Orlicz Classes Research Articles

Necessary and Sufficient Criteria for a Four-Weight Weak-Type Maximal Inequality in the Orlicz Class

Let Φi(i=1,2) be two N-functions, f be a μ-measurable function, and ωi(i=1,2,3,4) be four weight functions. This study presents necessary and sufficient conditions for weight functions (ω1,ω2,ω3,ω4) such that the inequality ∫{x:Mf(x)>λ}Φ1(λω1(x))ω2(x)dμ(x)≤c1∫XΦ2(c1|f(x)|ω3(x))ω4(x)dμ(x) holds, which extends several established results.

A Unified Version of Weighted Weak-Type Inequalities for the One-Sided Hardy–Littlewood Maximal Function in Orlicz Classes

A Unified Version of Weighted Weak-Type Inequalities for the One-Sided Hardy–Littlewood Maximal Function in Orlicz Classes

Criteria of a Two-Weight, Weak-Type Inequality in Orlicz Classes for Maximal Functions Defined on Homogeneous Spaces

In this study, some new necessary and sufficient conditions for a two-weight, weak-type maximal inequality of the form φ1(λ)∫{x∈X:Mf(x)>λ}ϱ(x)dμ(x)≤c∫Xφ2c|f(x)|σ(x)dμ(x) are obtained in Orlicz classes, where Mf is a Hardy–Littlewood maximal function defined on homogeneous spaces and ϱ is a weight function.

On Orlicz classes defined in terms of associated weight functions

N-functions and their growth and regularity properties are crucial in order to introduce and study Orlicz classes and Orlicz spaces. We consider N-functions which are given in terms of so-called associated weight functions. These functions are frequently appearing in the theory of ultradifferentiable function classes and in this setting additional information is available since associated weight functions are defined in terms of a given weight sequence. We express and characterize several known properties for N-functions purely in terms of weight sequences which allows to construct (counter-) examples. Moreover, we study how for abstractly given N-functions this framework becomes meaningful and finally we establish a connection between the complementary N-function and the recently introduced notion of the so-called dual sequence.

Sobolev embeddings in Musielak-Orlicz spaces

An embedding theorem for Sobolev spaces built upon general Musielak-Orlicz norms is offered. These norms are defined in terms of generalized Young functions which also depend on the x variable. Under minimal conditions on the latter dependence, a Sobolev conjugate is associated with any function of this type. Such a conjugate is sharp, in the sense that, for each fixed x, it agrees with the sharp Sobolev conjugate in classical Orlicz spaces. Both Sobolev inequalities in the whole Rn and Sobolev-Poincaré inequalities in domains are established. Compact Sobolev embeddings are also presented. In particular, optimal embeddings for standard Orlicz-Sobolev spaces, variable exponent Sobolev spaces, and double-phase Sobolev spaces are recovered and complemented in borderline cases. A key tool, of independent interest, in our approach is a new weak type inequality for Riesz potentials in Musielak-Orlicz spaces involving a sharp fractional-order Sobolev conjugate.

Uniqueness of solutions to some classes of anisotropic and isotropic curvature problems

In this paper, we apply various methods to establish the uniqueness of solutions to some classes of anisotropic and isotropic curvature problems. Firstly, by employing integral formulas derived by S. S. Chern [20], we obtain the uniqueness of smooth admissible solutions to a class of Orlicz-(Christoffel)-Minkowski problems. Secondly, inspired by Simon's uniqueness result [61], we then prove that the only smooth strictly convex solution to the following isotropic curvature problem(0.1)(Pk(W)Pl(W))1k−l=ψ(u,r)onSn must be an origin-centred sphere, where W=(∇2u+ug0), ∂1ψ≥0,∂2ψ≥0 and at least one of these inequalities is strict. As an application, we establish the uniqueness of solutions to the isotropic Gaussian-Minkowski problem. Finally, we derive the uniqueness result for the following isotropic Lp dual Minkowski problem(0.2)u1−prq−n−1det⁡(W)=1onSn, where −n−1<p≤−1 and n+1≤q≤n+12+14−(1+p)(n+1+p)n(n+2). This result utilizes the method developed by Ivaki and Milman [42] and generalizes a result due to Brendle, Choi and Daskalopoulos [10].

Existence of renormalized solutions to fully anisotropic and inhomogeneous elliptic problems

We will present the proof of existence and uniqueness of renormalized solutions to a broad family of strongly non-linear elliptic equations with lower order terms and data of low integrability. The leading part of the operator satisfies general growth conditions settling the problem in the framework of fully anisotropic and inhomogeneous Musielak–Orlicz spaces. The setting considered in this paper generalized known results in the variable exponents, anisotropic polynomial, double phase and classical Orlicz setting.

Duality and stable compactness in Orlicz-type modules

Orlicz-type modules are module analogues of classical Orlicz spaces. We study duality and stable compactness in Orlicz-type modules. We characterize the conditional Köthe dual of an Orlicz-type module as the space of all σ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sigma $$\\end{document}-order continuous module homomorphisms. We find an order continuity criterion for stable compactness in Orlicz-type modules. As an application, we obtain a robust representation result for conditional risk measures on Orlicz spaces.

Orlicz risks for assessing stochastic streamflow environments: a static optimization approach

This study applies novel risk measures, called Orlicz risks, to the risk and uncertainty evaluation of the streamflow discharge as a primary driver of hydrological and hydraulic processes of interest in civil and environmental engineering. We consider the mixed moving average process governing the discharge whose statistics are explicitly represented as some product of a time-scale characterizing the flow attenuation and a jump moment governing the size and frequency of jumps. The classical Orlicz risks are extended so that not only the upper tail risk but also the lower one of the jump size and attenuation of the discharge can be evaluated within a single mathematical framework. Further, the risk and uncertainty can be individually quantified in a tractable manner by the proposed Orlicz risks. Computing the Orlicz risks reduces to solving a pair of novel static optimization problems that are solvable semi-analytically. The risk and uncertainty involved in the streamflow dynamics can be consistently evaluated by specifying few user-dependent parameters. The associated Radon–Nikodym derivatives as the worst-case model uncertainties are obtained as byproducts. Sufficient conditions for the well-posedness of the Orlicz risks are discussed and numerical algorithms for computing them are presented. We finally apply the proposed framework to a statistical analysis of the streamflow discharge time series data collected at mountainous river environments.

A Generalized Version of the Lions-Type Lemma

Abstract In this short paper, I recall the history of dealing with the lack of compactness of a sequence in the case of an unbounded domain and prove the vanishing Lions-type result for a sequence of Lebesgue-measurable functions. This lemma generalizes some results for a class of Orlicz–Sobolev spaces. What matters here is the behavior of the integral, not the space.

Canonical Orlicz class 𝒦𝒮−𝜃[ℝJn

The objective of this paper is to construct canonical Orlicz class and study their fundamental properties. Also, we prove that this space contains Henstock–Kurzweil integrable functions.

A Lions Type Result for a Large Class of Orlicz–Sobolev Space and Applications

In this paper we prove a Lions type result for a large class of Orlicz-Sobolev space that can be nonreflexive and use this result to show the existence of solution for a large class of quasilinear problem on a nonreflexive Orlicz-Sobolev space.

Monotonicity properties and solvability of dominated best approximation problem in Orlicz spaces equipped with s-norms

In the paper, Wisła (J Math Anal Appl 483(2):123659, 2020, 10.1016/j.jmaa.2019.123659), it was proved that the classical Orlicz norm, Luxemburg norm and (introduced in 2009) p-Amemiya norm are, in fact, special cases of the s-norms defined by the formula left| xright| _{Phi ,s}=inf _{k>0}frac{1}{k}sleft( int _T Phi (kx)dmu right) , where s and Phi are an outer and Orlicz function respectively and x is a measurable real-valued function over a sigma -finite measure space (T,Sigma ,mu ). In this paper the strict monotonicity, lower and upper uniform monotonicity and uniform monotonicity of Orlicz spaces equipped with the s-norm are studied. Criteria for these properties are given. In particular, it is proved that all of these monotonicity properties (except strict monotonicity) are equivalent, provided the outer function s is strictly increasing or the measure mu is atomless. Finally, some applications of the obtained results to the best dominated approximation problems are presented.

The Boussinesq system revisited

In this work, we revisit the study by Schonbek (1981 J. Differ. Equ. 42 325–52) concerning the problem of existence of global entropic weak solutions for a specific Boussinesq system, as well as the study of the regularity of these solutions by Amick (1984 J. Differ. Equ. 54 231–47). We propose to study a regularized variant of this Boussinesq system, obtained by adding a ‘fractal’ operator (i.e. a differential operator defined by a Fourier multiplier of type ] 0, 2]) to the equation of the height of the water column. We first show that the regularized system is globally unconditionally well-posed in Sobolev spaces of type uniformly in the regularizing parameters ]0, 2]. As a consequence we obtain the global unconditional well-posedness of this Boussinesq system at this level of regularity as well as the convergence in the strong topology of the solution of the regularized system towards the solution of the Boussinesq system as the parameter ϵ goes to 0. Finally, we prove the existence of low regularity entropic solutions of the Boussinesq equations emanating from u 0 ∈ H 1 and ζ 0 in an Orlicz class as weak limits of regular solutions.

Compact Operators Under Orlicz Function

In this paper we investigate the compact operators under Orlicz function, named noncommutative Orlicz sequence space (denoted by Sϕ(ℌ)), where ℌ is a complex, separable Hilbert space. We will show that the space generalizes the Schatten classes Sp(ℌ) and the classical Orlicz sequence space respectively. After getting some relations of trace and norm, we will give some operator inequalities, such as Holder inequality and some other classical operator inequalities. Also we will give the dual space and reflexivity of Sϕ(ℌ) which generalizes the results of Sϕ(ℌ). Finally, as an application, we will show that the Toeplitz operator $${T_{1 — {{\left| z \right|}^2}}}$$ on the Bergman space $$L_\alpha ^2\left(\mathbb{R} \right)$$ belongs to some Sϕ(ℌ), and the norm satisfies $$1 = \sum\limits_{n \ge 0} {\varphi \left({{1 \over {\left({n + 2} \right){{\left\| {{T_{1 — {{\left| z \right|}^2}}}} \right\|}_\varphi}}}} \right)} $$ . Especially, if ϕ(T) = ∣T∣p, p > 1, the norm is $${\left\| {{T_{1 — {{\left| z \right|}^2}}}} \right\|_p} = {\left[{\sum\limits_{n \ge 0} {{1 \over {{{\left({n + 2} \right)}^p}}}}} \right]^{{1 \over p}}} = {\left({\zeta \left(p \right) — 1} \right)^{{1 \over p}}}$$ , where ζ(p) is the Riemann function.

Henstock–Orlicz space and its dense space

The motivation of this paper is to introduce Henstock–Orlicz space with non-absolute integrable functions. We prove that [Formula: see text] is dense in the Henstock–Orlicz space, which is not dense in the classical Orlicz space.

Criteria of a multi-weight weak type inequality in Orlicz classes for maximal functions defined on homogeneous type spaces

We obtain some new necessary and sufficient conditions for a multi-weight weak type maximal inequality of the form $$\begin{aligned} \int _{\{ {x: \mathcal {M} f(x) > \lambda } \}} {\varphi (\lambda {\omega _1}(x))} {\omega _2}(x) \,d\mu \le {c} \int _X \varphi ({c}f(x){\omega _3}(x)){\omega _4}(x) \,d\mu \end{aligned}$$ in Orlicz classes, where $$\mathcal {M} f$$ is a Hardy–Littlewood maximal function defined on homogeneous type spaces. Our main result extends some known results.

Orlicz spaces equipped with s-norms

On Orlicz spaces of measurable functions, the classical Orlicz and Luxemburg norm can be defined by use of the Amemiya formula: ‖x‖Φo=infk>0⁡1k(1+IΦ(kx)) and ‖x‖=infk>0⁡1kmax⁡{1,IΦ(kx)} respectively, where Φ is an Orlicz function and IΦ(x)=∫TΦ(x(t))dμ(t). Based on this observation, in the last few years a number of papers have been published that dealt with the geometrical properties of Orlicz spaces equipped with the so-called p-Amemiya norms defined by ‖x‖Φ,p=infk>0⁡1k(1+IΦp(kx))1/p, where 1≤p≤∞. The aim of this paper is to present a general and universal method of introducing norms in Orlicz spaces that will cover all the cases mentioned above. Namely, using the concept of outer function, s-norms ‖x‖Φ,s=infk>0⁡1ks(IΦ(kx)) are introduced. It is proved that to each outer function s we can associate an outer function s⁎ that is conjugate to s in the Hölder sense, i.e. u+v≤s(u)s⁎(v) for all u,v≥0. Moreover, it is proved, under some minor assumptions, that the Köthe dual of the subspace EΦ of the Orlicz space equipped with the s-norm ‖⋅‖Φ,s is an Orlicz space LΨ equipped with the s-norm ‖⋅‖Ψ,s⁎, where the outer function s⁎ is conjugate to s in the Hölder sense and the Orlicz function Ψ is complementary to Φ in the Young sense.

On Functions of Bounded (φ, k)-Variation

Abstract Given a φ-function φ and k ∈ ℕ, we introduce and study the concept of (φ, k)-variation in the sense of Riesz of a real function on a compact interval. We show that a function u :[a, b] → ℝ has a bounded (φ, k)-variation if and only if u ( k− 1) is absolutely continuous on [a, b]and u ( k ) belongs to the Orlicz class L φ [a, b]. We also show that the space generated by this class of functions is a Banach space. Our approach simultaneously generalizes the concepts of the Riesz φ-variation, the de la Vallée Poussin second-variation and the Popoviciu kth variation.

M‐constants in Orlicz–Lorentz function spaces

AbstractIn this paper some lower and upper estimates of M‐constants for Orlicz–Lorentz function spaces for both, the Luxemburg and the Amemiya norms, are given. Since degenerated Orlicz functions φ and degenerated weighted sequences ω are also admitted, this investigations concern the most possible wide class of Orlicz–Lorentz function spaces. M‐constants were defined in 1969 by E. A. Lifshits, and used by many authors in the study of lattice structures on Banach spaces, as well as in the fixed point theory.