Gähler’s neighbourhood condition for convergence approach spaces

We characterize convergence approach spaces that are approach spaces by generalizing a neighbourhood condition from the category of convergence spaces to the category of convergence approach spaces. We also study this condition in the categories of limit tower spaces and probabilistic convergence spaces.     

BMOA(R,m) and capacity density Bloch spaces on hyperbolic Riemann surfaces

We investigate a number of spaces of functions on Riemann surfaces which are related to Bloch spaces and functions of bounded mean oscillation (BMO). These spaces are defined using properties for the corresponding function spaces on the unit disk in the complex plane, and we show that, in general, different properties lead to different function spaces. We catalogue almost completely the various relationships between these spaces.     

Wavelets and local Triebel–Lizorkin spaces with the Lorentz index

In this paper, we apply wavelets to consider local norm function spaces with the Lorentz index. Triebel–Lizorkin–Lorentz spaces are based on the real interpolation of the Triebel–Lizorkin spaces. Triebel–Lizorkin–Morrey spaces are based on local norm of the Triebel–Lizorkin spaces. We give a unified depict of spaces that include these two kinds of spaces. Each index of the five index spaces represents a property of functions. We prove the wavelet characterization of the Triebel–Lizorkin–Lorentz–Morrey spaces and use such characterization to study some basic properties of these spaces.     

Trudinger–Moser Type Inequalities with Vanishing Weights in the Unit Ball

We introduce local and global generalized Herz spaces. As one of the main results we show that Morrey type spaces and complementary Morrey type spaces are included into the scale of these Herz spaces. We also prove the boundedness of a class of sublinear operators in generalized Herz spaces with application to Morrey type spaces and their complementary spaces, based on the mentioned inclusion.

     

Quasicrystallographic groups on Minkowski spaces

We generalize the quasicrystallographic groups in the sense of Novikov and Veselov from Euclidean spaces to pseudo-Euclidean and affine spaces. We prove that the quasicrystallographic groups on Minkowski spaces whose rotation groups satisfy an additional assumption are projections of crystallographic groups on pseudo-Euclidean spaces. An example shows that the assumption cannot be dropped. We prove that each quasicrystallographic group is a projection of a crystallographic group on an affine space.     

Pervasive and weakly pervasive pre-Riesz spaces

Pervasive pre-Riesz spaces are defined by means of vector lattice covers. To avoid the computation of a vector lattice cover, we give two distinct intrinsic characterizations of pervasive pre-Riesz spaces. We introduce weakly pervasive pre-Riesz spaces and observe that this property can be easily checked in examples. We relate weakly pervasive pre-Riesz spaces to pre-Riesz spaces with the Riesz decomposition property.     

The space of left-invariant metrics on a Lie group up to isometry and scaling

We study the spaces of left-invariant Riemannian metrics on a Lie group up to isometry, and up to isometry and scaling. In this paper, we see that such spaces can be identified with the orbit spaces of certain isometric actions on noncompact symmetric spaces. We also study some Lie groups whose spaces of left-invariant metrics up to isometry and scaling are small.     

Poletsky–Stessin Hardy Spaces on Complex Ellipsoids in $$\mathbb {C}^{n}$$

We study Poletsky–Stessin Hardy spaces on complex ellipsoids in \(\mathbb {C}^{n}\). Different from one variable case, classical Hardy spaces are strictly contained in Poletsky–Stessin Hardy spaces on complex ellipsoids so boundary values are not automatically obtained in this case. We have showed that functions belonging to Poletsky–Stessin Hardy spaces have boundary values and they can be approached through admissible approach regions in the complex ellipsoid case. Moreover, we have obtained that polynomials are dense in these spaces. We also considered the composition operators acting on Poletsky–Stessin Hardy spaces on complex ellipsoids and gave conditions for their boundedness and compactness.     

Points de namioka espaces normants applications à la théorie isométrique de la dualité

We study in this work various aspects of the isometric theory of duality. We show that in wide classes of Banach spaces, dual spaces are characterized by the existence of a retraction fromE″ ontoE. The predual of such spaces is then unique. We study the imbedding of regularly normed spaces into dual spaces. We better the known results on loss of regularity of the norm of dual spaces. We characterize the dual norms on an Asplund space in terms of “bad differentiability”.      

On countable connected locally connected almost regular Urysohn spaces

We construct connected, locally connected, almost regular, countable, Urysohn spaces. This answers a problem of G.X. Ritter. We show that there are 2^c such non-homeomorphic spaces. We also show that there are 2^c non-homeomorphic spaces which are further rigid. We discuss the group of homeomorphisms of such spaces.The following question was raised by G.X. Ritter: Does there exist a countable connected locally connected Urysohn space which is almost regular? We answer this question in the affirmative and in fact, show that not only are there as many as 2^c such spaces but that there are just as many rigid spaces with the same properties. Furthermore we show that every countable Urysohn space is a subspace of such a space. We also prove that every countable group is isomorphic to the group of autohomeomorphisms of some connected locally connected almost regular Urysohn space. Examples are given of groups of order c which can be represented in this manner.     

Multiply Hyperharmonic Functions

We consider multiply hyperharmonic functions on the product space of two harmonic spaces in the sense of Constantinescu and Cornea. Earlier multiply superharmonic and harmonic functions have been studied in Brelot spaces notably by GowriSankaran. Important examples of Brelot spaces are solutions of elliptic differential equations. The theory of general harmonic spaces covers in addition to Brelot spaces also solution of parabolic differential equations. A locally lower bounded function is multiply hyperharmonic on the product space of two harmonic spaces if it is a hyperharmonic function in each variable for every fixed value of the other. We prove similar results as in Brelot spaces, but our approach is different. We study sheaf properties of multiply hyperharmonic functions. Our main theorem states that multiply hyperharmonic functions are lower semicontinuous and satisfy the axiom of completeness with respect to products of relatively compact sets. We also study nearly multiply hyperharmonic functions.     

On quasilinear spaces of convex bodies and intervals

Algebraic systems abstracting properties of convex bodies and intervals, with respect to addition and multiplication by scalars, known as quasilinear spaces, are studied axiomatically. We discuss special quasilinear spaces with group structure called quasivector spaces. We show that every quasivector space is a direct sum of a vector space and a symmetric quasivector space. A complete characterization of symmetric quasivector spaces in the finite dimensional case is given, which permits to reduce computation in quasilinear spaces to computation in familiar vector spaces.     

Differentiable Distance Spaces

The distance function \({\varrho(p, q) ({\rm or} d(p, q))}\) of a distance space (general metric space) is not differentiable in general. We investigate such distance spaces over \({\mathbb{R}^n}\), whose distance functions are differentiable like in case of Finsler spaces. These spaces have several good properties, yet they are not Finsler spaces (which are special distance spaces). They are situated between general metric spaces (distance spaces) and Finsler spaces. We will investigate such curves of differentiable distance spaces, which possess the same properties as geodesics do in Finsler spaces. So these curves can be considered as forerunners of Finsler geodesics. They are in greater plenitude than Finsler geodesics, but they become geodesics in a Finsler space. We show some properties of these curves, as well as some relations between differentiable distance spaces and Finsler spaces. We arrive to these curves and to our results by using distance spheres, and using no variational calculus. We often apply direct geometric considerations.     

Extension conditions for bounded linear and sublinear operators with values in lindenstrauss spaces

We find extension conditions for linear and sublinear operators with values in four classes of spaces: the spaces of continuous functions on a compact set, Lindenstrauss spaces, and their separable parts. We prove that in all these cases the extension property for linear operators implies the extension property for sublinear operators, while in separable spaces the two properties are equivalent.     

On canonical almost geodesic mappings which preserve the weyl projective tensor

We study a partial case of canonical almost geodesic mappings of the first type of spaces with affine connection that preserve Weyl projective curvature tensor and certain other tensors. Main equations under consideration are reduced to a closed Cauchy system type in covariant derivatives. Therefore a general solution to these equations depends on a finite number of constants. We submit an example of above mappings between flat spaces. We establish that projective Euclidean and equiaffine spaces form closed classes of spaces with respect to these mappings.     

Convergence in distribution of random metric measure spaces (Λ-coalescent measure trees)

We consider the space of complete and separable metric spaces which are equipped with a probability measure. A notion of convergence is given based on the philosophy that a sequence of metric measure spaces converges if and only if all finite subspaces sampled from these spaces converge. This topology is metrized following Gromov’s idea of embedding two metric spaces isometrically into a common metric space combined with the Prohorov metric between probability measures on a fixed metric space. We show that for this topology convergence in distribution follows—provided the sequence is tight—from convergence of all randomly sampled finite subspaces. We give a characterization of tightness based on quantities which are reasonably easy to calculate. Subspaces of particular interest are the space of real trees and of ultra-metric spaces equipped with a probability measure. As an example we characterize convergence in distribution for the (ultra-)metric measure spaces given by the random genealogies of the Λ-coalescents. We show that the Λ-coalescent defines an infinite (random) metric measure space if and only if the so-called “dust-free”-property holds.     

Axiomatic theory of Sobolev spaces

We develop an axiomatic approach to the theory of Sobolev spaces on metric measure spaces and we show that this axiomatic construction covers the main known examples (Hajłasz Sobolev spaces, weighted Sobolev spaces, Upper-gradients, etc). We then introduce the notion of variational p-capacity and discuss its relation with the geometric properties of the metric space. The notions of p-parabolic and p-hyperbolic spaces are then discussed.     

Moduli spaces of arrangements of 10 projective lines with quadruple points

We classify moduli spaces of arrangements of 10 lines with quadruple points. We show that moduli spaces of arrangements of 10 lines with quadruple points may consist of more than 2 disconnected components, namely 3 or 4 distinct points. We also present defining equations to those arrangements whose moduli spaces are still reducible after taking quotients of complex conjugations.     

An estimate of the Banach-Mazur distances between Hilbert spaces and Banach spaces

We discuss a geometric property that provides tools for the estimate of the Δ₂ on spaces of Rademacher functions with values on a Banach space for particular sequences of vectors. We obtain in this way a technique to study those Banach spaces that are isomorphic to Hilbert spaces. In particular, an estimate of the Banach-Mazur distance between such these spaces is obtained.     

Straight nearness spaces

Straight spaces are spaces for which a continuous map defined on the space which is uniformly continuous on each set of a finite closed cover is then uniformly continuous on the whole space. Previously, straight spaces have been studied in the setting of metric spaces. In this paper, we present a study of straight spaces in the more general setting of nearness spaces. In a subcategory of nearness spaces somewhat more general than uniform spaces, we relate straightness to uniform local connectedness. We investigate category theoretic situations involving straight spaces. We prove that straightness is preserved by final sinks, in particular by sums and by quotients, and also by completions.