectively. In this paper we characterize the approach regions for boundary functions in two general classes of Orlicz spaces. The first of these classes contains spaces L Φ having the property L ∞ ⊂ L Φ L p , 1 ⩽ p > ∞. The second contains spaces L Φ that resemble L p spaces.

Weighted inequalities for Cesàro maximal operators in Orlicz spaces

Let 0 < α ≤ 1 and let M+α be the Cesàro maximal operator of order α defined by In this work we characterize the pairs of measurable, positive and locally integrable functions (u, v) for which there exists a constant C > 0 such that the inequality holds for all λ > 0 and every f in the Orlicz space LΦ(v). We also characterize the measurable, positive and locally integrable functions w such that the integral inequality holds for every f ∈ LΦ(w). The discrete versions of this results allow us, by techniques of transference, to prove weighted inequalities for the Cesàro maximal ergodic operator associated with an invertible measurable transformation, T, which preserves the measure.Finally, we give sufficient conditions on w for the convergence of the sequence of Cesàro-α ergodic averages for all functions in the weighted Orlicz space LΦ(w).

Composition operators in Orlicz spaces

AbstractComposition operators Cτ between Orlicz spaces Lϕ (Ω, Σ, μ) generated by measurable and nonsingular transformations τ from Ω into itself are considered. We characterize boundedness and compactness of the composition operator between Orlicz spaces in terms of properties of the mapping τ, the function ϕ and the measure space (Ω, Σ, μ). These results generalize earlier results known for Lp-spaces.

On the Nonsquare Constants of Orlicz Spaces with Orlicz Norm

AbstractLet lΦ and LΦ(Ω) be the Orlicz sequence space and function space generated by N-function Φ(u) with Orlicz norm. We give equivalent expressions for the nonsquare constants CJ(lΦ), CJ(LΦ(Ω)) in sense of James and CS(lΦ), CS(LΦ(Ω)) in sense of Schäffer. We are devoted to get practical computational formulas giving estimates of these constants and to obtain their exact value in a class of spaces lΦ and LΦ(Ω).

Geometry of Modulus Spaces

Abstract Let ϕ be a modulus function, i.e., continuous strictly increasing function on [0, ∞), such that ϕ(0) = 0, ϕ(1) = 1, and ϕ(𝑥+𝑦) ≤ ϕ(𝑥)+ϕ(𝑦) for all 𝑥, 𝑦 in [0, ∞). It is the object of this paper to characterize, for any Banach space 𝑋, extreme points, exposed points, and smooth points of the unit ball of the metric linear space ℓ ϕ (𝑋), the space of all sequences (𝑥𝑛), 𝑥𝑛 ∈ 𝑋, 𝑛 = 1, 2, . . . , for which ∑ϕ(‖𝑥𝑛‖) < ∞. Further, extreme, exposed, and smooth points of the unit ball of the space of bounded linear operators on ℓ𝑝, 0 < 𝑝 < 1, are characterized.

Maximal Functions over Hypersurfaces with Flat Points

For n≥4, let S be a flat hypersurface in Rn, and let dμ=ψdσ, where ψ∈C∞0(Rn) and σ is the surface area measure on S. Then the maximal functions Mf associated to S and μ by Mf(x)=supt>0|∫Sf(x−tξ)dμ(ξ)|, f∈S(Rn), are bounded on certain Orlicz spaces LΦ(Rn).

Weighted Inequalities for Certain Maximal Functions in Orlicz Spaces

Let Mg be the maximal operator defined by $$M_g f\left( x \right) = \sup \frac{{\int_a^b {f\left( y \right)g\left( y \right){\text{d}}y} }}{{\int_a^b {g\left( y \right){\text{d}}y} }}$$ , where g is a positive locally integrable function on R and the supremum is taken over all intervals [a,b] such that 0≤a≤x≤b/η(b−a), here η is a non-increasing function such that η (0) = 1 and $$\mathop {{\text{lim}}}\limits_{t \to {\text{ + }}\infty } \eta \left( t \right) = 0$$ η (t) = 0. This maximal function was introduced by H. Aimar and L. L. Forzani [AF]. Let Φ be an N - function such that Φ and its complementary N - function satisfy Δ2. It gives an A′Φ(g) type characterization for the pairs of weights (u,v) such that the weak type inequality $$u\left( {\left\{ {x \in {\text{R}}\left| {M_g f\left( x \right) >\lambda } \right.} \right\}} \right) \leqslant \frac{C}{{\Phi \left( \lambda \right)}}\int_{\text{R}} {\Phi \left( {\left| f \right|v} \right)} $$ holds for every f in the Orlicz space LΦ(v). And, there are no (nontrivial) weights w for which (w,w) satisfies the condition A′Φ(g).

Another extension of Orlicz‐Sobolev spaces to metric spaces

We propose another extension of Orlicz‐Sobolev spaces to metric spaces based on the concepts of the Φ‐modulus and Φ‐capacity. The resulting space is a Banach space. The relationship between and (the first extension defined in Aïssaoui (2002)) is studied. We also explore and compare different definitions of capacities and give a criterion under which is strictly smaller than the Orlicz space LΦ.

Embeddings of Rearrangement Invariant Spaces that are not Strictly Singular

We give partial answers to the following conjecture: the natural embedding of a rearrangement invariant space E into L 1([0,1]) is strictly singular if and only if G does not embed into E continuously, where G is the closure of the simple functions in the Orlicz space L Φ with Φ(x) = exp(x2)-1.

Connections on Non-Parametric Statistical Manifolds by Orlicz Space Geometry

The non-parametric version of Information Geometry has been developed in recent years. The first basic result was the construction of the manifold structure on ℳμ, the maximal statistical models associated to an arbitrary measure μ (see Ref. 48). Using this construction we first show in this paper that the pretangent and the tangent bundles on ℳμ are the natural domains for the mixture connection and for its dual, the exponential connection. Second we show how to define a generalized Amari embedding AΦ:ℳμ→SΦ from the Exponential Statistical Manifold (ESM) ℳμ to the unit sphere SΦ of an arbitrary Orlicz space LΦ. Finally we show that, in the non-parametric case, the α-connections ∇α(α∈(-1,1)) must be defined on a suitable α-bundle ℱα over ℳμ and that the bundle-connection pair (ℱα, ∇α) is simply (isomorphic to) the pull-back of the Amari embedding Aα: ℳμ→S2/1-α were the unit sphere S2/1-αcL2/1-α is equipped with the natural connection.

Global limiting embeddings of logarithmic Bessel potential spaces

The paper is a continuation of [EGO II-IV], where it was shown that Bessel potential spaces HσY(Rn) , modelled upon appropriate generalized Lorentz-Zygmund spaces Y(Rn) may be embedded into Orlicz spaces LΦ(Ω) , where Φ(t) = exp(exp(. . . exp tν) . . .)) for large t , ν > 0 , and Ω is a subset of Rn with finite volume. Using weighted Hardy inequalities, we modify the Young function Φ near the origin so that the above embedding holds with Ω replaced by Rn . The resulting Young function dominates globally the Young function Ψ(t) = tq , t > 0 , for q sufficiently large and consequently, HσY(Rn) ↪→ Lq(Rn) . We also obtain an estimate of the norms of the last embeddings which is sharp in their dependence upon q provided that q is large enough. Mathematics subject classification (1991): 46E35, 46E30, 26D15.

Geometric properties in Köthe-Bochner spaces

KOTHE-BOCHNER spaces are important generalizations of the Lebesgue-Bochner spaces L p( μ , Χ ) and Bochner-Orlicz spaces LΦ ( X ). More and more people take interest in studying theory of Kothe-Bochner spaces in recent years. It is one of the main problems of whether or not a geometric property can be lifted from X to E ( X )....

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.

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.

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.