On Weighted α-Besov Spaces and α-Bloch Spaces of Quaternion-Valued Functions

In this paper, we give the definitions of weighted α-Besov-type spaces and α-Bloch spaces of quaternion-valued functions, then we obtain characterizations of these quaternion α-Bloch spaces by quaternion α-Besov-type spaces. Relations between Q _p norms and weighted α-Besov norms are also considered. The role of ρ?^α sequences in securing non-Bloch functions is highlighted in quaternion sense.     

The Equivalence Theory of Native Spaces

In this paper some equivalence definitions are given for native spaces which were introduced by Madych and Nelson and have become influential in the theory of radial basis functions. The abstract elements in native spaces are interpreted. Moreover, Weinrich and Iske's theories are unified.     

The Equivalence Theory of Native Spaces

In this paper some equivalence definitions are given for native spaces which were introduced by Madych and Nelson and have become influential in the theory of radial basis functions. The abstract elements in native spaces are interpreted. Moreover, Weinrich and Iske's theories are unified.     

On Some Almost-Periodicity Problems in Various Metrics

The Bohr-type and the Bochner-type definitions for almost periodic functions are examined in various metrics (Stepanov, Weyl and Besicovitch). The correct definitions of Besicovitch-like multifunctions are given. Weak almost-periodic solutions are proved for differential equations and inclusions. This problem is also discussed as a fixed-point problem in function spaces.     

Contractivity of Leader type and fixed points in uniform spaces with generalized pseudodistances

Recently, Jachymski and Jó?wik proved that among various classes of contractions which are introduced and studied in the metric fixed point theory, the Leader contractions are greatest general contractions. In this article, we want to show how generalized pseudodistances in uniform spaces can be used to obtain new and general results of Leader type without complete graph assumptions about maps and without sequentially complete assumptions about spaces, which was not done in the previous publications on this subject. The definitions, results and methods presented here are new for maps in uniform and locally convex spaces and even in metric spaces. Examples showing a difference between our results and the well-known ones are given.     

Morse theory, Higgs fields, and Yang–Mills–Higgs functionals

In this mostly expository paper we describe applications of Morse theory to moduli spaces of Higgs bundles. The moduli spaces are finite-dimensional analytic varieties but they arise as quotients of infinite-dimensional spaces. There are natural functions for Morse theory on both the infinite-dimensional spaces and the finite-dimensional quotients. The first comes from the Yang?CMills?CHiggs energy, while the second is provided by the Hitchin function. After describing what Higgs bundles are, we explore these functions and how they may be used to extract topological information about the moduli spaces.     

The divergence equation in weighted- and L p(��)-spaces

We study the solvability of the divergence equation in weighted spaces and Lebesgue spaces with variable exponents, where the weights are so called Muckenhoupt weights. The question of constructing divergence free test functions, which can be used for problems arising in fluid dynamics, is also addressed. The approach is based on an explicit representation formula for solutions of the divergence equation due to Bogovski? and the theory of singular integral operators. The developed methods are used to prove an existence result for fluids which satisfy a p(·)-growth condition.     

The characterization of the Triebel-Lizorkin spaces forp=∞

We establish the characterization of the weighted Triebel-Lizorkin spaces for p=∞ by means of a “generalized” Littlewood-Paley function which is based on a kernel satisfying “minimal” moment and Tauberian conditions. This characterization completes earlier work by Bui et al. The definitions of the ? _∞,q ^α spaces are extended in a natural way to ? _∞,∞ ^α and it is proven that this is the same space as ? _∞,∞ ^α , which justifies the standard convention in which the two spaces are defined to be equal. As a consequence, we obtain a new characterization of the Hölder-Zygmund space ? _∞,∞ ^α .     

On Kaufmanʼs theorem

Kaufman?s theorem (Kaufman, 1978 [9]) on representing closed linear operators as quotients of bounded operators is given a new constructive proof, and is extended to operators between two Hilbert spaces. Two different definitions of operator quotients existing in the literature are compared.     

Spaces of Besov-Sobolev type and a problem on nonlinear approximation

We study fractional variants of the quasi-norms introduced by Brezis, Van Schaftingen, and Yung in the study of the Sobolev space ${\dot{W}}^{1,p}$. The resulting spaces are identified as a special class of real interpolation spaces of Sobolev-Slobodecki? spaces. We establish the equivalence between Fourier analytic definitions and definitions via difference operators acting on measurable functions. We prove various new results on embeddings and non-embeddings, and give applications to harmonic and caloric extensions. For suitable wavelet bases we obtain a characterization of the approximation spaces for best n-term approximation from a wavelet basis via smoothness conditions on the function; this extends a classical result by DeVore, Jawerth and Popov.     

Variable exponent Morrey and Campanato spaces

The classical Morrey spaces and Campanato spaces are generalized to the variable exponent case. The definitions and some basic properties of the variable exponent Morrey and Campanato spaces are presented. A concept of the p(⋅)-average of the functions is introduced.     

几个多复变数加权全纯函数空间的边界值与对偶性

该文将几个全纯函数空间的定义从单复变数推广到多复变数, 这些空间中全纯函数的增长性依赖于某个权函数. 作者研究了它们的增长性与边界值的关系以及这些空间相互之间的对偶关系, 所用方法和技巧与单复变数有不少区别. 所得到的结论是单复变数许多已知结果的推广.     

Hardy空间的等价刻画

设Ｇ为局部域Ｋ上的２ｎ＋１维Ｈｅｉｓｅｎｂｅｒｇ群．本文证明了通过各种极大函数定义的Ｈａｒｄｙ空间的等价性，给出了Ｈａｒｄｙ空间的原子分解．利用这些等价定义及一些卷积算子定理，给出了Ｈａｒｄｙ空间的平方函数刻画——Ｌｕｚｉｎ面积函数刻画与Ｌｉｔｌｅｗｏｏｄ－Ｐａｌｅｙ平方函数刻画     

THE DERIVATIVES AND INTEGRALS OF FRACTIONAL ORDER ON \alpha-ADIC GROUPS

This paper, the author studies fractional derivatives and integrals of fractional order \alpha > 0 for functions in L^r(G_\alpha) and Fourier transform for distributions. Under these definitions, the author obtains the formula $x_y^<\alhpa>(x)=|y|^\alpha X_y(x)$ for characters X_y and $(D^{(\alpha)}f)^{\Gamma}=|\cdot|^\alphaf^{\Gamma}$, discusses the existence of the fractional derivatives of test functions, gives relationships between some function spaces, and proves that the fractional derivatives and the fractional integrals are inverse operations one another.     

关于概率内积空间定义的平凡性

研究了分布函数拟逆的性质,进而利用这些性质,讨论了概率内积空间中半内积族之间的关系,并证明了几种定义下的概率内积空间等同于通常的内积空间.     

Infinite dimensional holomorphy via categorical differential calculus

We establish that the category of hological spaces is equipped for calculus with complex scalars. This provides a theory of infinite dimensional holomorphy which allows maps to have nonconvex domains with empty interior. Some relatively elementary functions, hitherto excluded by the restrictive definitions of other theories, emerge as holomorphic maps.KOSEF aided.NSERC aided.     

Products of compact spaces and the axiom of choice II

This is a continuation of [2]. We study the Tychonoff Compactness Theorem for various definitions of compactness and for various types of spaces (first and second countable spaces, Hausdorff spaces, and subspaces of ?^K). We also study well ordered Tychonoff products and the effect that the multiple choice axiom has on such products.     

Convolution Hankel Transforms on the Zemanian Spaces

In this paper we investigate the convolution Hankel transforms on the Zemanian spaces of Hankel transformable functions and distributions. The convolution Hankel transform is defined on generalized functions by using the adjoint method. Our new definition includes as special cases other known definitions of the convolution Hankel transform of distributions. Finally we establish a distributional inversion formula for the transformation under consideration involving Bessel differential operators.     

叙列空间上的二级囿变函数(Ⅱ)

作者在[1]中引入了叙列空间上的二级囿变函数,本文是它的继续。首先在[1]的基础上讨论叙列空间上二级囿变函数相应的凸函数分解,二级变差函数以及两个二级囿变函数的运算等问题,这也就是§1.§2再引进二级强囿变与二级弱囿变等概念。最后的§3研究可距离化和可赋范的完备(按kthe意义:λ=λ)空间,即(KF)和(KB)空间上的二级