Orlicz Space LΦ Research Articles

Monotonicities of Quasi-Normed Orlicz Spaces

In this paper, we introduce a new Orlicz function, namely a b-Orlicz function, which is not necessarily convex. The Orlicz spaces LΦ generated by the b-Orlicz function Φ equipped with a Luxemburg quasi-norm contain both classical spaces Lp(p≥1) and Lp(0<p<1). The Orlicz spaces LΦ are quasi-Banach spaces. Some basic properties in quasi-normed Orlicz spaces are discussed, and the criteria that a quasi-normed Orlicz space is strictly monotonic and lower (upper) locally uniformly monotonic are given.

SOLVABILITY OF QUADRATIC HADAMARD-TYPE FRACTIONAL INTEGRAL EQUATIONS IN ORLICZ SPACES

We demonstrate some properties of Hadamard fractional operators such as boundedness, monotonicity, continuity, and acting conditions in Orlicz spaces Lφ. We apply these properties with a proper measure of noncompactness (MNC) to inspect the existence of monotonic solutions of some general, but abstract, form of quadratic Hadamard-type fractional integral equations in Lφ. We also discuss the uniqueness of the solution.

Examples of weakly compact sets in Orlicz spaces

This paper provides a number of examples of relatively weakly compact sets in Orlicz spaces. We show some results arising from these examples. Particularly, we provide a criterion which ensures that some Orlicz function is increasing more rapidly than another (in a sense of T. Ando). In addition, we point out that if a bounded subset K of the Orlicz space LΦ is not bounded by the modular Φ, then it is possible for a set K to remain unbounded under any modular Ψ increasing more rapidly than Φ.

Fourier transforms of convolution operators on orlicz spaces

AbstractIn this paper, we study convolution operators on an Orlicz spaceLΦ(G) commuting with left translations, where Φ is an N-function andGis a locally compact group. We also present some basic properties of the Fourier transform of a Φ-convolution operator in the context of locally compact abelian groups.

Diameter two properties and the Radon–Nikodým property in Orlicz spaces

Some necessary and sufficient conditions are found for Banach function lattices to have the Radon–Nikodým property. Consequently it is shown that an Orlicz function space Lφ over a non-atomic σ-finite measure space (Ω,Σ,μ), not necessarily separable, has the Radon–Nikodým property if and only if φ is an N-function at infinity and satisfies the appropriate Δ2 condition. For an Orlicz sequence space ℓφ, it has the Radon–Nikodým property if and only if φ satisfies the Δ20 condition. In the second part a relationship between uniformly ℓ12 points of the unit sphere of a Banach space and the diameter of the slices are studied. Using these results, a quick proof is given that an Orlicz space Lφ has the Daugavet property only if φ is linear, so when Lφ is isometric to L1. Another consequence is that Orlicz spaces equipped with the Orlicz norm generated by N-functions never have the local diameter two property, while it is well-known that when equipped with the Luxemburg norm, it may have that property. Finally, it is shown that the local diameter two property, the diameter two property, and the strong diameter two property are equivalent in Orlicz function and sequence spaces with the Luxemburg norm under appropriate conditions on φ.

Some Estimates for Maximal Bochner—Riesz Means on Musielak—Orlicz Hardy Spaces

Let φ: ℝn × [0, ∞) → [0, ∞) satisfy that φ(x, ·), for any given x ∈ Rn, is an Orlicz function and φ(·, t) is a Muckenhoupt A∞ weight uniformly in t ∈ (0, ∞). The Musielak–Orlicz Hardy space Hφ(ℝn) is defined to be the space of all tempered distributions whose grand maximal functions belong to the Musielak–Orlicz space Lφ(ℝn). In this paper, the authors establish the boundedness of maximal Bochner–Riesz means T*δ from Hφ(ℝn) to WLφ(ℝn) or Lφ(ℝn). These results are also new even when φ(x, t):= Φ(t) for all (x, t) ∈ ℝn × [0, ∞), where Φ is an Orlicz function.

Optimal extension of the Fourier transform and convolution operator on compact groups

Let G be a compact group (not necessarily abelian) and let Φ be a Young function satisfying the Δ2-condition. We determine the optimal domain and the associated extended operator for both Fourier transform and the convolution operator defined on the Orlicz spaces LΦ(G).

New molecular characterizations of anisotropic Musielak–Orlicz Hardy spaces and their applications

Let φ:Rn×[0,∞)→[0,∞) be an anisotropic growth function and A a general expansive matrix on Rn. Let HAφ(Rn) be the anisotropic Musielak–Orlicz Hardy space associated with A defined via the non-tangential grand maximal function. In this article, the authors establish a new molecular characterization of HAφ(Rn) via its known atomic characterizations, which positively answer a question mentioned by Li et al. (2016) [34]. As an application, the authors obtain a criterion on the boundedness of linear operators on HAφ(Rn), which further implies the boundedness of integral anisotropic Calderón–Zygmund operators on HAφ(Rn). In addition, the boundedness of these operators from HAφ(Rn) to the Musielak–Orlicz space Lφ(Rn) is also presented.

New martingale inequalities and applications to Fourier analysis

Let (Ω,F,P) be a probability space and φ:Ω×[0,∞)→[0,∞) be a Musielak–Orlicz function. In this article, the authors prove that the Doob maximal operator is bounded on the Musielak–Orlicz space Lφ(Ω). Using this and extrapolation method, the authors then establish a Fefferman–Stein vector-valued Doob maximal inequality on Lφ(Ω). As applications, the authors obtain the dual version of the Doob maximal inequality and the Stein inequality for Lφ(Ω), which are new even in weighted Orlicz spaces. The authors then establish the atomic characterizations of martingale Musielak–Orlicz Hardy spaces Hφs(Ω), Pφ(Ω), Qφ(Ω), HφS(Ω) and HφM(Ω). From these atomic characterizations, the authors further deduce some martingale inequalities between different martingale Musielak–Orlicz Hardy spaces, which essentially improve the corresponding results in Orlicz space case and are also new even in weighted Orlicz spaces. By establishing the Davis decomposition on HφS(Ω) and HφM(Ω), the authors obtain the Burkholder–Davis–Gundy inequality associated with Musielak–Orlicz functions. Finally, using the previous martingale inequalities, the authors prove that the maximal Fejér operator is bounded from Hφ[0,1) to Lφ[0,1), which further implies some convergence results of the Fejér means; these results are new even for the weighted Hardy spaces.

The Rao–Reiter Criterion for the Amenability of Homogeneous Spaces

We prove that a homogeneous space G/H, with G a locally compact group and H a closed subgroup of G, is amenable in the sense of Eymard–Greenleaf if and only if the quasiregular action πΦ of G on the unit sphere of the Orlicz space LΦ(G/H) for some N-function Φ ∈ Δ2 satisfies the Rao–Reiter condition (PΦ).

Applications of the Theory of Orlicz Spaces to Vector Measures

Let (Ω, Σ, λ) be a finite complete measure space, (E, ξ) be a sequentially complete locally convex Hausdorff space and E′ be its topological dual. Let caλ (Σ, E) stand for the space of all λ-absolutely continuous measures m: Σ → E. We show that a uniformly bounded subset M of caλ (Σ, E) is uniformly λ-absolutely continuous if and only if for every equicontinuous subset D of E′, there exists a submultiplicative Young function φ such that the set $$\left\{ {\frac{{d\left( {e'om} \right)}}{{d\lambda }}:m \in M,e' \in D} \right\}$$ is relatively weakly compact in the Orlicz space Lφ(λ). As a consequence, we present a generalized Vitali–Hahn–Saks theorem on the setwise limit of a sequence of λ-absolutely continuous vector measures in terms of Orlicz spaces.

English

We are concerned with the boundedness of generalized fractional integral operators Iϱ,τ from Orlicz spaces LΦ(X) near L1(X) to Orlicz spaces LΨ(X) over metric measure spaces equipped with lower Ahlfors Q-regular measures, where Φ is a function of the form Φ(r) = rl(r) and l is of log-type. We give a generalization of paper by Mizuta et al. (2010), in the Euclidean setting. We deal with both generalized Riesz potentials and generalized logarithmic potentials.

English

We prove that the associate space of a generalized Orlicz space Lϕ(·) is given by the conjugate modular ϕ* even without the assumption that simple functions belong to the space. Second, we show that every weakly doubling Φ-function is equivalent to a doubling Φ-function. As a consequence, we conclude that Lϕ(·) is uniformly convex if ϕ and ϕ* are weakly doubling.

A generalization of Cesàro sequence spaces in the Orlicz space

In this paper, we generalize the Cesàro sequence spaces in the classic Banach space Lp to the generalized Orlicz space Lφ. We construct the space by replacing the norm in Lp with modular ρφ in Lφ. This generalization has lead to the use of the Luxemburg norm to discuss some topological properties of the spaces. We prove results regarding to modular and norm convergence. We also describe some properties of the spaces and a closed subspaces of the space.

Kadec–Klee property for convergence in measure of noncommutative Orlicz spaces

This paper studies the Kadec–Klee property for convergence in measure of noncommutative Orlicz spaces Lφ(M˜,τ), where M˜ is the space of τ-measurable operators, and φ is an Orlicz function. We show that Lφ(M˜,τ) has the Kadec–Klee property in measure if and only if the φ satisfies the Δ2(∞) condition. As a corollary, the dual space and reflexivity of Lφ(M˜,τ) are given.

An inclusion property of Orlicz-Morrey spaces

The Orlicz-Morrey spaces Lϕ,Φ (where Φ is a Young function and ϕ is a parameter for the Morrey spaces) are generalizations of Orlicz spaces and Morrey spaces. Inclusion properties between Orlicz spaces LΦ and between Morrey spaces are well known. In this study we will investigate the inclusion relation between Orlicz-Morrey spaces Lϕ1, Φ1 and Lϕ2, Φ2 with respect to Young functions Φ1, Φ2 and parameters ϕ1, ϕ2. Also, we give sufficient and necessary conditions for the inclusion property of these spaces, which are obtained through norm estimates for the characteristic functions of balls in ℝn. In addition, we shall give a sufficient and necessary condition for generalized Hölder’s inequality.

Normed Orlicz function spaces which can be quasi-renormed with easily calculable quasinorms

We are interested in the widest possible class of Orlicz functions Φ such that the easily calculable quasinorm [f]Φ,p:=‖f‖E{IΦ(f‖f‖E)}1p if f≠0 and [f]Φ,p=0 if f=0, on the Orlicz space LΦ(Ω,Σ,μ) generated by Φ, is equivalent to the Luxemburg norm ‖⋅‖Φ. To do this, we use a suitable Δ2-condition, lower and upper Simonenko indices pSa(Φ) and qSa(Φ) for the generating function Φ, numbers p∈[1,pSa(Φ)] satisfying qSa(Φ)−p≤1, and an embedding of LΦ(Ω,Σ,μ) into a suitable Köthe function space E=E(Ω,Σ,μ). We take as E the Lebesgue spaces Lr(Ω,Σ,μ) with r∈[1,pSl(Φ)], when the measure μ is nonatomic and finite, and the weighted Lebesgue spaces Lωr(Ω,Σ,μ), with r∈[1,pSa(Φ)] and a suitable weight function ω, when the measure μ is nonatomic infinite but σ-finite. We also use condition ∇3 if pSa(Φ)=1 and condition ∇2 if pSa(Φ)>1, proving their necessity in most of the considered cases. Our results seem important for applications of Orlicz function spaces.

Property [formula omitted] and property [formula omitted] for Orlicz spaces LΦ

An Orlicz space LΦ(Ω) is a Banach function space defined by using a Young function Φ, which generalizes the Lp spaces. We show, for an Orlicz space LΦ([0,1]) which is not isomorphic to L∞([0,1]), if a locally compact second countable group has property (TLΦ([0,1])), which is a generalization of Kazhdan's property (T) for linear isometric representations on LΦ([0,1]), then it has Kazhdan's property (T). We also show, for a separable complex Orlicz space LΦ(Ω) with gauge norm, Ω=R,[0,1],N, if a locally compact second countable group has Kazhdan's property (T), then it has property (TLΦ(Ω)). We prove, for a finitely generated group Γ and a Banach space B whose modulus of convexity is sufficiently large, if Γ has Kazhdan's property (T), then it has property (FB), which is a fixed point property for affine isometric actions on B. Moreover, we see that, for a hyperbolic group Γ (which may have Kazhdan's property (T)) and an Orlicz sequence space ℓΦΨ with gauge norm such that the Young function Ψ sufficiently rapidly increases near 0, Γ doesn't have property (FℓΦΨ). These results are generalizations of the results for Lp-spaces.

Estimates for the norms of monotone operators on weighted Orlicz–Lorentz classes

A monotone operator P mapping the Orlicz–Lorentz class to an ideal space is considered. The Orlicz–Lorentz class is the cone of measurable functions on R+ =(0, ∞) whose decreasing rearrangements with respect to the Lebesgue measure on R+ belong to the weighted Orlicz space LΦ,ν. Reduction theorems are proved, which make it possible to reduce estimates of the norm of the operator P: ΛΦ,ν →Y to those of the norm of its restriction to the cone of nonnegative step functions in LΦ,ν. The application of these results to the identity operator from ΛΦ,ν to the weighted Lebesgue space Y = L1(R+; g) gives exact descriptions of associated norms for ΛΦ,ν.

Multivalued Extended Best Φ-Polynomial Approximation Operator

ABSTRACTIn this article, we consider the best multivalued polynomial approximation operator, defined in an Orlicz Space LΦ(B), and its extension to Lφ(B), where φ is the derivative of Φ which is not an N-function. Thus, the extension of the best polynomial approximation operator from L1(B) to L0(B) arises as a particular case of this work, taking Φ(x) = x.