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.     

Variable Lebesgue norm estimates for BMO functions

In this paper, we are going to characterize the space BMO(? ⁿ ) through variable Lebesgue spaces and Morrey spaces. There have been many attempts to characterize the space BMO(? ⁿ ) by using various function spaces. For example, Ho obtained a characterization of BMO(? ⁿ ) with respect to rearrangement invariant spaces. However, variable Lebesgue spaces and Morrey spaces do not appear in the characterization. One of the reasons is that these spaces are not rearrangement invariant. We also obtain an analogue of the well-known John-Nirenberg inequality which can be seen as an extension to the variable Lebesgue spaces.     

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.     

Hardy space and Bergman space on the Octonions

In this paper, square-integrable Octonion-valued function spaces on R³ and R⁷ are decomposed into the direct sum of Octonion Hardy and conjugate Hardy spaces, and square-integrable Octonion function spaces on the upper half spaces R ₊ ⁴ and R ₊ ⁸ are decomposed into infinity direct sum of subspaces in which the first components are just the Octonion Bergman spaces. The Octonion Hardy Spaces on R³ and R⁷ are characterized.     

New abstract Hardy spaces

The aim of this paper is to propose an abstract construction of spaces which keep the main properties of the (already known) Hardy spaces H¹. We construct spaces through an atomic (or molecular) decomposition. We prove some results about continuity from these spaces into L¹ and some results about interpolation between these spaces and the Lebesgue spaces. We also obtain some results on weighted norm inequalities. Finally we present partial results in order to understand a characterization of the duals of Hardy spaces.     

INTERPOLATION SPACES BETWEEN H^1 AND L^∞ ON SPACES OF HOMOGENEOUS TYPE

Using the maximal function characterization of Hardy spaces,we study the interpolation spaces between H^1 and L^∞on spaces of homogeneous type.     

Sharp sobolev embeddings and related hardy inequalities: The critical case

The paper deals with sharp embeddings of the Sobolev spaces H^s_p(IRⁿ) and the Besov spaces B^s_p,p(IRⁿ) into rearrangement—invariant spaces and related Hardy inequalities. Here 1 < p < ∞ and s = n/p.     

Hardy spaces for α-harmonic functions in regular domains

We investigate Hardy spaces H ^p for singular α-harmonic functions in bounded domains with regular boundaries. We show the correspondence between these spaces and suitable L ^p spaces and measure spaces.     

Hausdorff Operators on Function Spaces

The authors mainly study the Hausdorff operators on Euclidean space Rn.They establish boundedness of the Hausdorff operators in various function spaces,such as Lebesgue spaces,Hardy spaces,local Hardy ...     

Double solid twistor spaces: the case of arbitrary signature

In a recent paper ([9]) we constructed a series of new Moishezon twistor spaces which are a kind of variant of the famous LeBrun twistor spaces. In this paper we explicitly give projective models of another series of Moishezon twistor spaces on n CP ² for arbitrary n≥3, which can be regarded as a generalization of the twistor spaces of ‘double solid type’ on 3CP ² studied by Kreußler, Kurke, Poon and the author. Similarly to the twistor spaces of ‘double solid type’ on 3CP ², projective models of the present twistor spaces have a natural structure of double covering of a CP ²-bundle over CP ¹. We explicitly give a defining polynomial of the branch divisor of the double covering, whose restriction to fibers is degree four. If n≥4 these are new twistor spaces, to the best of the author’s knowledge. We also compute the dimension of the moduli space of these twistor spaces. Differently from [9], the present investigation is based on analysis of pluri-(half-)anticanonical systems of the twistor spaces.     

Isometries between spaces of homogeneous polynomials

We derive Banach-Stone theorems for spaces of homogeneous polynomials. We show that every isometric isomorphism between the spaces of homogeneous approximable polynomials on real Banach spaces E and F is induced by an isometric isomorphism of E^′ onto F^′. With an additional geometric condition we obtain the analogous result in the complex case. Isometries between spaces of homogeneous integral polynomials and between the spaces of all n-homogeneous polynomials are also investigated.     

A maximal function characterization of Hardy spaces on spaces of homogeneous type

A new maximal funtion is introduced in the dual spaces of test function spaces on spaces of homogeneous type. Using this maximal function, we get new characterization of atomic H^p spaces. This work is supported by NSF.     

LOCAL COMPACTNESS IN SEMI-UNIFORM CONVERGENCE SPACES

Abstract

Local compactness is studied in the highly convenient setting of semi-uniform convergence spaces which form a common generalization of (symmetric) limit spaces (and thus of symmetric topological spaces) as well as of uniform limit spaces (and thus of uniform spaces). It turns out that it leads to a cartesian closed topological category and, in contrast to the situation for topological spaces, the local compact spaces are exactly the compactly generated spaces. Furthermore, a one-point Hausdorff compactification for noncompact locally compact Hausdorff convergence spaces is considered.¹     

Hankel operators on the Paley-Wiener space in ℝ d

The basic theory of Besov spaces inI ^d of Paley-Wiener type is developed. This kind of Besov spaces turns out to be quite a success to characterize the Schatten-von Neumann ideal criteria for Hankel operators acting on Paley-Wiener spaces inI ^d.     

Proper forcing axiom and selective separability

We continue the study of Selectively Separable (SS) and, a game-theoretic strengthening, strategically selectively separable spaces (SS⁺) (see Barman, Dow (2011) [1]). The motivation for studying SS⁺ is that it is a property possessed by all separable subsets of Cp(X) for each σ-compact space X. We prove that the winning strategy for countable SS⁺ spaces can be chosen to be Markov. We introduce the notion of being compactlike for a collection of open sets in a topological space and with the help of this notion we prove that there are two countable SS⁺ spaces such that the union fails to be SS⁺, which contrasts the known result about SS spaces. We also prove that the product of two countable SS⁺ spaces is again countable SS⁺. One of the main results in this paper is that the proper forcing axiom, PFA, implies that the product of two countable Fréchet spaces is SS, a statement that was shown in Barman, Dow (2011) [1] to consistently fail. An auxiliary result is that it is consistent with the negation of CH that all separable Fréchet spaces have π-weight at most ω₁.     

Intrinsic characterizations of Besov spaces on Lipschitz domains

The aim of this paper is to study the equivalence between quasi‐norms of Besov spaces on domains. We suppose that the domain Ω ? ?ⁿ is a bounded Lipschitz open subset in ?ⁿ. First, we define Besov spaces on Ω as the restrictions of the corresponding Besov spaces on ?ⁿ. Then, with the help of equivalent and intrinsic characterizations (the Peetre‐type characterization 3.10 and the characterization via local means 3.13) of these spaces, we get another equivalent and intrinsic quasi‐norm using, this time, generalized differences and moduli of smoothness. We extend the well‐known characterization of Besov spaces on ?ⁿ described in Theorem 2.4 to the case of Lipschitz domains.     

A characterization of the interpolation spaces ofH 1 andL ∞ on the line

The Calderón-Mitjagin theorem characterizes all interpolation spaces of the pair of Lebesgue spaces (L ¹,L ) as the rearrangement-invariant spaces. The results of this paper show that the interpolation spaces ofH ¹(R) andL (R) consist of elements whose nontangential maximal functions lie in rearrangement-invariant spaces.Communicated by Jaak Peetre.     

Vanishing moment conditions for wavelet atoms in higher dimensions

We provide explicit criteria for wavelets to give rise to frames and atomic decompositions in L²(?^d), but also in more general Banach function spaces. We consider wavelet systems that arise by translating and dilating the mother wavelet, with the dilations taken from a suitable subgroup of GL(?^d), the so-called dilation group.The paper provides a unified approach that is applicable to a wide range of dilation groups, thus giving rise to new atomic decompositions for homogeneous Besov spaces in arbitrary dimensions, but also for other function spaces such as shearlet coorbit spaces. The atomic decomposition results are obtained by applying the coorbit theory developed by Feichtinger and Gröchenig, and they can be informally described as follows: Given a function ψ ∈ L²(?^d) satisfying fairly mild decay, smoothness and vanishing moment conditions, any sufficiently fine sampling of the translations and dilations will give rise to a wavelet frame. Furthermore, the containment of the analyzed signal in certain smoothness spaces (generalizing the homogeneous Besov spaces) can be decided by looking at the frame coefficients, and convergence of the frame expansion holds in the norms of these spaces. We motivate these results by discussing nonlinear approximation.