Lebesgue Property Research Articles

Improved energy decay estimate for Dir-stationary Q-valued functions and its applications

In this paper, we establish an improved decay estimate for the Dirichlet energy of Dir-stationary Q-valued functions. As a direct application of this estimate, we derive a Liouville-type theorem for bounded Dir-stationary Q-valued functions defined on Rm. Additionally, in an attempt to establish the continuity of Dir-stationary Q-valued functions, we confirm that such functions exhibit the Lebesgue property at every point within their domain.

Banach spaces with the Lebesgue property of Riemann integrability

A Banach space is said to have the Lebesgue property if every Riemann-integrable function f:[0,1]→X is Lebesgue almost everywhere continuous. We give a characterization of the Lebesgue property in terms of a new sequential asymptotic structure that is strictly between the notions of spreading and asymptotic models. We also reproduce an apparently lost theorem of Pelczynski and da Rocha Filho that a subspace X⊂L1[0,1] has the Lebesgue property if every spreading model of X is equivalent to the unit vector basis of ℓ1.

On the Lebesgue property in uniform spaces

On the Lebesgue property in uniform spaces

Existence Results for a Dirichlet Boundary Value Problem Involving the p(x)-Laplacian Operator

The aim of this paper is to establish the existence of weak solutions, in$W_0^{1,p(x)}(\Omega)$, for a Dirichlet boundary value problem involving the$p(x)$-Laplacian operator. Our technical approach is based on the Berkovits topologicaldegree theory for a class of demicontinuous operators of generalized $(S_+)$ type. Wealso use as a necessary tool the properties of variable Lebesgue and Sobolev spaces, andspecially properties of $p(x)$-Laplacian operator. In order to usethis theory, we will transform our problem into an abstractHammerstein equation of the form $v+S\circ Tv=0$ in the reflexiveBanach space $W^{-1,p'(x)}(\Omega)$ which is the dual space of$W_0^{1,p(x)}(\Omega)$. Note also that the problem can be seen as anonlinear eigenvalue problem of the form$Au=\lambda u,$ where$Au:=-\Div(|\nabla u|^{p(x)-2}\nabla u)-f(x,u)$. When this problemadmits a~non-zero weak solution $u$, $\lambda$ is an eigenvalue ofit and $u$ is an associated eigenfunction.

TOWARDS A CHARACTERIZATION OF THE PROPERTY OF LEBESGUE

There are three main contributions in this work. First, the proof that every stabilized asymptotic-ℓ1 Banach space has the Property of Lebesgue is generalized to the coordinate-free case. Second, the proof that every Banach space with the Property of Lebesgue has a unique ℓ1 spreading model is generalized to cover a particular class of asymptotic models. Third, a characterization of the Property of Lebesgue is derived that applies to those Banach spaces with bases that admit in a strong sense favorable block bases. These results are significant because they demonstrate not only the efficacy of characterizing the Property of Lebesgue in terms of a connection between the local and the global asymptotic structures of certain Banach spaces, but also the possibility of finding a more general characterization in similar terms.

Refined probabilistic global well-posedness for the weakly dispersive NLS

We continue our study of the cubic fractional NLS with very weak dispersion α>1 and data distributed according to the Gibbs measure. We construct the natural strong solutions for α>α0=31−23314≈1.124 which is strictly smaller than 87, the threshold beyond which the first nontrivial Picard iteration has no longer the Sobolev regularity needed for the deterministic well-posedness theory. This also improves our previous result in Sun and Tzvetkov (2020). We rely on recent ideas of Bringmann (2021) and Deng et al. (2019). In particular we adapt to our situation the new resolution ansatz in Deng et al. (2019) which captures the most singular frequency interaction parts in the Xs,b type space. To overcome the difficulties caused by the weakly dispersive effect, our specific strategy is to benefit from the “almost” transport effect of these singular parts and to exploit their L∞ as well as the Fourier–Lebesgue property in order to inherit the random feature from the linear evolution of high frequency portions.

Riemann intergability versus continuity for vector-valued functions

The interplay between Riemann integrability and continuity is an interesting topic of modern analysis. In this paper, Riemann integrability of vector-valued continuous functions, property of Lebesgue and weak property of Lebesgue are surveyed and discussed. We also prove that `1(N, X) has the property of Lebesgue.

Quantum Anomalies via Differential Properties of Lebesgue–Feynman Generalized Measures

We address the problem concerning the origin of quantum anomalies, which has been the source of disagreement in the literature. Our approach is novel as it is based on the differentiability properties of families of generalized measures. To this end, we introduce a space of test functions over a locally convex topological vector space, and define the concept of logarithmic derivatives of the corresponding generalized measures. In particular, we show that quantum anomalies are readily understood in terms of the differential properties of the Lebesgue–Feynman generalized measures (equivalently, of the Feynman path integrals). We formulate a precise definition for quantum anomalies in these terms.

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.

Contraction, Lebesgue and Common Fixed Point Property of Fuzzy Metric Spaces

In this paper, we using contraction and contraction functions in complete fuzzy metric space and establish sequential characterization properties of Lebesgue fuzzy metric space and common fixed on it. Then first introduce a new type of Lebesgue fuzzy metric space, which is generalization of fuzzy metric space, second we study the topological properties of Lebesgue fuzzy metric space, third a relation between Lebesgue and weak G-complete, compact fuzzy metrics and Lebesgue integral mappings finally established characterization properties on it. We prove the existence of common fixed point and contraction mapping in fuzzy metric space using the property of Lebesgue fuzzy metric space and integral type of mappings. On the basis of these properties we are getting common fixed point of two mappings, three mappings and four mappings in a easy way as compared to old method like Banach contraction fixed point. Also coincidence fixed point theorem for two mapping, three mappings and four mappings using Lebesgue fuzzy metric space and integral type of mappings. Also contraction mappings property in fuzzy metric space is helpful to determine common fixed point in Lebesgue fuzzy metric space. We also discuss the Lebesgue property of several well-known fuzzy metric spaces in this paper and conclude uniqueness of common fixed point.

Mackey constraints for James's compactness theorem and risk measures

Let A be a closed, convex, bounded and non weakly compact subset of a Banach space E with 0∉A. Let us fix a convex and weakly compact subset W of E, a functional z0⁎∈E⁎ with z0⁎(A)>0, and ϵ>0. Then there is a linear form x0⁎∈BpW(z0⁎,ϵ), i.e. x0⁎∈E⁎ andsupw∈W⁡|x0⁎(w)−z0⁎(w)|<ϵ, which does not attain its infimum on A, and such that x0⁎(A)>0. As a consequence we show that a Fatou coherent monetary utility function u1≠ ess.inf has the Lebesgue property if, and only if, the convolution u1□u2 is Fatou for every Fatou coherent monetary utility function u2.

Stability in locally L0-convex modules and a conditional version of James' compactness theorem

Locally L0-convex modules were introduced in Filipovic et al. (2009) [10] as the analytic basis for the study of conditional risk measures. Later, the algebra of conditional sets was introduced in Drapeau et al. (2016) [8]. In this paper we study locally L0-convex modules, and find exactly which subclass of locally L0-convex modules can be identified with the class of locally convex vector spaces within the context of conditional set theory. Second, we provide a version of the classical James' theorem of characterization of weak compactness for conditional Banach spaces. Finally, we state a conditional version of the Fatou and Lebesgue properties for conditional convex risk measures and, as application of the developed theory, we establish a version of the so-called Jouini–Schachermayer–Touzi theorem for robust representation of conditional convex risk measures defined on a L∞-type module.

Riemann integrability versus weak continuity

In this paper we focus on the relation between Riemann integrability and weak continuity. A Banach space X is said to have the weak Lebesgue property if every Riemann integrable function from [0,1] into X is weakly continuous almost everywhere. We prove that the weak Lebesgue property is stable under ℓ1-sums and obtain new examples of Banach spaces with and without this property. Furthermore, we characterize Dunford–Pettis operators in terms of Riemann integrability and provide a quantitative result about the size of the set of τ-continuous nonRiemann integrable functions, with τ a locally convex topology weaker than the norm topology.

Lebesgue quasi-uniformity on textures

This is a continuation of the work where the notions&lt;br /&gt; of Lebesgue uniformity and Lebesgue quasi uniformity in a&lt;br /&gt;texture space were introduced. It is well known that the quasi&lt;br /&gt;uniform space with a compact topology has the Lebesgue property.&lt;br /&gt;This result is extended to direlational quasi uniformities and&lt;br /&gt;dual dicovering quasi uniformities. Additionally we discuss the&lt;br /&gt;completeness of lebesgue di-uniformities and dual dicovering&lt;br /&gt;lebesgue di-uniformities.

Approximation and its implementation process of the stochastic hybrid fuzzy system

The stochastic Mamdani and Takagi–Sugeno fuzzy systems are firstly unified in a random environment, and the resulting stochastic hybrid fuzzy system is established according to some stochastic parameters. Secondly, A canonical representation of the stochastic process with orthogonal increments is presented by the properties of the Lebesgue–Stieltjes measure and stochastic integral, the approximation of the stochastic hybrid fuzzy system in the mean square sense is proved. Finally, an implementation process of this system is described through a simulation example, and the surface figure of the covariance function shows that the stochastic hybrid fuzzy system has excellent approximation capability.

Bounded subharmonic functions possess the Lebesgue property at each point

It is proved that the restriction of a bounded subharmonic function in a domain D ⊂ ℂ to any real line l ⊂ℂ possesses the Lebesgue property at each point of l ∩ D.

Maximum Lebesgue extension of monotone convex functions

Given a monotone convex function on the space of essentially bounded random variables with the Lebesgue property (order continuity), we consider its extension preserving the Lebesgue property to as big solid vector space of random variables as possible. We show that there exists a maximum such extension, with explicit construction, where the maximum domain of extension is obtained as a (possibly proper) subspace of a natural Orlicz-type space, characterized by a certain uniform integrability property. As an application, we provide a characterization of the Lebesgue property of monotone convex function on arbitrary solid spaces of random variables in terms of uniform integrability and a “nice” dual representation of the function.

On the Lebesgue property of monotone convex functions

The Lebesgue property (order-continuity) of a monotone convex function on a solid vector space of measurable functions is characterized in terms of (1) the weak inf-compactness of the conjugate function on the order-continuous dual space, (2) the attainment of the supremum in the dual representation by order-continuous linear functionals. This generalizes and unifies several recent results obtained in the context of convex risk measures.

Lebesgue property for convex risk measures on Orlicz spaces

We present a robust representation theorem for monetary convex risk measures \({\rho : \mathcal{X} \rightarrow \mathbb{R}}\) such that $$\lim_n\rho(X_n) = \rho(X)\,{\rm whenever}\,(X_n)\,{\rm almost\,surely\,converges\,to}\,X,$$ \({|Xn| \leq Z \in \mathcal{X}, {\rm for\,all}\,n \in \mathbb{N}}\) and \({\mathcal{X}}\) is an arbitrary Orlicz space. The separable \({\langle\mathbb{L}^{1}, \mathbb{L}^{\infty}\rangle}\) case of Jouini et al. (Adv Math Econ 9:49–71 2006), as well as the non-separable version of Delbaen [7], are contained as a particular case here. We answer a natural question posed by Biagini and Fritelli [2]. Our approach is based on the study for unbounded sets, as the epigraph of a given penalty function associated with ρ, of the celebrated weak compactness Theorem due to James (Isr J Math 13:289–300 1972).

Lebesgue property of convex risk measures for bounded Càdlàg processes

In this paper, we study the Lebesgue property for convex risk measures on the class of bounded c`adl`ag processes. For that, we characterize the compact subsets of a family of bounded variation processes, which is, of course, the topological dual of the bounded c`adl`ag processes, in an appropriate topology. We show that the Lebesgue property can be characterized in several equivalent ways.