Orlicz Function Spaces Research Articles

Lower Local Uniform Monotonicity in F-Normed Musielak–Orlicz Spaces

Lower strict monotonicity points and lower local uniform monotonicity points are considered in the case of Musielak–Orlicz function spaces LΦ endowed with the Mazur–Orlicz F-norm. The findings outlined in this study extend the scope of geometric characteristics observed in F-normed Orlicz spaces, as well as monotonicity properties within specific F-normed lattices. They are suitable for the Orlicz spaces of ordered continuous elements, specifically in relation to the Mazur–Orlicz F-norm. In addition, in this paper presents results that can be used to derive certain monotonicity properties in F-normed Musielak–Orlicz spaces.

On the convexity coefficient of Musielak–Orlicz function spaces equipped with the Orlicz norm

AbstractIn Hudzik and Landes, the convexity coefficient of Musielak–Orlicz function spaces over a non‐atomic measure space equipped with the Luxemburg norm is computed whenever the Musielak–Orlicz functions are strictly convex see [6]. In this paper, we extend this result to the case of Musielak–Orlicz spaces equipped with the Orlicz norm. Also, a characterization of uniformly convex Musielak–Orlicz function spaces as well as k‐uniformly convex Musielak–Orlicz spaces equipped with the Orlicz norm is given.

Points of monotonicity in F-normed Orlicz function spaces

Points of monotonicity in F-normed Orlicz function spaces

Two-weight inequalities for multilinear maximal functions in Orlicz spaces

In this note, we provide various two-weight norm estimates of the multi-linear fractional maximal function and weighted maximal function between different Orlicz spaces. More precisely, we obtain Sawyer-type characterizations and norm estimates for these operators.

On Various Notions of Representability of $l_r$-Spaces in Orlicz Function Spaces

On Various Notions of Representability of $l_r$-Spaces in Orlicz Function Spaces

Exposed points of Orlicz function spaces equipped with p-Amemiya norms

Exposed points of Orlicz function spaces equipped with p-Amemiya norms

Bernstein-Jackson-type inequalities with exact constants in Orlicz spaces

We establish the Bernstein and Jackson type inequalities with exact constants for estimations of best approximations by exponential type functions in Orlicz spaces $L_M(\mathbb{R}^n)$. For this purpose, we use a special scale of approximation spaces $\mathcal{B}_\tau^s(M)$ that are interpolation spaces between the subspace $\mathscr{E}_M$ of exponential type functions and the space $L_M(\mathbb{R}^n)$. These approximation spaces are defined using a functional $E\left(t,f\right)$ that plays a similar role as the module of smoothness. The constants in obtained inequalities are expressed using a normalization factor $N_{\vartheta,q}$ that is determined by the parameters $\tau$ and $s$ of the approximation space $\mathcal{B}_\tau^s(M)$.

On a Range of Exponents for Absence of Lavrentiev Phenomenon for Double Phase Functionals

For a class of functionals having the (p, q)-growth, we establish an improved range of exponents p, q for which the Lavrentiev phenomenon does not occur. The proof is based on a standard mollification argument and Young convolution inequality. Our contribution is two-fold. First, we observe that it is sufficient to regularise only bounded functions. Second, we exploit the L^{infty } bound on the function rather than the L^p estimate on the gradient. Our proof does not rely on the properties of minimizers to variational problems but it is rather a consequence of the underlying Musielak–Orlicz function spaces. Moreover, our method works for unbounded boundary data, the variable exponent functionals and vectorial problems. In addition, the result seems to be optimal for pleqq d.

Uniform monotonicity of Orlicz spaces equipped with the Mazur–Orlicz F‐norm and dominated best approximation in F‐normed Köthe spaces

AbstractWe collect several useful equivalent conditions for order continuity in F‐normed Köthe spaces. Next, we study uniform monotonicity of Orlicz function (sequence) spaces equipped with the Mazur–Orlicz F‐norm. We show necessary and sufficient conditions, separately. Moreover, we study the best dominated approximation problems in F‐normed Köthe spaces showing that order continuity and strict monotonicity are crucial tools in these problems. Finally, we apply the general results in Orlicz spaces. We focus mainly on these methods and techniques which are essentially different and more delicate in comparison to the case of normed Köthe spaces.

An improvement of Bernstein’s inequality for functions in Orlicz spaces with smooth Fourier image

Let Φ:[0,+∞)→[0,+∞] be an arbitrary Young function and K be a compact set in ℝn having (O)-property. Then there exists a constant CK,Φ<∞ independent of f such that ∥Dαf∥(Φ)≤CK,Φsupz∈K|zα|∥f∥ℒK,3 for all α∈ℤ+n and f∈ℒK,3, where ℒK,3={f∈𝒮′(ℝn):suppf^⊂K,D(3,3,…,3)f^∈C(ℝn)}, ∥f∥ℒK,3=∥D(3,3,…,3)f^∥∞, f^ is the Fourier transform of f and ∥⋅∥(Φ) is the Luxemburg norm. As an application, a new Paley–Wiener theorem is given.

Kadec-Klee Property in Orlicz Function Spaces Equipped with S-Norms

Using some new techniques, the necessary and sufficient conditions for Kadec-Klee property of Orlicz function spaces equipped with s-norms are presented. An original method that was used in the process of inquiry and the obtained results also systematically complete and broaden the characterization of Kadec-Klee property of Orlicz spaces.

Some spectral formulas for functions generated by differential and integral operators in Orlicz spaces

In this paper, we investigate the behavior of the sequence of $L^\Phi$-norm of functions, which are generated by differential and integral operators through their spectra (the support of the Fourier transform of a function $f$ is called its spectrum and denoted by sp$(f)$). With $Q$ being a polynomial, we introduce the notion of $Q$-primitives, which will return to the notion of primitives if ${Q}(x)= x$, and study the behavior of the sequence of norm of $Q$-primitives of functions in Orlicz space $L^\Phi(\mathbb R^n)$. We have the following main result: let $\Phi $ be an arbitrary Young function, ${Q}({\bf x} )$ be a polynomial and $(\mathcal{Q}^mf)_{m=0}^\infty \subset L^\Phi(\mathbb R^n)$ satisfies $\mathcal{Q}^0f=f, {Q}(D)\mathcal{Q}^{m+1}f=\mathcal{Q}^mf$ for $m\in\mathbb{Z}_+$. Assume that sp$(f)$ is compact and $sp(\mathcal{Q}^{m}f)= sp(f)$ for all $m\in \mathbb{Z}_+.$ Then $$ \lim\limits_{m\to \infty } \|\mathcal{Q}^m f\|_{\Phi}^{1/m}= \sup\limits_{{\bf x} \in sp(f)} \bigl|1/ {Q}({\bf x}) \bigl|. $$ The corresponding results for functions generated by differential operators and integral operators are also given.

The Modular Inequalities for Hardy-type Operators on Monotone Functions in Orlicz Space

This paper is devoted to the study of modular inequality for general Hardy–Copson type operator restricted on the cone of monotone functions from weighted Orlicz space with general weight.

Smoothness of Orlicz function spaces equipped with the p-Amemiya norm

In this paper, we will use the convex modular rho ^{*}(f) to investigate Vert fVert _{Psi ,q}^{*} on (L_{Phi })^{*} defined by the formula Vert fVert _{Psi ,q}^{*}=inf _{k>0}frac{1}{k}s_{q}(rho ^{*}(kf)), which is the norm formula in Orlicz dual spaces equipped with p-Amemiya norm. The attainable points of dual norm Vert fVert _{Psi ,q}^{*} are discussed, the interval for dual norm Vert fVert _{Psi ,q}^{*} attainability is described. By presenting the explicit form of supporting functional, we get sufficient and necessary conditions for smooth points. As a result, criteria for smoothness of L_{Phi ,p}~(1le ple infty ) is also obtained. The obtained results unify, complete and extended as well the results presented by a number of paper devoted to studying the smoothness of Orlicz spaces endowed with the Luxemburg norm and the Orlicz norm separately.

Strongly Proximinal Subspaces in Orlicz Function Space

In this paper, we prove that if Y is a separable proximinal subspace of , then Yis strongly proximinal in if and only if L(U,Y) is strongly L(U,Y) proximinal in. where L(U,Y) is an Orlicz function space with Luxemburg norm.

Strongly extreme points of Orlicz function spaces equipped with \u03a6-Amemiya norm

In this paper, the criterion that points of Orlicz function spaces equipped with Φ-Amemiya norm generated by an Orlicz function are strongly extreme is given. As a corollary, the sufficient and necessary conditions of midpoint local uniform rotundity of Orlicz function spaces equipped with Φ-Amemiya norm are obtained.

Entropic regularization of continuous optimal transport problems

We analyze continuous optimal transport problems in the so-called Kantorovich form, where we seek a transport plan between two marginals that are probability measures on compact subsets of Euclidean space. We consider the case of regularization with the negative entropy with respect to the Lebesgue measure, which has attracted attention because it can be solved by the very simple Sinkhorn algorithm. We first analyze the regularized problem in the context of classical Fenchel duality and derive a strong duality result for a predual problem in the space of continuous functions. However, this problem may not admit a minimizer, which prevents obtaining primal-dual optimality conditions. We then show that the primal problem is naturally analyzed in the Orlicz space of functions with finite entropy in the sense that the entropically regularized problem admits a minimizer if and only if the marginals have finite entropy. We then derive a dual problem in the corresponding dual space, for which existence can be shown by purely variational arguments and primal-dual optimality conditions can be derived. For marginals that do not have finite entropy, we finally show Gamma-convergence of the regularized problem with smoothed marginals to the original Kantorovich problem.

Hypercyclicity of Composition Operators on Orlicz Function Spaces

Abstract In this paper, we discuss the hypercyclic properties of composition operators on Orlicz function spaces. We give some different conditions under which a composition operator on Orlicz spaces is hyper-cyclic or not. Similarly, multiplication operators are considered. It is shown that there is no hypercyclic multiplication operator on Orlicz spaces.

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.

Non-$$l_n^{(1)}$$ point and uniformly non-$$l_n^{(1)}$$ point of Orlicz–Bochner function spaces

Let $$L_M$$ be an Orlicz function space endowed with the Orlicz norm or the Luxemburg norm, and let X be a Banach space. In this paper we characterize the non-$$l_n^{(1)}$$ point and the uniformly non-$$l_{n}^{(1)}$$ point of Orlicz–Bochner function space $$L_M(\mu ,X)$$. As the immediate consequences some criteria for non-square point and uniformly non-square point of $$L_M(\mu ,X)$$ are obtained.