Analysis on Besov spaces II: Embedding and duality theorems

This paper gives several results on Besov spaces of holomorphic functions on a very large class of domains D in Cn. They include duality theorem, embedding theorem, best growth estimate, and boundedness of multiplication operators on Besov spaces.     

Hardy spaces for the strip

In this paper we shall study Hardy spaces of analytic functions in a strip S. Our main result is on one hand an intrinsic characterization of the spaces and on the second that polynomials are dense. We also present an orthogonal (in H²(S)) basis of polynomials.     

Lim's theorems for multivalued mappings in CAT(0) spaces

Let X be a complete CAT(0) space. We prove that, if E is a nonempty bounded closed convex subset of X and a nonexpansive mapping satisfying the weakly inward condition, i.e., there exists p∈E such that ∀x∈E, ∀α∈[0,1], then T has a fixed point. In Banach spaces, this is a result of Lim [On asymptotic centers and fixed points of nonexpansive mappings, Canad. J. Math. 32 (1980) 421-430]. The related result for unbounded R-trees is given.     

Tightness and distinguished Fréchet spaces

Valdivia invented a nondistinguished Fréchet space whose weak bidual is quasi-Suslin but not K-analytic. We prove that Grothendieck/Köthe's original nondistinguished Fréchet space serves the same purpose. Indeed, a Fréchet space is distinguished if and only if its strong dual has countable tightness, a corollary to the fact that a (DF)-space is quasibarrelled if and only if its tightness is countable. This answers a Cascales/K?kol/Saxon question and leads to a rich supply of (DF)-spaces whose weak duals are quasi-Suslin but not K-analytic, including the spaces Cc(κ) for κ a cardinal of uncountable cofinality. Our level of generality rises above (DF)- or even dual metric spaces to Cascales/Orihuela's class G. The small cardinals b and d invite a novel analysis of the Grothendieck/Köthe example, and are useful throughout.     

Multilinear commutators of Calderón-Zygmund operators on Hardy-type spaces with non-doubling measures

Under the assumption that μ is a non-negative Radon measure on Rd which only satisfies some growth condition, the authors obtain the boundedness in some Hardy-type spaces of multilinear commutators generated by Calderón-Zygmund operators or fractional integrals with RBMO(μ) functions, where the Hardy-type spaces are some appropriate subspaces, associated to the considered RBMO(μ) functions, of the Hardy space H¹(μ) of Tolsa.     

Lushness, numerical index one and duality

We construct an example of a Banach space which is not lush, but whose dual space is lush. This example shows that lushness is not equivalent to numerical index one. A characterization of lushness for some quotient spaces of L₁(μ) spaces and new results on C-rich subspaces of (scalar- or vector-valued) C(K) spaces are also presented.     

The role of directional compactness in the existence and computation of contingent epiderivatives

This paper underlines the role of directional compactness in the scalarization of graphical derivatives of set-valued maps taking values in infinite-dimensional spaces. Two main theorems are given. The first one states the equivalence of contingent epiderivatives and τw-contingent epiderivatives for directionally compact maps. The second main result proves a variational characterization for the contingent epiderivative of stable and directionally compact maps taking values in general image spaces, extending known results in finite-dimensional and reflexive Banach spaces. The hypotheses given are minimal as is shown by means of several examples. Connections of these theorems with other results of the literature are also provided.     

On the Alexandrov problem of distance preserving mapping

In this paper the author has studied the Alexandrov problem of area preserving mappings in linear 2-normed spaces and has provided some remarks for the generalization of earlier results of H.Y. Chu, C.G. Park and W.G. Park.In addition the author has introduced the concept of linear (2,p)-normed spaces and for such spaces he has solved the Alexandrov problem.     

Growth conditions, compact perturbations and operator subdecomposability, with applications to generalized Cesàro operators

We adapt recent results of Albrecht and Ricker to obtain conditions under which growth constraints on the left resolvent of a Banach space operator are preserved under suitable perturbations. As an application, we establish Bishop's property (β) for certain generalized Cesàro operators on the classical Hardy spaces Hp, 1<p<∞. Our methods also apply to unilateral weighted shifts whose weight sequence converges sufficiently rapidly as well as to perturbations of restrictions of a class of generalized scalar operators.     

On some iterated weighted spaces

It is proved that the Hörmander and spaces (Ω₁⊂Rn, Ω₂⊂Rm open sets, 1?p<∞, ki Beurling-Björck weights, k=k₁⊗k₂) are isomorphic whereas the iterated spaces and are not if 1<p≠q<∞. A similar result for weighted Lp-spaces of entire analytic functions is also obtained. Finally a result on iterated Besov spaces is given: and are not isomorphic when 1<q≠2<∞.     

Means in metric spaces and the center of mass

In this article k-convex metric spaces are considered where a several variable mapping is provided as a limit point of an iteration scheme based on the midpoint map in the metric space itself. This mapping, considered as a mean of its variables, has some properties which relates it to the center of mass of these variables in the metric space. Sufficient conditions are given here for the two points to be identical, as well as upper bounds on their distances from one another. The asymptotic rate of convergence of the iterative process defining the mean is also determined here. The case of the symmetric space on the convex cone of positive definite matrices related to the geometric mean and the special orthogonal group are also studied here as examples of k-convex metric spaces.     

On operator valued sequences of multipliers and R-boundedness

In recent papers (cf. [J.L. Arregui, O. Blasco, (p,q)-Summing sequences, J. Math. Anal. Appl. 274 (2002) 812-827; J.L. Arregui, O. Blasco, (p,q)-Summing sequences of operators, Quaest. Math. 26 (2003) 441-452; S. Aywa, J.H. Fourie, On summing multipliers and applications, J. Math. Anal. Appl. 253 (2001) 166-186; J.H. Fourie, I. Röntgen, Banach space sequences and projective tensor products, J. Math. Anal. Appl. 277 (2) (2003) 629-644]) the concept of (p,q)-summing multiplier was considered in both general and special context. It has been shown that some geometric properties of Banach spaces and some classical theorems can be described using spaces of (p,q)-summing multipliers. The present paper is a continuation of this study, whereby multiplier spaces for some classical Banach spaces are considered. The scope of this research is also broadened, by studying other classes of summing multipliers. Let E(X) and F(Y) be two Banach spaces whose elements are sequences of vectors in X and Y, respectively, and which contain the spaces c₀₀(X) and c₀₀(Y) of all X-valued and Y-valued sequences which are eventually zero, respectively. Generally spoken, a sequence of bounded linear operators (un)⊂L(X,Y) is called a multiplier sequence from E(X) to F(Y) if the linear operator from c₀₀(X) into c₀₀(Y) which maps (xi)∈c₀₀(X) onto (unxn)∈c₀₀(Y) is bounded with respect to the norms on E(X) and F(Y), respectively. Several cases where E(X) and F(Y) are different (classical) spaces of sequences, including, for instance, the spaces Rad(X) of almost unconditionally summable sequences in X, are considered. Several examples, properties and relations among spaces of summing multipliers are discussed. Important concepts like R-bounded, semi-R-bounded and weak-R-bounded from recent papers are also considered in this context.     

Numerical index of absolute sums of Banach spaces

We study the numerical index of absolute sums of Banach spaces, giving general conditions which imply that the numerical index of the sum is less or equal than the infimum of the numerical indices of the summands and we provide some examples where the equality holds covering the already known case of c₀-, ?₁- and ?_∞-sums and giving as a new result the case of E-sums where E has the RNP and n(E)=1 (in particular for finite-dimensional E with n(E)=1). We also show that the numerical index of a Banach space Z which contains a dense union of increasing one-complemented subspaces is greater or equal than the limit superior of the numerical indices of those subspaces. Using these results, we give a detailed short proof of the already known fact that, for a fixed p, the numerical indices of all infinite-dimensional Lp(μ)-spaces coincide.     

Ball remotality in Banach spaces

In this paper, we continue our study of ball remotality of subspaces of Banach spaces. In particular, we study the problem in classical sequence spaces c₀, c, ?₁ and ?_∞; and also ball remotality of a Banach space in its bidual.     

Fixed point theorems for better admissible multimaps on almost convex sets

We obtain new fixed point theorems on multimaps in the class Bp defined on almost convex subsets of topological vector spaces. Our main results are applied to deduce various fixed point theorems, coincidence theorems, almost fixed point theorems, intersection theorems, and minimax theorems. Consequently, our new results generalize well-known works of Kakutani, Fan, Browder, Himmelberg, Lassonde, and others.     

A counterexample to Wood's conjecture

We prove that if the one-point compactification of a locally compact, noncompact Hausdorff space L is the topological space called pseudoarc, then C₀(L,C) is almost transitive. We also obtain two necessary conditions on a metrizable locally compact Hausdorff space L for C₀(L) being almost transitive.     

Tractability results for weighted Banach spaces of smooth functions

We study the L_∞-approximation problem for weighted Banach spaces of smooth d-variate functions, where d can be arbitrarily large. We consider the worst case error for algorithms that use finitely many pieces of information from different classes. Adaptive algorithms are also allowed. For a scale of Banach spaces we prove necessary and sufficient conditions for tractability in the case of product weights. Furthermore, we show the equivalence of weak tractability with the fact that the problem does not suffer from the curse of dimensionality.     

The quantitative difference between countable compactness and compactness

We establish here some inequalities between distances of pointwise bounded subsets H of RX to the space of real-valued continuous functions C(X) that allow us to examine the quantitative difference between (pointwise) countable compactness and compactness of H relative to C(X). We prove, amongst other things, that if X is a countably K-determined space the worst distance of the pointwise closure of H to C(X) is at most 5 times the worst distance of the sets of cluster points of sequences in H to C(X): here distance refers to the metric of uniform convergence in RX. We study the quantitative behavior of sequences in H approximating points in . As a particular case we obtain the results known about angelicity for these Cp(X) spaces obtained by Orihuela. We indeed prove our results for spaces C(X,Z) (hence for Banach-valued functions) and we give examples that show when our estimates are sharp.     

The inclusion relation between Sobolev and modulation spaces

The inclusion relations between the Lp-Sobolev spaces and the modulation spaces is determined explicitly. As an application, mapping properties of unimodular Fourier multiplier eiα|D| between Lp-Sobolev spaces and modulation spaces are discussed.     

Dirichlet and VMOA domains via Schwarzian derivative

Short proofs of the following results concerning a bounded conformal map g of the unit disc D are presented: (1) logg^′ belongs to the Dirichlet space if and only if the Schwarzian derivative Sg of g satisfies Sg(z)(1−²|z|)∈L²(D); (2) logg^′∈VMOA if and only if ²|Sg(z)|³(1−²|z|) is a vanishing Carleson measure on D. Analogous results for Besov and Qp,0 spaces are also given.