Spaces of Densely Continuous Forms, USCO and Minimal USCO Maps

Our paper studies the topology of uniform convergence on compact sets on the space of densely continuous forms (introduced by Hammer and McCoy (1997)), usco and minimal usco maps. We generalize and complete results from Hammer and McCoy (1997) concerning the space D(X,Y) of densely continuous forms from X to Y. Let X be a Hausdorff topological space, (Y,d) be a metric space and D _k (X,Y) the topology of uniform convergence on compact sets on D(X,Y). We prove the following main results: D _k (X,Y) is metrizable iff D _k (X,Y) is first countable iff X is hemicompact. This result gives also a positive answer to question 4.1 of McCoy (1998). If moreover X is a locally compact hemicompact space and (Y,d) is a locally compact complete metric space, then D _k (X,Y) is completely metrizable, thus improving a result from McCoy (1998). We study also the question, suggested by Hammer and McCoy (1998), when two compatible metrics on Y generate the same topologies of uniform convergence on compact sets on D(X,Y). The completeness of the topology of uniform convergence on compact sets on the space of set-valued maps with closed graphs, usco and minimal usco maps is also discussed.     

Densely continuous forms, pointwise topology and cardinal functions

We consider the space D(X, Y) of densely continuous forms introduced by Hammer and McCoy [5] and investigated also by Holá [6]. We show some additional properties of D(X, Y) and investigate the subspace D*(X) of locally bounded real-valued densely continuous forms equipped with the topology of pointwise convergence τ- _p . The largest part of the paper is devoted to the study of various cardinal functions for (D*(X), τ _p ), in particular: character, pseudocharacter, weight, density, cellularity, diagonal degree, π-weight, π-character, netweight etc. This work was supported by Science and Technology Assistance Agency under the contract No. APVT-51-006904. The authors would like to thank Ľubica Holá for suggestions and comments.     

Densely Continuous Forms in Vietoris Hyperspaces

For countably paracompact normal spaces X and locally compact separable metric spaces Y, a characterization is given for the closure of the set of densely continuous forms from X to Y in the hyperspace of nonempty closed subsets of X × Y under the Vietoris topology. This shows that for such X having no isolated points, every closed subset of X × R that is dense over X can be Vietoris approximated by a semicontinuous function on X.     

Some notes on densely continuous forms

We consider a special space of set-valued functions (multifunctions), the space of densely continuous forms D(X, Y) between Hausdorff spaces X and Y, defined in [HAMMER, S. T.—McCOY, R. A.: Spaces of densely continuous forms, Set-Valued Anal. 5 (1997), 247–266] and investigated also in [HOLá, L’.: Spaces of densely continuous forms, USCO and minimal USCO maps, Set-Valued Anal. 11 (2003), 133–151]. We show some of its properties, completing the results from the papers [HOLY, D.—VADOVIČ, P.: Densely continuous forms, pointwise topology and cardinal functions, Czechoslovak Math. J. 58(133) (2008), 79–92] and [HOLY, D.—VADOVIČ, P.: Hausdorff graph topology, proximal graph topology and the uniform topology for densely continuous forms and minimal USCO maps, Acta Math. Hungar. 116 (2007), 133–144], in particular concerning the structure of the space of real-valued locally bounded densely continuous forms D _p ^*(X) equipped with the topology of pointwise convergence in the product space of all nonempty-compact-valued multifunctions. The paper also contains a comparison of cardinal functions on D _p ^*(X) and on real-valued continuous functions C _p (X) and a generalization of a sufficient condition for the countable cellularity of D _p ^*(X). This work was supported by Science and Technology Assistance Agency under the contract No. APVT-51-006904 and by the Eco-Net (EGIDE) programme of the Laboratoire de Mathématiques de l’Université de Saint-Etienne (LaMUSE), France.     

Uniform convergence on spaces of multifunctions

We study spaces of multifunctions with closed values, multifunctions with closed graphs, USCO multifunctions, minimal USCO multifunctions and the space of densely continuous forms as metric spaces, equipped with the topology of uniform convergence. We give conditions under which these metric spaces are complete.      

Poisson Brackets of Even Symplectic Forms on the Algebra of Differential Forms

Given a symplectic form and a pseudo-Riemannian metric on a manifold, a nondegenerate even Poisson bracket on the algebra of differential forms is defined and its properties are studied. A comparison with the Koszul–Schouten bracket is established.     

有界闭区间上连续函数列一致收敛的充要条件

判别函数列一致收敛的方法有函数列一致收敛定义、Cauchy一致收敛准则、limn→∞supx∈D|fn(x)-f(x)|=0及Dini定理,本文由函数列的等度连续性,可得出几个有界闭区间上连续函数列一致收敛的充要条件,推广了Dini定理.     

Tessellations of Homogeneous Spaces of Classical Groups of Real Rank Two

Let H be a closed, connected subgroup of a connected, simple Lie group G with finite center. The homogeneous space G/H has a tessellation if there is a discrete subgroup of G, such that acts properly discontinuously on G/H, and the double-coset space \G/H is compact. Note that if either H or G/H is compact, then G/H has a tessellation; these are the obvious examples.It is not difficult to see that if G has real rank one, then only the obvious homogeneous spaces have tessellations. Thus, the first interesting case is when G has real rank two. In particular, Kulkarni and Kobayashi constructed examples that are not obvious when G=SO(2, 2n)° or SU(2, 2n). Oh and Witte constructed additional examples in both of these cases, and obtained a complete classification when G=SO(2, 2n)°. We simplify the work of Oh-Witte, and extend it to obtain a complete classification when G=SU(2, 2n). This includes the construction of another family of examples.The main results are obtained from methods of Benoist and Kobayashi: we fix a Cartan decomposition G=K A ⁺ K, and study the intersection (K H K)A ⁺. Our exposition generally assumes only the standard theory of connected Lie groups, although basic properties of real algebraic groups are sometimes also employed; the specialized techniques that we use are developed from a fairly elementary level.     

抽象度量空间上的一致连续函数空间

本文主要讨论抽象度量空间上的一致连续函数空间的Banach空间结构,代数结构和格结构．     

Musielak-Orlicz-Sobolev空间中的序连续性质和最佳逼近算子的连续性质

借鉴Orlicz-Sobolev空间中的受控最佳逼近算子问题的研究结果,抓住Musielak-Orlicz-Sobolev空间的构成特点,利用△_2条件及其否定,给出了MusielakOrlicz-Sobolev空间具有序连续性的充要条件,同时研究了该空间中最佳逼近算子的连续性.     

Mappings on Spaces of Continuous Functions

Let X and Y be locally compact Hausdorff spaces and T : C₀(X) C₀(Y) a ring homomorphism. We completely characterize such homomorphisms and show that if T is R-linear, then T is either C-linear or C-antilinear. In any case T is continuous and there is a continuous map : Y X such that Tf = f o , f C₀(X) (if T is C-linear) or (if T is C-antilinear). Thus, extending a result of Mólnar, we also derive the general form of an isometry T.AMS Subject Classification (2000): primary 46J05, 46E25(deceased) Passed away on 24 May 1999.     

Continuous Function Characterizations of Stratifiable Spaces

The insertion of a continuous function between pairs of semicontinuous functions in a monotone manner is investigated. In particular, it is established that a topological space is stratifiable if and only if for every pair of semicontinuous functions......     

Regularity of Continuous Linear Operators on Banach Function Spaces

In this paper we present some characterizations of Banach function spaces on which every continuous linear operator is regular.     

Bilipschitz Embeddings of Metric Spaces into Space Forms

The paper describes some basic geometric tools to construct bilipschitz embeddings of metric spaces into (finite-dimensional) Euclidean or hyperbolic spaces. One of the main results implies the following: If X is a geodesic metric space with convex distance function and the property that geodesic segments can be extended to rays, then X admits a bilipschitz embedding into some Euclidean space iff X has the doubling property, and X admits a bilipschitz embedding into some hyperbolic space iff X is Gromov hyperbolic and doubling up to some scale. In either case the image of the embedding is shown to be a Lipschitz retract in the target space, provided X is complete.     

Function Spaces as Path Spaces of Feller Processes

Let {X_t}_{t ≥ 0} be a Feller process with infinitesimal generator (A, D(A)). If the test functions are contained in D(A), —A |C^∞_c (ℝⁿ) is a pseudo–differential operator p(x, D) withsymbol p(x, ξ). We investigate local and global regularity properties of the sample paths t ↦ X_t in terms of (weighted) Besov B^s_pq (ℝ, ρ) and Triebel–Lizorkin F^s_pq (ℝ, ρ) spaces. The parameters for these spaces are determined by certain indices that describe the asymptotic behaviour of the symbol p(x, ξ). Our results improve previous papers on Lévy [5, 9] and Feller processes [22].     

Mapping Properties of the Laplacian in Sobolev Spaces of Forms on Complete Hyperbolic Manifolds

For a complete manifold M with constant negative curvature, weprove that the rough Laplacian _R defines topological isomorphisms in the scale of Sobolev spaces H _p ^s (M) ofp-forms for all p, 0 < p< n. For the de Rham Laplacian and M= ⁿ , the Poincaréhyperbolic space, this is shown too for 0 pn,pn/2, p (n± 1)/2.     

Local Intersection Cohomology of Baily�CBorel Compactifications

The local intersection cohomology of a point in the Baily–Borel compactification (of a Hermitian locally symmetric space) is shown to be canonically isomorphic to the weighted cohomology of a certain linear locally symmetric space (an arithmetic quotient of the associated self-adjoint homogeneous cone). Explicit computations are given for the symplectic group in four variables.     

Convergence of Discrete Dirichlet Forms to Continuous Dirichlet Forms on Fractals

It is well known that a Dirichlet form on a fractal structure can be defined as the limit of an increasing sequence of discrete Dirichlet forms, defined on finite subsets which fill the fractal. The initial form is defined on V ⁽⁰⁾, which is a sort of boundary of the fractal, and we have to require that it is an eigenform, i.e., an eigenvector of a particular nonlinear renormalization map for Dirichlet forms on V ⁽⁰⁾. In this paper, I prove that, provided an eigenform exists, even if the form on V ⁽⁰⁾ is not an eigenform, the corresponding sequence of discrete forms converges to a Dirichlet form on all of the fractal, both pointwise and in the sense of -convergence (but these two limits can be different). The problem of -convergence was first studied by S. Kozlov on the Gasket.     

On the Volume of Flowers in Space Forms

Let f(x ₁,..., x _N ) be a lattice polynomial in N variables, in which each variable occurs exactly once, B ₁,..., B _N be smoothly moving balls in the hyperbolic, Euclidean, or spherical space. Introducing a suitable modification of the Dirichlet–Voronoi decomposition, we prove a formula for the derivative of the volume of the domain f(B ₁,..., B _N ). As an application of the formula, we show that the volume increases if the balls move continuously in such a way that the functions _ij d _ij increase for all 1 i < j N, where _ij is a sign depending on f, d _ij is the distance between the centers of B _I and B _j .