Jesus Araujo
It is proved that every linear biseparating map between spaces of vector-valued differentiable functions is a weighted composition map. As a consequence, such a map is always continuous.
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In this paper we study the problem of extending functions with values in a locally convex Hausdorff space $E$ over a field $\mathbb{K}$, which have weak extensions in a weighted Banach space $\mathcal{F}ν(Ω,\mathbb{K})$ of scalar-valued functions on a set $Ω$, to functions in a vector-valued counterpart $\mathcal{F}ν(Ω,E)$ of $\mathcal{F}ν(Ω,\mathbb{K})$. Our findings rely on a description of vector-valued functions as linear continuous operators and extend results of Frerick, Jordá and Wengenroth. As an application we derive weak-strong principles for continuously partially differentiable fun…
On Continuously Differentiable Vector-Valued Functions of Non-Integer Order
本文研究阶数 $\alpha>0$ 的连续分数阶可微函数空间，考虑 Riemann-Liouville 和 Caputo 导数，探讨其基本性质并与 Hölder 连续函数空间比较，旨在建立严格理论框架。
This paper deals with the new concept of interval whose both the bounds themselves are also intervals. We name this new type of interval as Type-2 interval. Here we have introduced Type-2 interval-valued function and its properties. To serve this purpose, we have defined a distance on the set of all Type-2 intervals, named as extended Moore distance for Type-2 intervals which is a metric on the set of all Type-2 intervals. Then we have shown that the space of Type-2 interval is a complete metric space with respect to extended Moore distance. Then we have introduced the concept of limit-continu…
We present a characterization for a positive definite operator valued kernel to be universal or $C_{0}$-universal, and apply these characterizations to a family of operator valued kernels that are shown to be well behaved. Later, we obtain a characterization for an operator valued differentiable kernel to be $C^{q}$-universal and $C_{0}^{q}$-universal. In order to obtain such characterization and examples we generalize some well known results concerning the structure of differentiable kernels to the operator valued context. On the examples is given an emphasis on the radial kernels on Euclidea…
This paper treats the variation of sets. We attempt to formulate convergence and continuity of set-valued functions in a different way from the theories on sequences of sets and correspondence. In the final section, we also attempt to define differentiation of set-valued functions in a Euclidean space by bijection between two sets.
A pair of functions defined on a set X with values in a vector space E is said to be disjoint if at least one of the functions takes the value 0 at every point in X. An operator acting between vector-valued function spaces is disjointness preserving if it maps disjoint functions to disjoint functions. We characterize compact and weakly compact disjointness preserving operators between spaces of Banach space-valued differentiable functions.
We show that, on any given finite Borel measure space with the ambient space being a Polish metric space, every Borel real-valued function is almost a bounded, uniformly continuous function in the sense that for every $\varepsilon > 0$ there is some bounded, uniformly continuous function such that the set of points at which they would not agree has measure $< \varepsilon$. In particular, this result complements the known result of almost uniform continuity of Borel real-valued functions on a finite Radon measure space whose ambient space is a locally compact metric space. As direct application…
Let $\A$ be the operator which assigns to each $m \times n$ matrix-valued function on the unit circle with entries in $H^\infty + C$ its unique superoptimal approximant in the space of bounded analytic $m \times n$ matrix-valued functions in the open unit disc. We study the continuity of $\A$ with respect to various norms. Our main result is that, for a class of norms satifying certain natural axioms, $\A$ is continuous at any function whose superoptimal singular values are non-zero and is such that certain associated integer indices are equal to 1. We also obtain necessary conditions for cont…
We study notions of absolute continuity for functions defined on $\mathbb{R}^n$similar to the notion of $α$-absolute continuity in the sense of Bongiorno. We confirm a conjecture of Malý that 1-absolutely continuous functions do not need to be differentiable a.e., and we show several other pathological examples of functions in this class. We establish containment relations of the class $1-AC_{\rm WDN}$ which consits of all functions in $1-AC$ which are in the Sobolev space $W^{1,2}_{loc}$, are differentiable a.e. and satisfy the Luzin (N) property, with previously studied classes of absolutely…
Towards a Theory of Dobrakov-Sobolev Spaces
本文基于Dobrakov积分引入Sobolev空间的推广，针对有限维空间上Banach值函数与Fomin可微算子值测度，建立可微实值测度理论的类比结果，并嵌入向量值分布理论框架。
The aim of this expository article is to present recent developments in the centuries old discussion on the interrelations between continuous and differentiable real valued functions of one real variable. The truly new results include, among others, the $D^n$-$C^n$ interpolation theorem: {\em For every $n$-times differentiable $f\colon\R\to\R$ and perfect $P\subset \R$ there is a $C^n$ function $g\colon\R\to\R$ such that $f\restriction P$ and $g\restriction P$ agree on an uncountable set} and an example of a differentiable function $F\colon\R\to\R$ (which can be nowhere monotone) and of compac…
We prove that a biseparating map between spaces of vector-valued continuous functions is usually automatically continuous. However, we also discuss special cases when it is not true.
We prove an extension theorem (with non-tangential limits) for vector-valued Baire one functions. Moreover, at every point where the function is continuous (or bounded), the continuity (or boundedness) is preserved. More precisely: Let $H$ be a closed subset of a metric space $X$ and let $Z$ be a normed vector space. Let $f: H\to Z$ be a Baire one function. We show that there is a continuous function $g: (X\setminus H) \to Z$ such that, for every $a\in \partial H$, the non-tangential limit of $g$ at a equals $f(a)$ and, moreover, if $f$ is continuous at $a\in H$ (respectively bounded in a neig…
We give a necessary and sufficient condition for non-local functionals on vector-valued Lebesgue spaces to be weakly sequentially lower semi-continuous. Here a non-local functional shall have the form of a double integral of a density which depends on the function values at two different points. The characterisation we get is essentially that the density has to be convex in one variable if we integrate over the other one with an arbitrary test function in it. Moreover, we show that this condition is in the case of non-local functionals on real-valued Lebesgue spaces (up to some equivalence in …
Continuity equation on metric spaces via measure-valued derivations and BV-Wasserstein curves
本文引入度量空间上的连续性方程概念，基于测度值导子刻画配备1-瓦瑟斯坦距离的概率测度空间中的BV-曲线，其公式与Almi等人的相关工作一致，还扩展了此前的概率表示研究。
In this work, various versions of the so-called Omega-Lemma are provided, which ensure differentiability properties of pushforwrds between spaces of C^r-sections (or compactly supported C^r-sections) in vector bundles over finite-dimensional base manifolds whose fibres are (possibly infinite-dimensional) locally convex spaces. Applications are given, including the proof of continuity for some natural module multiplications on spaces of sections and the construction of certain infinite-dimensional Lie groups of Lie group-valued maps.
Continuity of Metric Projection Operator from C[0, 1] onto Pn with Applications to Mordukhovich Derivatives
本文证明了从Banach空间C[0, 1]到多项式子空间Pn的度量投影算子是单值且连续的，并利用该连续性研究了其Mordukhovich导数的Gateaux方向导数、性质及不动点性质。
We establish exponential laws for certain spaces of differentiable functions over a valued field K. For example, we show that the topological vector spaces C^{r,s}(U x V,E) and C^r(U,C^s(V,E)) are isomorphic if U and V are open subsets of K^n and K^m, respectively, E is a topological K-vector space, and r,s are degrees of differentiability. As a first application, we study the density of locally polynomial functions in spaces of partially differentiable functions over an ultrametric field (thus solving an open problem by E. Nagel), and also global approximations by polynomial functions. As a s…
Let X be an infinite-dimensional Banach space. In 1995, settling a long outstanding problem of Pettis, Dilworth and Girardi constructed an X-valued Pettis integrable function on [0; 1] whose primitive is nowhere weakly differentiable. Using their technique and some new ideas we show that ND, the set of strongly measurable Pettis integrable functions with nowhere weakly differentiable primitives, is lineable, i.e., there is an infinite dimensional vector space whose nonzero vectors belong to ND.
We propose a unified approach to the study of isometries on algebras of vector-valued Lipschitz maps and those of continuously differentiable maps by means of the notion of natural $C(Y)$-valuezations that take values in unital commutative $C^*$-algebras. A precise proof of a theorem of Jarosz \cite{ja} is exhibited.
In this paper we consider the problem of optimization of approximate integration of set-valued functions from the class defined by given majorant of their moduli of continuity, using values of the functions at $n$ fixed or free points of their domain. We consider the cases of exact information and information with error.
In this paper, we introduce restricted products for families of locally convex spaces and formulate criteria ensuring that mappings into such products are continuous or smooth. As a special case, can define restricted products of weighted function spaces and obtain results concerning continuity and differentiability properties of natural non-linear mappings between such spaces. These concepts and results are the basis for the study of weighted vector fields on Riemannian manifolds in a subsequent work (see [B. Walter, "Weighted diffeomorphism groups of Riemannian manifolds", arXiv: 1601.02834]…
A necessary condition on a singular kernel for the continuity of an integral operator in Hölder spaces
该研究在含非加倍测度的度量测度空间框架下，证明了奇异积分算子极大函数有界性是其在Hölder空间连续的必要条件（当算子不降低函数正则性时），并给出紧微分流形上积分算子的应用。
Strong Bochner type integrals with values in locally convex spaces are introduced. It is shown that the strong integral exists in the same cases as the weak (Gelfand-Pettis) integral is known to exist. The strong integral has better continuity properties that the weak integral.
Some optimal recovery problems for operators on classes of $L$-space valued functions
本文解决了由连续性模控制函数定义的L-空间值函数类上算子的三个最优恢复问题，L-空间可统一处理多值、模糊值、随机过程等非实值函数。
We propose a unified approach to the study of isometries on algebras of vector-valued Lipschitz maps and those of continuously differentiable maps by means of the notion of admissible quadruples. We describe isometries on function spaces of some admissible quadruples that take values in unital commutative $C^*$-algebras. As a consequence we confirm the statement of \cite[Example 8]{jp} on Lipschitz algebras and show that isometries on such algebras indeed take the canonical form.
For a compact Hausdorff space $S$, we prove that the closed unit ball of a closed linear subalgebra of the space of real-valued continuous functions on $S$, denoted by $C(S)$, satisfies property-$(P_1)$ (the set-valued generalization of strong proximinality) for the non-empty closed bounded subsets of the bidual of $C(S)$. Various stability results related to property-$(P_1)$ and semi-continuity properties of restricted Chebyshev-center maps are also established. As a consequence, we derive that if $Y$ is a proximinal finite co-dimensional subspace of $c_0$ then the closed unit ball of $Y$ sat…
Let $X$ and $Y$ be Banach or normed linear spaces and $F\subset X$ a closed set. We apply our recent extension theorem for vector-valued Baire one functions arXiv:1512.03717 to obtain an extension theorem for vector-valued functions $f\colon F\to Y$ with pre-assigned derivatives, with preservation of differentiability (at every point where the pre-assigned derivative is actually a derivative), preservation of continuity, preservation of (pointwise) Lipschitz property and (for finite dimensional domain $X$) preservation of strict differentiability and global (eventually local) Lipschitz continu…
In this paper the continuity of the set valued map $p\rightarrow B_{Ω,\mathcal{X},p}(r),$ $p\in (1,+\infty),$ is proved where $B_{Ω,\mathcal{X},p}(r)$ is the closed ball of the space $L_{p}\left(Ω,Σ,μ; \mathcal{X}\right)$ centered at the origin with radius $r,$ $\left(Ω,Σ,μ\right)$ is a finite and positive measure space, $\mathcal{X}$ is separable Banach space. An application to input-output system described by Urysohn type integral operator is discussed.
We introduce vectorial and topological continuities for functions defined on vector metric spaces and illustrate spaces of such functions. Also, we describe some fundamental classes of vector valued functions and extension theorems.
In this paper we estimate the Kuratowski and the Hausdorff measures of noncompactness of bounded subsets of spaces of vector-valued bounded functions and of vector-valued bounded differentiable functions. To this end, we use a quantitative characteristic modeled on a new equiconti\-nuity-type concept and classical quantitative characteristics related to pointwise relative compactness. We obtain new regular measures of noncompactness in the spaces taken into consideration. The established inequalities reduce to precise formulas in some classes of subsets. We derive Ascoli-Arzel\` a type compact…
We prove an implicit function theorem for C^k-maps from arbitrary topological vector spaces over valued fields to Banach spaces (for k at least 2). As a tool, we show the C^k-dependence of fixed points on parameters for suitable families of contractions of a Banach space. Similar results are obtained for k times strictly differentiable maps, and for k times Lipschitz differentiable maps. In the real case, our results subsume an implicit function theorem for Keller C^k_c-maps from arbitrary topological vector spaces to Banach spaces.
Let $\mathcal{N}$ be a non-Archimedean ordered field extension of the real numbers that is real closed and Cauchy complete in the topology induced by the order, and whose Hahn group is Archimedean. In this paper, we first review the properties of weakly locally uniformly differentiable (WLUD) functions, $k$ times weakly locally uniformly differentiable (WLUD$^k$) functions, and WLUD$^\infty$ functions at a point or on an open subset of $\mathcal{N}$. Then we show under what conditions a WLUD$^\infty$ function at a point $x_0\in\mathcal{N}$ is analytic in an interval around $x_0$, that is, it h…
On Matrix Valued Schrödinger Operators on the Discrete Real Line: Resolvent Boundary Values, Limiting Absorption Principle, Hölder Regularity and Dispersive Estimates
本文研究了离散实线上矩阵值Schrödinger算子的谱理论和时间演化，通过构造Jost解并利用Volterra方程和转置算子，推导了解析算子的显式核表示，证明了边界值的Hölder连续性，并建立了色散估计。
The recent extensions of domain theory have proved particularly efficient to study lattice-valued maxitive measures, when the target lattice is continuous. Maxitive measures are defined analogously to classical measures with the supremum operation in place of the addition. Building further on the links between domain theory and idempotent analysis highlighted by Lawson (2004), we investigate the concept of domain-valued linear forms on an idempotent (semi)module. In addition to proving representation theorems for continuous linear forms, we address two applications: the idempotent Radon--Nikod…
We show that Sobolev maps with values in a dual Banach space can be characterized in terms of weak derivatives in a weak* sense. Since every metric space embeds isometrically into a dual Banach space, this implies a characterization of metric space valued Sobolev maps in terms of such derivatives. Furthermore, we investigate for which target spaces Sobolev maps are weak* differentiable almost everywhere.
In this paper, we give analytic methods for finding m (and m+ω) times continuously differentiable solutions of a finite system of linear equations. Along the way, we also solve a generalized Whitney problem for vector-valued functions and jets.
Classifications of $\rm{SL}(n)$ covariant function-valued valuations are established with some assumptions of continuity. New valuations, for example, weighted moment functions, are introduced and our classifications give unified characterizations of the Laplace transform on convex bodies, $L_p$ moment bodies, $L_p$ difference bodies, and polar $L_p$ moment bodies ($L_{-p}$ intersection bodies). Using the new classifications, we also establish some Euler-type relations.
Based on Colombeau's theory of algebras of generalized functions we introduce the concepts of generalized functions taking values in differentiable manifolds as well as of generalized vector bundle homomorphisms. We study their basic properties, in particular with respect to some new point value concepts for generalized functions and indicate applications of the resulting theory in general relativity.
Let $t\mapsto A(t)$ for $t\in T$ be a $C^M$-mapping with values unbounded operators with compact resolvents and common domain of definition which are self-adjoint or normal. Here $C^M$ stands for $C^\om$ (real analytic), a quasianalytic or non-quasianalytic Denjoy-Carleman class, $C^\infty$, or a Hölder continuity class $C^{0,\al}$. The parameter domain $T$ is either $\mathbb R$ or $\mathbb R^n$ or an infinite dimensional convenient vector space. We prove and review results on $C^M$-dependence on $t$ of the eigenvalues and eigenvectors of $A(t)$.
Automatic time continuity of positive matrix and operator semigroups
本文研究无可测性假设的正矩阵半群，证明其在原点附近有界时自动连续，并将结果推广到序列空间上的正算子半群。
A class of stochastic delay equations in Banach space $E$ driven by cylindrical Wiener process is studied. We investigate two concepts of solutions: weak and generalised strong, and give conditions under which they are equivalent. We present an evolution equation approach in a Banach space $\Ep{p}:=E\times L^p(-1,0;E)$ proving that the solutions can be reformulated as $\Ep{p}$-valued Markov processes. Based on the Markovian representation we prove the existence and continuity of the solutions. The results are applied to stochastic delay partial differential equations with an application to neu…
Based on the concept of manifold valued generalized functions we initiate a study of nonlinear ordinary differential equations with singular (in particular: distributional) right hand sides in a global setting. After establishing several existence and uniqueness results for solutions of such equations and flows of singular vector fields we compare the solution concept employed here with the purely distributional setting. Finally, we derive criteria securing that a sequence of smooth flows corresponding to a regularization of a given singular vector field converges to a measurable limiting flow…
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