Continuity in Vector Metric Spaces

度量空间中曲线的速度测度与绝对连续性

The speed measure and absolute continuity for curves in metric spaces

本文研究了度量空间中曲线的速度测度及其绝对连续性，通过速度测度与勒贝格测度的导数关系，推广了巴纳赫-扎雷斯基定理。

度量空间中曲线绝对连续性的准则

Criterion for the absolute continuity of curves in metric spaces

本文证明了度量空间中参数化曲线绝对连续的充要条件，即其与任意Lipschitz函数的复合也绝对连续。

向量格与连续函数格上的序连续性和正则性

Order continuity and regularity on vector lattices and on lattices of continuous functions

本文刻画 Archimedean 向量格间序连续同态，借助连续函数格上的复合算子研究 $C(X)$ 子格性质，并给出正则子格刻画及 Banach 格中闭包保持正则性的结论。

Cone monotone mappings: continuity and differentiability

We generalize some results of Borwein, Burke, Lewis, and Wang to mappings with values in metric (resp. ordered normed linear) spaces. We define two classes of monotone mappings between an ordered linear space and a metric space (resp. ordered linear space): $K$-monotone dominated and cone-to-cone monotone mappings. First we show some relationships between these classes. Then, we study continuity and differentiability (also in the metric and $w^*$ senses) of mappings in these classes.

从C[0, 1]到Pn的度量投影算子的连续性及其在Mordukhovich导数中的应用

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。

Three superposition principles: currents, continuity equations and curves of measures

We establish a general superposition principle for curves of measures solving a continuity equation on metric spaces without any smooth structure nor underlying measure, representing them as marginals of measures concentrated on the solutions of the associated ODE defined by some algebra of observables. We relate this result with decomposition of acyclic normal currents in metric spaces. As an application, a slightly extended version of a probabilistic representation for absolutely continuous curves in Kantorovich-Wasserstein spaces, originally due to S. Lisini, is provided in the metric frame…

On Almost Uniform Continuity of Borel Functions on Polish Metric Spaces

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…

度量图与分形上的连续性方程与输运型方程研究

On equations of continuity and transport type on metric graphs and fractals

该研究针对鞅维数1的度量空间上一阶连续性与输运方程弱解不唯一的问题，借助边界四元组与半群理论证明无散场下的适定性，为分形空间标量解一阶方程适定性研究提供了首个结果。

Extensions of vector-valued Baire one functions with preservation of points of continuity

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…

Continuity properties of vectors realizing points in the classical field of values

For an $n$-by-$n$ matrix $A$, let $f_A$ be its "field of values generating function" defined as $f_A\colon x\mapsto x^*Ax$. We consider two natural versions of the continuity, which we call strong and weak, of $f_A^{-1}$ (which is of course multi-valued) on the field of values $F(A)$. The strong continuity holds, in particular, on the interior of $F(A)$, and at such points $z \in \partial F(A)$ which are either corner points, belong to the relative interior of flat portions of $\partial F(A)$, or whose preimage under $f_A$ is contained in a one-dimensional set. Consequently, $f_A^{-1}$ is cont…

Continuity of bilinear maps on direct sums of topological vector spaces

We prove a criterion for continuity of bilinear maps on countable direct sums of topological vector spaces. As a first application, we get a new proof for the fact (due to Hirai et al. 2001) that the map taking a pair of test functions on R^n to their convolution is continuous. The criterion also allows an open problem by K.-H. Neeb to be solved: If E is a locally convex space, regard the tensor algebra T(E) as the locally convex direct sum of the projective tensor powers T^j(E) of E. We show that T(E) is a topological algebra if and only if every sequence of continuous seminorms on E has an u…

On a continuity theorem for constructive functions

One proves that any everywhere defined constructive mapping from a complete metric space into a complete metric space which preserves the property of precompacity of subsets is locally uniformly continuous. This fact can be viewed as interpretation of L. E. J. Brower's fan theorem in terms of A. A. Markov's constructive analysis.

Well posedness of Lagrangian flows and continuity equations in metric measure spaces

We establish, in a rather general setting, an analogue of DiPerna-Lions theory on well-posedness of flows of ODE's associated to Sobolev vector fields. Key results are a well-posedness result for the continuity equation associated to suitably defined Sobolev vector fields, via a commutator estimate, and an abstract superposition principle in (possibly extended) metric measure spaces, via an embedding into $\mathbb{R}^\infty$. When specialized to the setting of Euclidean or infinite dimensional (e.g. Gaussian) spaces, large parts of previously known results are recovered at once. Moreover, the …

通过测度值导子与BV-瓦瑟斯坦曲线建立度量空间上的连续性方程

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。

Automatic continuity of biseparating maps

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.

谱邻近性下谱三元组的Dirac算子谱的连续性

Continuity of the Spectrum of Dirac Operators of Spectral Triples for the Spectral Propinquity

该研究证明度量谱三元组按谱邻近性收敛时其Dirac算子谱也收敛，通过有界连续函数演算收敛推导得出，还给出作用泛函连续等应用，形式化了不同C*-对应上可伴算子的收敛。

Another look at the continuity of moduli of noncompact convexity

In this article we introduce the new modulus $\triangle'_{X,ϕ}(\varepsilon)$, for which we prove that in the general case is different from the classical modulus of noncompact convexity. The main result of the paper is showing the continuity of the modulus of noncompact convexity for arbitrary minimalizable (strictly minimalizable) measure of noncompactness on arbitrary metric space.

Order continuity from a topological perspective

We study three types of order convergence and related concepts of order continuous maps in partially ordered sets, partially ordered abelian groups and partially ordered vector spaces, respectively. An order topology is introduced such that in the latter two settings under mild conditions order continuity is a topological property. We present a generalisation of the Ogasawara theorem on the structure of the set of order continuous operators.

Jackson theorem and modulus of continuity in Hilbert spaces and on homogeneous manifolds

We consider a Hilbert space ${\bf H}$ equipped with a set of strongly continuous bounded semigroups satisfying certain conditions. The conditions allow to define a family of moduli of continuity $Ω^{r}(s,f),\>r\in \mathbb{N}, s>0,$ of vectors in ${\bf H}$ and a family of Paley-Wiener subspaces $PW_σ$ parametrized by bandwidth $σ>0$. These subspaces are explored to introduce notion of the best approximation $\mathcal{E}(σ, f)$ of a general vector in ${\bf H}$ by Paley-Wiener vectors of a certain bandwidth $σ>0$. The main objective of the paper is to prove the so-called Jackson-type estimate $\m…

Relatively Uniformly Continuous Semigroups on Vector Lattices

In this paper we study continuous semigroups of positive operators on general vector lattices equipped with the relative uniform topology $τ_{ru}$. We introduce the notions of strong continuity with respect to $τ_{ru}$ and relative uniform continuity for semigroups. These notions allow us to study semigroups on non-locally convex spaces such as $L^p(\mathbb{R})$ for $0<p<1$ and non-complete spaces such as $Lip(\mathbb{R})$, $UC(\mathbb{R})$, and $C_c(\mathbb{R})$. We show that the (left) translation semigroup on the real line, the heat semigroup and some Koopman semigroups are relatively unifo…

Continuity of LF-algebra representations associated to representations of Lie groups

Let G be a Lie group and E be a locally convex topological G-module. If E is sequentially complete, then E and its space of smooth vectors are modules for the algebra D(G) of compactly supported smooth functions on G. However, the module multiplication need not be continuous. The pathology can be ruled out if E is (or embeds into) a projective limit of Banach G-modules. Moreover, in this case the space of analytic vectors is a module for the algebra A(G) of superdecaying analytic functions introduced by Gimperlein, Kroetz and Schlichtkrull. We prove that the space of analytic vectors is a topo…

关于热核的Lipschitz连续性

On the Lipschitz continuity of the heat kernel

本文研究度量测度空间上强连续半群积分核的连续性，利用半群光滑化性质和抽象Morrey型不等式给出Hölder或Lipschitz连续的充分条件，并应用于微分算子、分形上的Laplacian及结构阻尼波动方程。

Extensions of vector-valued functions with preservation of derivatives

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…

Banach空间上线性余切Oseledets子空间的Hölder连续性

Hölder continuity of Oseledets subspaces for linear cocycles on Banach spaces

本文研究了Banach空间上线性余切的Oseledets子空间，在可逆与不可逆情形下证明了其Hölder连续性，推广了已有结果。

Continuity properties of Schrödinger semigroups with magnetic fields

The objects of the present study are one-parameter semigroups generated by Schrödinger operators with fairly general electromagnetic potentials. More precisely, we allow scalar potentials from the Kato class and impose on the vector potentials only local Kato-like conditions. The configuration space is supposed to be an arbitrary open subset of multi-dimensional Euclidean space; in case that it is a proper subset, the Schrödinger operator is rendered symmetric by imposing Dirichlet boundary conditions. We discuss the continuity of the image functions of the semigroup and show local-norm-contin…