Fatou Property Research Articles

Köthe Amalgams: The Ideal Type of Infinite Direct Sums

We study a special type of infinite direct sums E(X)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$E({\\mathcal {X}})$$\\end{document} which can be seen as the amalgam spaces characterized by a local component given by a countable family X=Xαα∈I\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {X}}=\\left( X_{\\alpha }\\right) _{\\alpha \\in I}$$\\end{document} of quasi-normed function spaces and by a global component E, which is a quasi-normed sequence space. We characterize some fundamental properties of E(X)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$E({\\mathcal {X}})$$\\end{document} such as completeness, Köthe-duality, order continuity and the Fatou property. We also provide its Banach function space characterization. Then, we apply our general results to the appropriate amalgamations of Lorentz (Orlicz) function spaces and Lebesgue sequence spaces. Moreover, for the Lorentz-type amalgams, we derive interpolation results and prove the boundedness of a class of sublinear integral operators whose kernels satisfy a size condition.

Core decreasing functions

Given a measure space and a totally ordered collection of measurable sets, called an ordered core, the notion of a core decreasing function is introduced and used to define the down space of a Banach function space. This is done using a variant of the Köthe dual restricted to core decreasing functions. To study down spaces, the least core decreasing majorant construction and the level function construction, already known for functions on the real line, are extended to this general setting. These are used to give concrete descriptions of the duals of the down spaces and, in the case of universally rearrangement invariant (u.r.i.) spaces, of the down spaces themselves.The down spaces of L1 and L∞ are shown to form an exact Calderón couple with divisibility constant 1; a complete description of the exact interpolation spaces for the couple is given in terms of level functions; and the down spaces of u.r.i. spaces are shown to be precisely those interpolation spaces that have the Fatou property. The dual couple is also an exact Calderón couple with divisibility constant 1; a complete description of the exact interpolation spaces for the couple is given in terms of least core decreasing majorants; and the duals of down spaces of u.r.i. spaces are shown to be precisely those interpolation spaces that have the Fatou property.

Uniqueness of Convex-Ranged Probabilities and Applications to Risk Measures and Games

We revisit Marinacci’s uniqueness theorem for convex-ranged probabilities and its applications. Our approach does away with both the countable additivity and the positivity of the charges involved. In the process, we uncover several new equivalent conditions, which lead to a novel set of applications. These include extensions of the classic Fréchet–Hoeffding bounds as well as of the automatic Fatou property of law-invariant functionals. We also generalize existing results of the “collapse to the mean”-type concerning capacities and α-MEU preferences.

Algebraic and context-free subsets of subgroups

We study the relation between the structure of algebraic and context-free subsets of a group G and that of a finite index subgroup H. Using these results, we prove that a kind of Fatou property, previously studied by Berstel and Sakarovitch in the context of rational subsets and by Herbst in the context of algebraic subsets, holds for context-free subsets if and only if the group is virtually free. We also exhibit a counterexample to a question of Herbst concerning this property for algebraic subsets.

Stability of Quartic Functional Equation in Modular Spaces via Hyers and Fixed-Point Methods

In this work, we introduce a new type of generalised quartic functional equation and obtain the general solution. We then investigate the stability results by using the Hyers method in modular space for quartic functional equations without using the Fatou property, without using the Δb-condition and without using both the Δb-condition and the Fatou property. Moreover, we investigate the stability results for this functional equation with the help of a fixed-point technique involving the idea of the Fatou property in modular spaces. Furthermore, a suitable counter example is also demonstrated to prove the non-stability of a singular case.

Solution and Stability of Quartic Functional Equations in Modular Spaces by Using Fatou Property

We propose a novel generalized quartic functional equation and investigate its Hyers–Ulam stability in modular spaces using a fixed point technique and the Fatou property in this paper.

Automatic Fatou property of law-invariant risk measures

In the paper we investigate automatic Fatou property of law-invariant risk measures on a rearrangement-invariant function space X other than L∞. The main result is the following characterization: Every real-valued, law-invariant, coherent risk measure on X has the Fatou property at every random variable X∈X whose negative tails have vanishing norm (i.e., limn⁡‖X1{X≤−n}‖=0) if and only if X satisfies the Almost Order Continuous Equidistributional Average (AOCEA) property, namely, d(CL(X),Xa)=0 for any X∈X+, where CL(X) is the convex hull of all random variables having the same distribution as X and Xa={X∈X:limn⁡‖X1{|X|≥n}‖=0}. As a consequence, we show that under the AOCEA property, every real-valued, law-invariant, coherent risk measure on X admits a tractable dual representation at every X∈X whose negative tails have vanishing norm. Furthermore, we show that the AOCEA property is satisfied by most classical model spaces, including Orlicz spaces, and therefore the foregoing results have wide applications.

On the Lattice Structure of the Space of all Bochner Integrable Banach Lattice-Valued Functions

Suppose $(X,\Sigma,\mu)$ is a finite measure space, $E$ is a Banach lattice, and $B(X,E,\mu)$ is the space of all Bochner integrable $E$-valued functions. In this note, we show that $B(X,E,\mu)$ is a $KB$-space or has the sequential Fatou property if and only if so is $E$. Among this, some results about Bochner integral convergence in $B(X,E,\mu)$, using order structure of $E$, have been proved, as well.

Common fixed point of generalized weakly contractive mappings on orthogonal modular spaces with applications

In this paper, we provide certain conditions under which guarantee the existence of a common fixed point for weakly contractive mappings defined on orthogonal modular spaces. Also, Banach fixed point theorem on an orthogonal modular space without completeness is obtained. To prove much stronger and more applicable results, some strong assumptions such as the convexity and the Fatou property of a modular are relaxed.

The Fatou Property for General Approximate Identities on Metric Measure Spaces

The Fatou Property for General Approximate Identities on Metric Measure Spaces

Some Fixed Point Results of Kannan Maps on the Nakano Sequence Space

In the recent past, some researchers studied some fixed point results on the modular variable exponent sequence spaceℓr.ψ, whereψv=∑a=0∞1/ravaraandra≥1. They depended on their proof that the modularψhas the Fatou property. But we have explained that this result is incorrect. Hence, in this paper, the concept of the premodular, which generalizes the modular, on the Nakano sequence space such as its variable exponent in1,∞and the operator ideal constructed by this sequence space ands-numbers is introduced. We construct the existence of a fixed point of Kannan contraction mapping and Kannan nonexpansive mapping acting on this space. It is interesting that several numerical experiments are presented to illustrate our results. Additionally, some successful applications to the existence of solutions of summable equations are introduced. The novelty lies in the fact that our main results have improved some well-known theorems before, which concerned the variable exponent in the aforementioned space.

The Fatou Property for General Approximate Identities on Metric Measure Spaces

В работе рассматриваются абстрактные аппроксимативные единицы на метрических пространствах с мерой. Находятся точные условия на геометрию областей, для которых имеет место сходимость аппроксимативных единиц почти всюду для функций из пространств $L^p$, $p\geqslant 1$. Результаты иллюстрируются на примерах ядер Пуассона и их степеней в единичном шаре в $\mathbb{R}^n$ или $\mathbb{C}^n$, а также свертки с растяжениями на $\mathbb{R}^n$. Во всех этих примерах найденные условия являются точными. Библиография: 20 названий.

Kannan Prequasi Contraction Maps on Nakano Sequence Spaces

In this article, we explore the concept of the prequasi norm on Nakano special space of sequences (sss) such that its variable exponent in 0 , 1 . We evaluate the sufficient setting on it with the definite prequasi norm to configuration prequasi Banach and closed (sss). The Fatou property of different prequasi norms on this (sss) has been investigated. Moreover, the existence of a fixed point of Kannan prequasi norm contraction maps on the prequasi Banach (sss) and the prequasi Banach operator ideal constructed by this (sss) and s − numbers have been examined.

Modular Stabilities of a Reciprocal Second Power Functional Equation

In the present work, we propose a dierent reciprocal second power Functional Equation (FE) which involves the arguments of functions in rational form and determine its stabilities in the setting of modular spaces with and without using Fatou property. We also prove the stabilities in beta-homogenous spaces. As an application, we associate this equation with the electrostatic forces of attraction between unit charges in various cases using Coloumb's law.

On the extension property of dilatation monotone risk measures

Abstract Let 𝒳 be a subset of L 1 L^{1} that contains the space of simple random variables ℒ and ρ : X → ( - ∞ , ∞ ] \rho\colon\mathcal{X}\to(-\infty,\infty] a dilatation monotone functional with the Fatou property. In this note, we show that 𝜌 extends uniquely to a σ ⁢ ( L 1 , L ) \sigma(L^{1},\mathcal{L}) lower semicontinuous and dilatation monotone functional ρ ¯ : L 1 → ( - ∞ , ∞ ] \overline{\rho}\colon L^{1}\to(-\infty,\infty] . Moreover, ρ ¯ \overline{\rho} preserves monotonicity, (quasi)convexity and cash-additivity of 𝜌. We also study conditions under which ρ ¯ \overline{\rho} preserves finiteness on a larger domain. Our findings complement extension and continuity results for (quasi)convex law-invariant functionals. As an application of our results, we show that transformed norm risk measures on Orlicz hearts admit a natural extension to L 1 L^{1} that retains robust representations.

Stability properties of Haezendonck–Goovaerts premium principles

We investigate a variety of stability properties of Haezendonck–Goovaerts premium principles on their natural domain, namely Orlicz spaces. We show that such principles always satisfy the Fatou property. This allows to establish a tractable dual representation without imposing any condition on the reference Orlicz function. In addition, we show that Haezendonck–Goovaerts principles satisfy the stronger Lebesgue property if and only if the reference Orlicz function fulfills the so-called Δ2 condition. We also discuss (semi)continuity properties with respect to Φ-weak convergence of probability measures. In particular, we show that Haezendonck–Goovaerts principles, restricted to the corresponding Young class, are always lower semicontinuous with respect to the Φ-weak convergence.

Two multi-cubic functional equations and some results on the stability in modular spaces

In this article, we study n-variable mappings which are cubic in each variable. We also show that such mappings can be described by an equation, say, multi-cubic functional equation. Furthermore, we study the stability of such functional equations in the modular space X_{rho } by applying Delta _{2}-condition and the Fatou property (in some cases) on the modular function ρ. Finally, we show that, under some mild conditions, one of these new multi-cubic functional equations can be hyperstable.

Rosenthal’s inequalities: ${\Delta }$-norms and quasi-Banach symmetric sequence spaces

Let $X$ be a symmetric quasi-Banach function space with the Fatou property and let $E$ be an arbitrary symmetric quasi-Banach sequence space. Suppose that $(f_k)_{k\geq 0}\subset X$ is a sequence of independent random variables. We present a necessary and

A few remarks on bounded homomorphisms acting on topological lattice groups and topological rings

Suppose G is a locally solid lattice group. It is known that there are non-equivalent classes of bounded homomorphisms on G which have topological structures. In this paper, our attempt is to assign lattice structures on them. More precisely, we use of a version of the remarkable Riesz-Kantorovich formulae and Fatou property for bounded order bounded homomorphisms to allocate the desired structures. Moreover, we show that unbounded convergence on a locally solid lattice group is topological and we investigate some applications of it. Also, some necessary and sufficient conditions for completeness of different types of bounded group homomorphisms between topological rings have been obtained, as well.

Functional equation and its modular stability with and without Δp-condition

Mixed type is a further step of development in functional equations. In this paper, the authors made an attempt to introduce such equation of the following form with its general solution h(py + z) + h(py-z) + h(y + pz) + h(y-pz) = (p + p2)[h(y + z) + h(y-z)] + 2h(py)- 2(p2 + p-1)h(y) for all y,z ? R, p ? 0,?1. Also, without Fatou property authors investigate its various stabilities related to Ulam problem in modular space by considering with and without ?p-condition.