A New Characterization of the Continuous Functions on a Locale

Within the category W of archimedean lattice-ordered groups with weak order unit, we show that the objects of the form C(L), the set of continuous real-valued functions on a locale L, are precisely those which are divisible and complete with respect to a variant of uniform convergence, here termed indicated uniform convergence. We construct the corresponding completion of a W-object A purely algebraically in terms of Cauchy sequences. This completion can be variously described as c³A, the ``closed under countable composition hull of A,' as C(Y_lA), where Y_lA is the Yosida locale of A, and as the largest essential reflection of A.     

Conductor and capacitary inequalities for functions on topological spaces and their applications to Sobolev-type imbeddings

In 1972 the author proved the so-called conductor and capacitary inequalities for the Dirichlet-type integrals of a function on a Euclidean domain. Both were used to derive necessary and sufficient conditions for Sobolev-type inequalities involving arbitrary domains and measures.The present article contains new conductor inequalities for nonnegative functionals acting on functions defined on topological spaces. Sharp capacitary inequalities, stronger than the classical Sobolev inequality, with the best constant and the sharp form of the Yudovich inequality (Soviet Math. Dokl. 2 (1961) 746) due to Moser (Indiana Math. J. 20 (1971) 1077) are found.     

A note on weighted Banach spaces of holomorphic functions

For every open subset G of and for every continuous, strictly positive weight v on G, the Banach space of all the holomorphic functions f on G such that vanishes at infinity on G, endowed with the natural weighted sup-norm, is isomorphic to a closed subspace of the Banach space c₀; hence it is reflexive if and only if it is finite dimensional.Received: 30 September 2002     

Analytic functions of bounded mean oscillation and the Bloch space

In this paper we show that the closure of the space BMOA of analytic functions of bounded mean oscillation in the Bloch spaceB is the image P(U) of space of all continuous functions on the maximal ideal space ofH under the Bergman projection P. It is proved that the radial growth of functions in P(U) is slower than the iterated logarithm studied by Makarov. So some geometric conditions are given for functions in P(U), which we can easily use to construct many Bloch functions not in P(U).     

Characterizations of Besov-type and Triebel-Lizorkin-type spaces via maximal functions and local means

Let s∈R. In this paper, the authors first establish the maximal function characterizations of the Besov-type space with and τ∈[0,∞), the Triebel-Lizorkin-type space with p∈(0,∞), q∈(0,∞] and τ∈[0,∞), the Besov-Hausdorff space with p∈(1,∞), q∈[1,∞) and and the Triebel-Lizorkin-Hausdorff space with and , where t^′ denotes the conjugate index of t∈[1,∞]. Using this characterization, the authors further obtain the local mean characterizations of these function spaces via functions satisfying the Tauberian condition and establish a Fourier multiplier theorem on these spaces. All these results generalize the existing classical results on Besov and Triebel-Lizorkin spaces by taking τ=0 and are also new even for Q spaces and Hardy-Hausdorff spaces.     

Weakly compact operators onH ∞

We prove that a weakly compact operator fromH or any of its even duals into an arbitrary Banach space is uniformly convexifying. By using this, we establish three dicothomies: (1) every operator defined onH or any of its even duals either fixes a copy ofl or factors through a Banach space having the Banach-Saks property; (2) every quotient ofH or any of its even duals either contains a copy ofl or is super-reflexive; (3) every subspace ofL ¹/H ₀ ¹ or any of its even duals either contains a complemented copy ofl ¹ or is super-reflexive.     

Extremal structure and Samuel compactification

Let M be a uniform space and X the Banach space of bounded and uniformly continuous functions from M into R, provided with its supremum norm.The aim of this paper is to discuss the connection between the geometry of X and the nature of M. In particular, we will prove that certain reconstructions of the unit ball of X by means of its extreme points admit a translation in terms of extension of uniformly continuous functions. We also analyze the impact of these properties on the Samuel compactification of M.     

Necessary and sufficient conditions for weighted Orlicz class inequalities for maximal functions and singular integrals. II

This paper continues the investigation of weight problems in Orlicz classes for maximal functions and singular integrals defined on homogeneous type spaces considered in [1].     

Criteria of weighted inequalities in Orlicz classes for maximal functions defined on homogeneous type spaces

The necessary and sufficient conditions are derived in order that a strong type weighted inequality be fulfilled in Orlicz classes for scalar and vector-valued maximal functions defined on homogeneous type spaces. A weak type problem with weights is solved for vector-valued maximal functions.     

Coding into Ramsey sets

In [6] W. T. Gowers formulated and proved a Ramsey-type result which lies at the heart of his famous dichotomy for Banach spaces. He defines the notion of weakly Ramsey set of block sequences of an infinite dimensional Banach space and shows that every analytic set of block sequences is weakly Ramsey. We show here that Gowers’ result follows quite directly from the fact that all G_δ sets are weakly Ramsey, if the Banach space does not contain c₀, and from the fact that all F_σδ sets are weakly Ramsey, in the case of an arbitrary Banach space. We also show that every result obtained by the application of Gowers’ theorem to an analytic set can also be obtained by applying the Theorem to a F_σδ set (or to a G_δ set if the space does not contain c₀). This fact explains why the only known applications of this technique are based on very low-ranked Borel sets (open, closed, F_σ, or G_δ).     

Necessary and sufficient conditions for weighted Orlicz class inequalities for maximal functions and singular integrals. I

Criteria of various weak and strong type weighted inequalities are established for singular integrals and maximal functions defined on homogeneous type spaces in the Orlicz classes.     

Hankel operators in Schatten ideals

We study membership to Schatten ideals S _E , associated with a monotone Riesz–Fischer space E, for the Hankel operators H _f defined on the Hardy space H ²(∂D). The conditions are expressed in terms of regularity of its symbol: we prove that H _f ∈S _E if and only if f∈B _E , the Besov space associated with a monotone Riesz–Fischer space E(dλ) over the measure space (D,dλ) and the main tool is the interpolation of operators. Received: December 17, 1999; in final form: September 25, 2000?Published online: July 13, 2001     

COMPLEMENTED COPIES OF c 0 IN THE SPACE OF PETTIS INTEGRABLE FUNCTIONS

Abstract

Let X be a Banach space containing a copy of c0, then the space of Pettis integrable functions defined from any perfect atomless measure space to X, contains a complemented copy of c0.     

Mixed Norm and Multidimensional Lorentz Spaces

In the last decade, the problem of characterizing the normability of the weighted Lorentz spaces has been completely solved ([16], [7]). However, the question for multidimensional Lorentz spaces is still open. In this paper, we consider weights of product type, and give necessary and sufficient conditions for the Lorentz spaces, defined with respect to the two-dimensional decreasing rearrangement, to be normable. To this end, it is also useful to study the mixed norm Lorentz spaces. Finally, we prove embeddings between all the classical, multidimensional, and mixed norm Lorentz spaces. Research partially supported by KAW 2000.0048 and STINT KU 2002-4025. Research partially supported by Grants MTM2004-02299, 2005SGR00556 and The Swedish Research Council no. 624-2003-571.     

Banach-stone theorems and separating maps

LetC(X,E) andC(Y,F) denote the spaces of continuous functions on the Tihonov spacesX andY, taking values in the Banach spacesE andF, respectively. A linear mapH:C(X,E)→C(Y,F) isseparating iff(x)g(x)=0 for allx inX impliesHf(y)Hg(y)=0 for ally inY. Some automatic continuity properties and Banach-Stone type theorems (i.e., asserting that isometries must be of a certain form) for separating mapsH between spaces of real- and complex-valued functions have already been developed. The extension of such results to spaces of vector-valued functions is the general subject of this paper. We prove in Theorem 4.1, for example, for compactX andY, that a linear isometryH betweenC(X,E) andC(Y,F) is a “Banach-Stone” map if and only ifH is “biseparating (i.e,H andH ⁻¹ are separating). The Banach-Stone theorems of Jerison and Lau for vector-valued functions are then deduced in Corollaries 4.3 and 4.4 for the cases whenE andF or their topological duals, respectively, are strictly convex. Research supported by the Fundació Caixa Castelló, MI/25.043/92     

Densely ball remotal subspaces of C(K)

We call a subspace Y of a Banach space X a DBR subspace if its unit ball By admits farthest points from a dense set of points of X. In this paper, we study DBR subspaces of C(K). In the process, we study boundaries, in particular, the Choquet boundary of any general subspace of C(K). An infinite compact Hausdorff space K has no isolated point if and only if any finite co-dimensional subspace, in particular, any hyperplane is DBR in C(K). As a consequence, we show that a Banach space X is reflexive if and only if X is a DBR subspace of any superspace. As applications, we prove that any M-ideal or any closed *-subalgebra of C(K) is a DBR subspace of C(K). It follows that C(K) is ball remotal in C(K)^**.     

Interpolating varieties for entire functions of minimal type

We obtain both necessary and sufficient conditions for a discrete variety in Cⁿ to be an interpolating variety for entire functions of minimal type.