Radon-Nikodym Property Research Articles

Examples and applications of the density of strongly norm attaining Lipschitz maps

We study the density of the set SNA (M,Y) of those Lipschitz maps from a (complete pointed) metric space M to a Banach space Y which strongly attain their norm (i.e., the supremum defining the Lipschitz norm is actually a maximum). We present new and somehow counterintuitive examples, and we give some applications. First, we show that SNA (\mathbb T,Y) is not dense in Lip _0(\mathbb T,Y) for any Banach space Y , where \mathbb T denotes the unit circle in the Euclidean plane. This provides the first example of a Gromov concave metric space (i.e., every molecule is a strongly exposed point of the unit ball of the Lipschitz-free space) for which the density does not hold. Next, we construct metric spaces M satisfying that SNA (M,Y) is dense in Lip _0(M,Y) regardless Y but which contain isometric copies of [0,1] and so the Lipschitz-free space \mathcal F(M) fails the Radon–Nikodym property, answering in the negative a question posed by Cascales et al. in 2019 and by Godefroy in 2015. Furthermore, an example of such M can be produced failing all the previously known sufficient conditions for the density of strongly norm attaining Lipschitz maps. Finally, among other applications, we show that if M is a boundedly compact metric space for which SNA (M,\mathbb R) is dense in Lip _0(M,\mathbb R) , then the unit ball of the Lipschitz-free space on M is the closed convex hull of its strongly exposed points. Further, we prove that given a compact metric space M which does not contain any isometric copy of [0,1] and a Banach space Y , if SNA (M,Y) is dense, then SNA (M,Y) actually contains an open dense subset.

Dual representation of expectile-based expected shortfall and its properties

<p style='text-indent:20px;'>An expectile can be considered a generalization of a quantile. While expected shortfall is a quantile-based risk measure, we study its counterpart—the expectile-based expected shortfall—where expectile takes the place of a quantile. We provide its dual representation in terms of a Bochner integral. Among other properties, we show that it is bounded from below in terms of the convex combination of expected shortfalls, and also from above by the smallest law invariant, coherent, and comonotonic risk measures, for which we give the explicit formulation of the corresponding distortion function. As a benchmark to the industry standard expected shortfall, we further provide its comparative asymptotic behavior in terms of extreme value distributions. Based on these results, we finally explicitly compute the expectile-based expected shortfall for selected classes of distributions.</p>

On the Radon-Nikodym Property for Vector Measures and Extensions of Transfunctions

Abstract If ( μ n ) n = 1 ∞ \left( {{\mu _n}} \right)_{n = 1}^\infty are positive measures on a measurable space (X, Σ) and ( v n ) n = 1 ∞ \left( {{v_n}} \right)_{n = 1}^\infty are elements of a Banach space 𝔼 such that ∑ n = 1 ∞ ‖ v n ‖ μ n ( X ) < ∞ \sum\nolimits_{n = 1}^\infty {\left\| {{v_n}} \right\|{\mu _n}\left( X \right)} < \infty , then ω ( S ) = ∑ n = 1 ∞ v n μ n ( S ) \omega \left( S \right) = \sum\nolimits_{n = 1}^\infty {{v_n}{\mu _n}\left( S \right)} defines a vector measure of bounded variation on (X, Σ). We show 𝔼 has the Radon-Nikodym property if and only if every 𝔼-valued measure of bounded variation on (X, Σ) is of this form. This characterization of the Radon-Nikodym property leads to a new proof of the Lewis-Stegall theorem. We also use this result to show that under natural conditions an operator defined on positive measures has a unique extension to an operator defined on 𝔼-valued measures for any Banach space 𝔼 that has the Radon-Nikodym property.

Nuclear operators on Banach function spaces

Let X be a Banach space and E be a perfect Banach function space over a finite measure space (Omega ,Sigma ,lambda ) such that L^infty subset Esubset L^1. Let E' denote the Köthe dual of E and tau (E,E') stand for the natural Mackey topology on E. It is shown that every nuclear operator T:Erightarrow X between the locally convex space (E,tau (E,E')) and a Banach space X is Bochner representable. In particular, we obtain that a linear operator T:L^infty rightarrow X between the locally convex space (L^infty ,tau (L^infty ,L^1)) and a Banach space X is nuclear if and only if its representing measure m_T:Sigma rightarrow X has the Radon-Nikodym property and |m_T|(Omega )=Vert TVert _{nuc} (= the nuclear norm of T). As an application, it is shown that some natural kernel operators on L^infty are nuclear. Moreover, it is shown that every nuclear operator T:L^infty rightarrow X admits a factorization through some Orlicz space L^varphi , that is, T=Scirc i_infty , where S:L^varphi rightarrow X is a Bochner representable and compact operator and i_infty :L^infty rightarrow L^varphi is the inclusion map.

Additive regression with Hilbertian responses

This paper develops a foundation of methodology and theory for the estimation of structured nonparametric regression models with Hilbertian responses. Our method and theory are focused on the additive model, while the main ideas may be adapted to other structured models. For this, the notion of Bochner integration is introduced for Banach-space-valued maps as a generalization of Lebesgue integration. Several statistical properties of Bochner integrals, relevant for our method and theory and also of importance in their own right, are presented for the first time. Our theory is complete. The existence of our estimators and the convergence of a practical algorithm that evaluates the estimators are established. These results are nonasymptotic as well as asymptotic. Furthermore, it is proved that the estimators achieve the univariate rates in pointwise, $L^{2}$ and uniform convergence, and that the estimators of the component maps converge jointly in distribution to Gaussian random elements. Our numerical examples include the cases of functional, density-valued and simplex-valued responses, demonstrating the validity of our approach.

On the continuity of the adjoint of evolution operators

By associating a quasi semigroup to a commutative evolution operator and using techniques from vector measures like the Gel’fand integral and the Radon- Nikodym theorem for Bochner integral, we prove the continuity on [0, +∞) of the adjoint of a certain class of evolution operators. Furthermore, we provide examples of evolution operators satisfying our conclusion. We also provide an application to optimal control theory; particularly, to an optimal control problem governed by a time-varying heat equation.

Some remarks on minimum norm attaining operators

In this paper we investigate continuous linear operators between Banach spaces which attain their minimum norms on the unit sphere. We characterize finite dimensionality of Banach spaces with the help of minimum norm attainment of continuous linear operators. We introduce a property called weak minimizing property for Banach spaces related to minimum norm attainment of continuous linear operators. We further study Lindentrauss type properties AM and BM of Banach spaces for minimum norm attaining operators and show that Radon-Nikodym property of Banach spaces is a sufficient condition for both the properties to hold. We also investigate a weaker version of minimum norm attaining operators.

Non-embedding theorems of nilpotent Lie groups and sub-Riemannian manifolds

We prove that there do not exist quasi-isometric embeddings of connected nonabelian nilpotent Lie groups equipped with left invariant Riemannian metrics into a metric measure space satisfying the RCD(0,N), with N > 1. In fact, we can prove that a subRiemannian manifold whose generic degree of nonholonomy is not smaller than 2 can not be biLipschitzly embedded in any Banach space with the Radon-Nikodym property. We also get that every regular sub-Riemannian manifold do not satisfy the CD(K,N) with N > 1. We also prove that the subRiemannian manifold is infinitesimally Hilbert space.

Optimal partitioning of an interval and applications to Sturm-Liouville eigenvalues

We study the optimal partitioning of a (possibly unbounded) interval of the real line into n subintervals in order to minimize the maximum of certain set-functions, under rather general assumptions such as continuity, monotonicity, and a Radon-Nikodym property. We prove existence and uniqueness of a solution to this minimax partition problem, showing that the values of the set-functions on the intervals of any optimal partition must coincide. We also investigate the asymptotic distribution of the optimal partitions as n tends to infinity. Several examples of set-functions fit in this framework, including measures, weighted distances and eigenvalues. We recover, in particular, some classical results of Sturm-Liouville theory: the asymptotic distribution of the zeros of the eigenfunctions, the asymptotics of the eigenvalues, and the celebrated Weyl law on the asymptotics of the counting function.

Interval-valued Choquet integral for set-valued mappings: definitions, integral representations and primitive characteristics

In this paper, a new kind of real-valued major Choquet integral, real-valued minor Choquet integral and interval-valued Choquet integrals for set-valued functions is introduced and investigated. The representations of the Choquet integral of set-valued functions with respect to a fuzzy measure are given. In particular, we focus on the case of the distorted Lebesgue measure as a fuzzy measure. Furthermore, the characteristics of the primitive of Choquet integral for set-valued functions are given as Radon-Nikodym property in some sense.

Strongly measurable functions and multipliers of $${\mathcal {M}}-$$ integrable functions

We investigate the integrability of Banach space valued strongly measurable functions defined on [0, 1]. In the case of functions given by $$f=\sum _{n=1}^{\infty }x_n \chi _{E_n}$$ where $$x_n\in X$$ and the sets $$E_n$$ are Lebesgue measurable and pairwise disjoint subsets of [0, 1], there are well known characterizations for Bochner and McShane integrability of f. The absolute (resp. unconditional) convergence of $$\sum _{n=1}^{\infty }x_n m(E_n)$$ is equivalent to Bochner (resp. McShane) integrability of f. We give some conditions for scalar McShane and weakly McShane integrability of f and their relation with unconditionally converging operators. We also study the space of vector valued multipliers of strongly McShane $$(\mathcal {SM})-$$ integrable functions. We prove that if X is a commutative Banach algebra, with identity e of norm one, satisfying Radon–Nikodym property and $$M: \mathcal {SM}\rightarrow \mathcal {SM}$$ is a bounded linear operator, then there exists $$g\in L^\infty ([0,1],X)$$ such that $$M(f)= fg$$ for all $$f\in \mathcal {SM}$$ . Some results on multipliers of McShane $$({\mathcal {M}})-$$ integrable functions are also derived.

Porosity and differentiability of Lipschitz maps from stratified groups to Banach homogeneous groups

Let f be a Lipschitz map from a subset A of a stratified group to a Banach homogeneous group. We show that directional derivatives of f act as homogeneous homomorphisms at density points of A outside a $$\sigma $$-porous set. At all density points of A, we establish a pointwise characterization of differentiability in terms of directional derivatives. These results naturally lead us to an alternate proof of almost everywhere differentiability of Lipschitz maps from subsets of stratified groups to Banach homogeneous groups satisfying a suitably weakened Radon–Nikodym property.

Coalitional extreme desirability in finitely additive economies with asymmetric information

We prove a coalitional core-Walras equivalence theorem for an asymmetric information exchange economy with a finitely additive measure space of agents, finitely many states of nature, and an infinite dimensional commodity space having the Radon–Nikodym property and whose positive cone has possibly empty interior. The result is based on a new cone condition, firstly developed in Centrone and Martellotti (2015), called coalitional extreme desirability. We also formulate a notion of incentive compatibility suitable for coalitional models and study it in relation to equilibria.

Integrated Semigroups on Fréchet Spaces I: Bochner Integral, Laplace Integral and Real Representation Theorems

In this articles we will study the integration of the vectorial functions in Fréchet spaces. Particularly we will introduce and we will study a new functional space and we will prove some theorems of representation.

Integrated Semigroups on Fréchet Spaces I: Bochner Integral, Laplace Integral and Real Representation Theorems

In this articles we will study the integration of the vectorial functions in Fréchet spaces. Particularly we will introduce and we will study a new functional space and we will prove some theorems of representation.

The Radon Nikodym Property and Multipliers of 𝓗𝓚-Integrable Functions

We study the space of vector valued multipliers of strongly Henstock-Kurzweil \((\mathcal{SHK})\) integrable functions. We prove that if \(X\) is a commutative Banach algebra, with identity \(e\) of norm one, satisfying Radon-Nikodym property and \(g:[a,b] \rightarrow X\) is of strong bounded variation, then the multiplication operator defined by \(M_g(f)=fg\) maps \(\mathcal{SHK}\) to \(\mathcal{SHK}.\) We also investigate the problems when the domain is \(\mathcal{HK}\) or when \(X\) satisfies weak Radon-Nikodym property.

Numerical solution of fractional elliptic stochastic PDEs with spatial white noise

Abstract The numerical approximation of solutions to stochastic partial differential equations with additive spatial white noise on bounded domains in $\mathbb{R}^d$ is considered. The differential operator is given by the fractional power $L^\beta $, $\beta \in (0,1)$ of an integer-order elliptic differential operator $L$ and is therefore nonlocal. Its inverse $L^{-\beta }$ is represented by a Bochner integral from the Dunford–Taylor functional calculus. By applying a quadrature formula to this integral representation the inverse fractional-order operator $L^{-\beta }$ is approximated by a weighted sum of nonfractional resolvents $( I + \exp(2 y_\ell) \, L )^{-1}$ at certain quadrature nodes $t_j> 0$. The resolvents are then discretized in space by a standard finite element method. This approach is combined with an approximation of the white noise, which is based only on the mass matrix of the finite element discretization. In this way an efficient numerical algorithm for computing samples of the approximate solution is obtained. For the resulting approximation the strong mean-square error is analyzed and an explicit rate of convergence is derived. Numerical experiments for $L=\kappa ^2-\Delta $, $\kappa> 0$ with homogeneous Dirichlet boundary conditions on the unit cube $(0,1)^d$ in $d=1,2,3$ spatial dimensions for varying $\beta \in (0,1)$ attest to the theoretical results.

On Complexification of Real Spaces and Its Manifestations in the Theory of Bochner and Pettis Integrals

This work is a continuation of our work [12] where we considered linear spaces in the following two situations: a real space admits a multiplication by complex scalars without changing the set itself; a real space is embedded into a wider set with a multiplication by complex scalars. We studied there also how they manifest themselves when the initial space possesses additional structures: topology, norm, inner product, as well as what happens with linear operators acting between such spaces. Changing the linearities of the linear spaces unmasks some very subtle properties which are not so obvious when the set of scalars is not changed. In the present work, we follow the same idea considering now Bochner and Pettis integrals for functions ranged in real and complex Banach and Hilbert spaces. Finally, this leads to the study of strong and weak random elements with values in real and complex Banach and Hilbert spaces, in particular, some properties of their expectations.

Limits of Sequences of Bochner Integrable Functions Over Sequences of Probability Measures Spaces

We prove limits of sequences of Bochner integrable functions over sequences of probability measures spaces. A sample result: Let X be a bounded closed convex set in a Banach space F, $$a\in X$$ and E a non-null Banach space. Let $$\left( \Omega _{n},\Sigma _{n},\mu _{n}\right) _{n\in {\mathbb {N}}}$$ be a sequence of probability measure spaces, $$\varphi _{n}:\Omega _{n}\rightarrow X$$ a sequence of $$\mu _{n}$$ -Bochner integrable functions. Then the following assertions are equivalent: $$\begin{aligned} \lim \limits _{n\rightarrow \infty }\int _{\Omega _{n}}f\left( \varphi _{n}\left( \omega _{n}\right) \right) d \mu _{n} (w_n)=f\left( a\right) \text { in norm of }E. \end{aligned}$$

Differentiability and Poincaré-type inequalities in metric measure spaces

We demonstrate the necessity of a Poincaré type inequality for those metric measure spaces that satisfy Cheeger's generalization of Rademacher's theorem for all Lipschitz functions taking values in a Banach space with the Radon–Nikodym property. This is done by showing the existence of a rich structure of curve fragments that connect nearby points, similar in nature to Semmes's pencil of curves for the standard Poincaré inequality. Using techniques similar to Cheeger–Kleiner [12], we show that our conditions are also sufficient.We also develop another characterization of RNP Lipschitz differentiability spaces by connecting points by curves that form a rich structure of partial derivatives that were first discussed in [5].