Tractable embeddings of Besov spaces into small Lebesgue spaces

This paper deals with dimension‐controllable (tractable) embeddings of Besov spaces on n‐dimensional torus into small Lebesgue spaces. Our techniques rely on the approximation structure of Besov spaces, extrapolation properties of small Lebesgue spaces and interpolation.     

ConvergenceLF spaces

In this paper we examineLF spaces, inductive limits of Fréchet spaces, in two different settings: the categoryCV S of convergence vector spaces and the categoryLC S of locally convex topological vector spaces. Special attention is given to permanence properties and retractivity properties in each case. Some interaction between properties ofLF spaces inCV S and other properties inLC S are investigated.R. Beattie's research was supported by NSERC grant OGP0005316.     

Metric tangent spaces to Euclidean spaces

We prove that the metric spaces pretangent to a finite-dimensional Euclidean or unitary space E are isometric to E. As a consequence of this result, we describe the metric pretangent spaces at the nonsingular points of smooth surfaces. It is also proved that there exist the spaces pretangent to the Hilbert space l ₂ , which are not isometric to it.     

Ultrametric spaces and related Hilbert spaces

We present simple proofs of the possibility of embedding ultrametric spaces in Hilbert spaces. The main part of the paper deals with ultrametric spaces that we call totally infinite spaces. Related Hilbert spaces, automorphisms of totally infinite spaces, and the corresponding linear operators are considered. Transplated fromMatematicheskie Zametki, Vol. 62, No. 2, pp. 223–237, August, 1997. Translated by V. E. Nazaikinskii     

S-paracompact spaces

Summary We introduce the class of S-paracompact spaces as a generalization of paracompact spaces. A space (X,T) is S-paracompact if every open cover of X has a locally finite semi-open refinement. We characterize S-paracompact spaces and study their basic properties. The relationships between S-paracompact spaces and other well-known spaces are investigated.     

Embeddings of anisotropic Besov spaces into Sobolev spaces

We study the embeddings of (homogeneous and inhomogeneous) anisotropic Besov spaces associated to an expansive matrix A into Sobolev spaces, with a focus on the influence of A on the embedding behavior. For a large range of parameters, we derive sharp characterizations of embeddings.     

Generalized Hardy spaces

Hardy spaces with generalized parameter are introduced following the maximal characterization approach. As particular cases, they include the classical Hp spaces and the Hardy-Lorentz spaces H^p,q. Real interpolation results with function parameter are obtained, Based on them, the behavior of some classical operators is studied in this generalized setting.     

Natural examples of Valdivia compact spaces

We collect examples of Valdivia compact spaces, their continuous images and associated classes of Banach spaces which appear naturally in various branches of mathematics. We focus on topological constructions generating Valdivia compact spaces, linearly ordered compact spaces, compact groups, L¹ spaces, Banach lattices and noncommutative L¹ spaces.     

Characterizations of Banach spaces whose duals areL 1 spaces

We characterize those Banach spaces whose duals are isometric toL ₁ spaces in terms of the structure of the spaces of absolutely summing, integral, nuclear, and fully nuclear operators from or into these spaces.     

Strongly irreducible operators on Banach spaces

This paper firstly discusses the existence of strongly irreducible operators on Banach spaces. It shows that there exist strongly irreducible operators on Banach spaces with w*-separable dual. It also gives some properties of strongly irreducible operators on Banach spaces. In particular, if T is a strongly irreducible operator on an infinite-dimensional Banach space, then T is not of finite rank and T is not an algebraic operator. On Banach spaces with subsymmetric bases, including infinite-dimensional separable Hilbert spaces, it shows that quasisimilarity does not preserve strong irreducibility. In addition, we show that the strong irreducibility of an operator does not imply the strong irreducibility of its conjugate operator, which is not the same as the property in Hilbert spaces.     

Hilbert-generated spaces

We classify several classes of the subspaces of Banach spaces X for which there is a bounded linear operator from a Hilbert space onto a dense subset in X. Dually, we provide optimal affine homeomorphisms from weak star dual unit balls onto weakly compact sets in Hilbert spaces or in c₀(Γ) spaces in their weak topology. The existence of such embeddings is characterized by the existence of certain uniformly Gâteaux smooth norms.     

Counting metric spaces

If κ is a cardinal number, then any class of mutually non-homeomorphic metric spaces of size κ must be a set whose cardinality cannot exceed 2 ^κ . Our main result is a vivid construction of 2 ^κ mutually non-homeomorphic complete and both path connected and locally path connected metric spaces of size κ for each cardinal number κ from continuum up. Additionally we also deal with counting problems concerning countable metric spaces and Euclidean spaces.     

SeparableG spaces are isomorphic toC(K) spaces

We show that a wide class of separable preduals ofL ¹(μ) spaces, namely, theG spaces, introduced by Grothendieck, are isomorphic toC(K) spaces. This is a part of the author’s Ph.D. thesis prepared at the Hebrew University of Jerusalem under the supervision of Professor J. Lindenstrauss. I wish to thank Professor Lindenstrauss for his interest and advice.     

Uniqueness of unconditional bases in quasi-banach spaces with applications to Hardy spaces

We prove some general results on the uniqueness of unconditional bases in quasi-Banach spaces. We show in particular that certain Lorentz spaces have unique unconditional bases answering a question of Nawrocki and Ortynski. We then give applications of these results to Hardy spaces by showing the spacesH _p (T ⁿ ) are mutually non-isomorphic for differing values ofn when 0<p<1. The research of the first two authors was partially supported by NSF-grant DMS 8901636.     

The Fourier integral operators on hardy spaces associated with Herz spaces

In this paper, it is proved that the Fourier integral operators of order m, with −n < m ⩽ −(n−1)/2, are bounded from three kinds of Hardy spaces associated with Herz spaces to their corresponding Herz spaces.     

New hardy spaces associated with some anisotropic Herz spaces and their applications

In this paper, a class of anisotropic Herz-type Hardy spaces associated with a non-isotropic dilation on ℝ ⁿ are introduced, and the central atomic and molecular decomposition characterizations of those spaces are established. As some applications of the decomposition theory, the authors study the interpolation problem and the boundedness of the central δ-Calderón-Zygmund operators on the anisotropic Herz-type Hardy spaces. The research is supported by NSF of China (Grant Nos. 10571014 and 10571015) and SRFDP of China (Grant No. 20050027025)     

On K-groups of operator algebras on HD n spaces and QD n spaces

We obtain the K-groups of the operator ideals contained in the class of Riesz operators. And based on the results, we calculate the K-groups of the operator algebras on HD nspaces and QDn spaces.     

Filter spaces

The category FIL of filter spaces and cauchy maps is a topological universe. This paper establishes the foundation for a completion theory forT ₂ filter spaces.     

Onm-separated projection spaces

Closure operators in the category of projection spaces are investigated. It is shown that completeness, absolutes-closure ands-injectivity coincide in the subcategory of separated projection spaces and that there compactness with respect to projections implies completeness.Dedicated to Nico Pumplün on the occasion of his 60th birthdayPartial financial support by the Italian Ministry of Public Education is gratefully acknowledged.     

Onm-compact spaces

We give characterizations ofm-compact andm-closed spaces by means of nets and filterbases. We study the images ofm-compact andm-closed sets under various modifications ofM-continuous functions. We introduce the notion ofmc-compact space and investigate some relations betweenmc-compact spaces andM-continuous functions.