Hardy Spaces, Singular Integrals and The Geometry of Euclidean Domains of Locally Finite Perimeter

We study the interplay between the geometry of Hardy spaces and functional analytic properties of singular integral operators (SIO’s), such as the Riesz transforms as well as Cauchy–Clifford and harmonic double-layer operator, on the one hand and, on the other hand, the regularity and geometric properties of domains of locally finite perimeter. Among other things, we give several characterizations of Euclidean balls, their complements, and half-spaces, in terms of the aforementioned SIO’s.     

Tilings,packings, coverings,and the approximation of functions

A packing (resp. covering) ? of a normed space X consisting of unit balls is called completely saturated (resp. completely reduced) if no finite set of its members can be replaced by a more numerous (resp. less numerous) set of unit balls of X without losing the packing property (resp. covering property) of ?. We show that a normed space X admits completely saturated packings with disjoint closed unit balls as well as completely reduced coverings with open unit balls, provided that there exists a tiling of X with unit balls. Completely reduced coverings by open balls are of interest in the context of an approximation theory for continuous real‐valued functions that rests on so‐called controllable coverings of compact metric spaces. The close relation between controllable coverings and completely reduced coverings allows an extension of the approximation theory to non‐compact spaces. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Covering an ellipsoid with equal balls

The thinnest coverings of ellipsoids are studied in the Euclidean spaces of an arbitrary dimension n. Given any ellipsoid, our goal is to find the minimum number of unit balls needed to cover this ellipsoid. A tight asymptotic bound on the logarithm of this number is obtained.     

Weighted reproducing kernels and Toeplitz operators on harmonic Bergman spaces on the real ball

For the standard weighted Bergman spaces on the complex unit ball, the Berezin transform of a bounded continuous function tends to this function pointwise as the weight parameter tends to infinity. We show that this remains valid also in the context of harmonic Bergman spaces on the real unit ball of any dimension. This generalizes the recent result of C. Liu for the unit disc, as well as the original assertion concerning the holomorphic case. Along the way, we also obtain a formula for the corresponding weighted harmonic Bergman kernels.

     

“Potato kugel” for sub-Laplacians

In this note we extend, to the sub-Laplacian setting, a theorem of Aharonov, Schiffer and Zalcman regarding an inverse property for harmonic functions. As a byproduct, a harmonic characterization of the gauge balls is proved, thus extending a theorem of Kuran concerning the Euclidean balls.     

Instability for harmonic foliations on compact homogeneous spaces

In this paper we discuss the instability of harmonic foliations on compact submanifolds immersed in Euclidean spaces and compact homogeneous spaces. We obtain a sufficient condition for a harmonic foliation to be unstable on compact submanifolds in a Euclidean space and on compact isotropy irreducible homogeneous spaces. We also classify compact symmetric spaces which have no non-trivial stable harmonic foliation.     

DW Complexes and Their Underlying Topological Spaces

The naive concept of a DW complex is that of a differential space that can be built up from cells and whose differential structure is defined in terms of differential structures on Euclidean unit closed balls. This concept stems from an analogue in the category of topological spaces: the so-called CW complex. The paper goes as far as investigating the underlying topological space of a DW complex.     

Variational inequalities over Euclidean balls

This paper investigates solution stability of parametric variational inequalities over Euclidean balls in finite dimensional spaces. We provide exact formulas for computing required coderivatives of the normal cone mappings to Euclidean balls via the initial data. On the basis of these formulas, we establish necessary and sufficient conditions for Lipschitzian stability of the solution maps of the aforementioned variational inequalities.     

赋范空间单位球之间的1-Lipshcitz算子

证明了赋范空间单位球之间任意保一的1-Lipshcitz算子在假设像空间是严格凸的,或者算子是满的条件下是定义在全空间上的线性等距算子在单位球上的限制.同时,也给出了这个结果的一些应用,以及当两个赋范空间是严格凸时推广了的结果.     

Exposed and denting points in duals of operator spaces

We study the extremal structure of the dual unit balls of various operator spaces. Mainly, we show that the classes of [w*-] strongly exposed, [w*-] exposed, and denting points in the dual unit balls of spaces of compact operators between Banach spacesX andY are completely — and in a canonical way — determined by the corresponding classes of points in the unit balls of the (bi-)duals of the factor spacesX andY. Applications to the duality of operator spaces and differentiability properties of the norm in operator spaces are given.     

Superrigidity of Hyperbolic Buildings

In this paper, we study the behavior of harmonic maps into complexes with branching differentiable manifold structure. The main examples of such target spaces are Euclidean and hyperbolic buildings. We show that a harmonic map from an irreducible symmetric space of noncompact type other than real or complex hyperbolic into these complexes are non-branching. As an application, we prove rank-one and higher-rank superrigidity for the isometry groups of a class of complexes which includes hyperbolic buildings as a special case.     

Harmonic polynomial morphisms between Euclidean spaces

Two non-existence theorems on harmonic polynomial morphisms between Euclidean spaces have been shown.     

Mazur Sets in Normed Spaces

In this paper we investigate different questions concerning Mazur sets in normed spaces, which point out the close connections between geometric functional analysis and discrete geometry. Motivated by a result of Chen and Lin, we study the relationship between Mazur disks and weak* denting points of the dual unit ball. We prove that the only Mazur sets of the spaces l₁ⁿ are points and closed balls. Finally, a new stability property for the family of all sets which are intersections of closed balls is found.     

From the Kneser–Poulsen conjecture to ball-polyhedra

A very fundamental geometric problem on finite systems of spheres was independently phrased by Kneser [M. Kneser, Einige Bemerkungen über das Minkowskische Flächenmass, Arch. Math. 6 (1955) 382–390] and Poulsen [E.T. Poulsen, Problem 10, Math. Scand. 2 (1954) 346]. According to their well-known conjecture if a finite set of balls in Euclidean space is repositioned so that the distance between the centers of every pair of balls is decreased, then the volume of the union (resp., intersection) of the balls is decreased (resp., increased). In the first half of this paper we survey the state of the art of the Kneser–Poulsen conjecture in Euclidean, spherical as well as hyperbolic spaces with the emphases being on the Euclidean case. Based on that it seems very natural and important to study the geometry of intersections of finitely many congruent balls from the viewpoint of discrete geometry in Euclidean space. We call these sets ball-polyhedra. In the second half of this paper we survey a selection of fundamental results known on ball-polyhedra. Besides the obvious survey character of this paper we want to emphasize our definite intention to raise quite a number of open problems to motivate further research.     

On the Kneser–Poulsen conjecture in elliptic space

The Kneser–Poulsen conjecture claims that if some balls of Euclidean space are rearranged in such a way that the distances between their centers do not increase, then neither does the volume of the union of the balls. A special case of the conjecture, when the balls move continuously in such a way that the distances between the centers (weakly) decrease during the motion, is known to hold not only in Euclidean, but also in spherical and hyperbolic spaces. In the present paper, we show that this theorem cannot be extended to elliptic space by constructing three smoothly moving congruent balls with centers getting closer to one another in such a way that the volume of the union of the balls strictly increase during the motion. In spite of this counterexample, it is true that n + 1 balls in n-dimensional elliptic space cover maximal volume if the distances between the centers are all equal to the diameter π/2 of the space. The second part of the paper is devoted to the proof of this fact.

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The authors were supported by the Hung. Nat. Sci. Found. (OTKA), grant no. T047102 and T037752.     

On the Discrepancy of Point Distributions on Spheres and Hyperbolic Spaces

 Optimal lower bounds are given for the discrepancy of point distributions w.r.t. geodesic balls on spheres and hyperbolic spaces. The mean discrepancy is estimated below by using a non-commutative version of the Fourier transform method developed by Beck for Euclidean spaces.     

Purely periodic β-expansions in the Pisot non-unit case

It is well known that real numbers with a purely periodic decimal expansion are rationals having, when reduced, a denominator coprime with 10. The aim of this paper is to extend this result to beta-expansions with a Pisot base beta which is not necessarily a unit. We characterize real numbers having a purely periodic expansion in such a base. This characterization is given in terms of an explicit set, called a generalized Rauzy fractal, which is shown to be a graph-directed self-affine compact subset of non-zero measure which belongs to the direct product of Euclidean and p-adic spaces.     

On geometric vector fields of Minkowski spaces and their applications

As it is well-known, a Minkowski space is a finite dimensional real vector space equipped with a Minkowski functional F. By the help of its second order partial derivatives we can introduce a Riemannian metric on the vector space and the indicatrix hypersurface S:=F⁻¹(1) can be investigated as a Riemannian submanifold in the usual sense.Our aim is to study affine vector fields on the vector space which are, at the same time, affine with respect to the Funk metric associated with the indicatrix hypersurface. We give an upper bound for the dimension of their (real) Lie algebra and it is proved that equality holds if and only if the Minkowski space is Euclidean. Criteria of the existence is also given in lower dimensional cases. Note that in case of a Euclidean vector space the Funk metric reduces to the standard Cayley-Klein metric perturbed with a nonzero 1-form.As an application of our results we present the general solution of Matsumoto's problem on conformal equivalent Berwald and locally Minkowski manifolds. The reasoning is based on the theory of harmonic vector fields on the tangent spaces as Riemannian manifolds or, in an equivalent way, as Minkowski spaces. Our main result states that the conformal equivalence between two Berwald manifolds must be trivial unless the manifolds are Riemannian.     

Numerical index of some polyhedral norms on the plane

We give explicit formulae for the numerical index of some (real) polyhedral spaces of dimension two. Concretely, we calculate the numerical index of a family of hexagonal norms, two families of octagonal norms and the family of norms whose unit balls are regular polygons with an even number of vertices.     

Dense computability structures

We examine computability structures on a metric space and the relationships between maximal, separable and dense computability structures. We prove that in a computable metric space which has the effective covering property and compact closed balls for a given computable sequence which is a metric basis there exists a unique maximal computability structure which contains that sequence. Furthermore, we prove that each maximal computability structure on a convex subspace of Euclidean space is dense. We also examine subspaces of Euclidean space on which each dense maximal computability structure is separable and prove that spheres, boundaries of simplices and conics are such spaces.