A predual of a predual of B_σ and its applications to commutators

A predual of B_σ-spaces is investigated. A predual of a predual of B_σ-spaces is also investigated,which can be used to investigate the boundedness property of the commutators. The relation between Herz spaces and local Morrey spaces is discussed. As an application of the duality results, one obtains the boundedness of the singular integral operators, the Hardy-Littlewood maximal operators and the fractional integral operators, as well as the commutators generated by the bounded mean oscillation(BMO) and the singular integral operators.What is new in this paper is that we do not have to depend on the specific structure of the operators. The results on the boundedness of operators are formulated in terms of B_σ-spaces and B_σ-spaces together with the detailed comparison of the ones in Herz spaces and local Morrey spaces. Another application is the nonsmooth atomic decomposition adapted to B_σ-spaces.     

μ-Separations in generalized topological spaces

This paper takes some investigations on generalized topological spaces with some μ- separations. Some characterizations of μTi-spaces for i = 0,1,2,3,4, μTD-spaces and μR0-spaces are obtained and some relations among these spaces are established.     

一类无限矩阵拓扑代数

Let λ and μ be sequence spaces and have both the signed-weak gliding hump property, (λ,μ) the algebra of the infinite matrix operators which transform λ into μ. In this paper, it is proved that if λ and μ are β-spaces and λ^β and ,μ^β have also the signed-weak gliding hump property, then for any polar topology τ, ((λ,μ),τ) is always sequentially complete locally convex topological algebra.     

Remarks on B（D,λ）-refinability

In this paper, some equivalent versions of B(D,λ)-refinability are given. One of these equivalent versions, is that a space X is B(D, ωo)-refinable if and only if X is strongly quasi-paracompact. As an application of the above result, the author shows that weak θ-refinability is strictly weaker than strong quasi-paracompactness in T4-spaces, which answers a question posed by Jiang. In addition, the author proves that a weak version of B(D,λ) always implies weak θ-refinability for any λ<ω1, and also give a T4, B(D,ωo)-refinable (=strongly quasi-paracompact) space which is not θ-refinable.     

Fixed point theorems for better admissible multimaps on abstract convex spaces

In this paper, a better admissible class B+ is introduced and a new fixed point theorem for better admissible multimap is proved on abstract convex spaces. As a consequence, we deduce a new fixed point theorem on abstract convex Φ-spaces. Our main results generalize some recent work due to Lassonde, Kakutani, Browder, and Park.     

On approximation by reciprocals of polynomials with positive coefficients in Orlicz spaces

This paper discusses the approximation by reciprocals of polynomials with positive coefficients in Orlicz spaces and proved that if f(x) ∈ LM*[0,1], changes its sign at most once in (0,1), then there exists x0 ∈ (0,1) and a polynomial Pn ∈ Πn(+) such that f (x) -Pn (x)x-x0 M ≤ Cω( f,n-1/2 )M, where Πn(+) indicates the set of all polynomials of degree n with positive coefficients.     

Co-RS-compact Topologies

A topoogy R(r)is constructed from a given topology r on a set X.R(r)iscoarser than r,and the following are some results based on this topology:1.Continuity and RS-continuity are equivalent if the codomain is retopologized by R(τ).2.Each topological spase has a coarser extremely disconnected topology3.The class of semi-open sets with respect to R(r)is a topology4.T_2 and semi-T_2 properties are equivalent on a space whose topology is R(r).5.Minboal R_0-spaces are RS-compact.6.MaxAnalα-compact spaces are RS-compact.     

Co-RS-紧拓扑

A topoogy R(τ) is constructed from a given topology τ on a set X. R(τ) is coarser than τ, and the following are some results based on this topology:1. Continuity and RS-continuity are equivalent if the codomain is retopologized by R(τ).2. Each topological space has a coarser extremely disconnected topology.3. The class of semi-open sets with respect to R(τ) is a topology.4. T₂ and semi-T₂ properties are equivalent on a space whose topology is R(τ).5. Minimal R₀-spaces are RS-compact.6. Maximal a-compact spaces are RS-compact.     

复拟Musielak-Orlicz空间及其解析RN性质

设X是一个复拟Banach空间，M(ω，u)是一个满足△2条件的含参数ω的Φ函数．则LM(X)是具有连续拟范数的拟Banach空间，并且LM(X)具有解析Radon-Nikodym性质当且仅当X具有解析Radon-Nikodym性质．     

ON BOUNDED SETS IT INDUCTIVE LIMITS OF LOCAL LY CONVEXSPACES OF SOME CLASSES0

Let E_1■E_2■E_3■...be a sequence of locally convex spaces and (E,ξ)=indlim(E_n,ξ_n) be their Hausdorff inductive limit. In this paper, we discuss bounded sets in inductive limits,The main results are as follows. (1) When all the (E_n,ξ_n) are (DF)-spaces, each bounded set in (E,ξ) is contained in someE_n provided that: jor each n∈N. there is a neighborhood U_n of o in (E_n,ξ_n) and m(n)∈N such that U_n?E_m(n). (2) When all the (E_n,ξ_n) are C-barrelled spaces, each bounded set,in (E.ξ) is contained in some E_nprovided that: for each n∈N,there is an absolutely convex absorbing set W_n in E_n and m(n)∈N suchthat W_n~E?E_m(n) and W_n is absorbed by W_(n+1). These improve the relevant results in [3] and [4].     

Cc （X） Spaces with X Locally Compact

In this paper, we show, among other results, that if X is a [separable] locally compact space X [satisfying the first countability axiom] then the space Cc （X） has countable tightness [if and only if it has bounding tightness] if and only if it is Frechet-Urysohn, if and only if Cc （X） contains a dense （LM） subspace and if and only if X is a-compact.     

本刊英文版2021年64卷第7期(1331–1714)摘要（英文）

正On the vortex filament in 3-spaces and its generalizations Qing Ding Shiping Zhong Abstract In this article,we devote to a mathematical survey on the theory of the vortex filament in 3-dimensional spaces and its generalizations.We shall present some effective geometric tools applied in the study,such as the Schrodinger flow,the geometric Korteweg-de Vries (KdV) flow and the generalized bi-Schrodinger flow,as well as the complex and para-complex structures.It should be mentioned that the investigation in the imaginary part of the octonions looks very fascinating,since it relates to almost complex structures and the G_2 structure on S~6.As a new result in this survey,we describe the equation of generalized bi-Schrodinger flows from R~1 into a Riemannian surface.     

新的齐型BMO, Lipschitz空间上的奇异积分算子

Let(X, d, μ) be a space of homogeneous type, BMO_A(X) and Lip_A(β,X) be the space of BMO type,lipschitz type associated with an approximation to the identity {A_t}_t0 and introduced by Duong,Yan and Tang, respectively. Assuming that T is a bounded linear operator on L~2(X), we find the sufficient condition on the kernel of T so that T is bounded from BMO(X) to BMO_A(X) and from Lip(β, X) to Lip_A(β, X). As an application, the boundedness of Calderón-Zygmund operators with nonsmooth kernels on BMO(R~n) and Lip(β, R~n) are also obtained.     

ON UNIVALENT BLOCH FUNCTIONS

Let A = {z: lzl < 1} be the unit disk of complex plane C. For a 6 A, let W.(z) = isbe the M5biu8 transfOrlnation of A and let g(z, a) = log Ieel be the Green's functiou on Awith logarithlllic sillgularity at a E a. FOr 0 < p < oo, tl1e space Q. consists of all fuuctiolls fa11alytic il1 A for w11ichsup / I If'(z)l'(g(z, a))P dA(z) < oo, (1)aea j ja If'(z)l'(g(z, a))P dA(z) < oo, (1)wllere dA(z) is tlle Euclidcan area eIelllent o11 A.Q.-spaces have been i11vestigated by n1any authors (…     

收敛序列上的连续函数下方图形超空间

Let S={1,1/2,1/2 2,…,1/∞=0} and I = [0,1] be the unit interval. We use ↓USC(S) and ↓C(5) to denote the families of the regions below of all upper semi-continuous maps and of the regions below of all continuous maps from S to I and ↓C0(S) ={↓f∈↓C(S):f(0)=0}.↓USC(S) endowed with the Vietoris topology is a topological space. A pair of topological spaces (X, Y) means that X is a topological space and Y is its subspace. Two pairs of topological spaces (X, Y) and (A, B) are called pair-homeomorphic (≈) if there exists a homeomorphism h:X→A from X onto A such that h{Y) = B. It is proved that, (↓USC(S), ↓C0(S))≈(Q,s) and (↓USC(5), ↓C(5)\↓C0(S))≈(Q,c0), where Q =[-1, l]ω is the Hilbert cube and s=(-1,1)ω, c0 = {(xn)∈Q:lim n→∞ xn=0}. But we do not know what (↓USC(S),↓C(5)) is.     

与截面相关的Morrey空间与奇异积分

In this paper, we define the Morrey spaces M_F~(p,q) (Rn) and the Campanato spaces E_F~(p,q) (R~n) associated with a family F of sections and a doubling measure μ, where F is closely related to the Monge-Amp`ere equation. Furthermore, we obtain the boundedness of the Hardy-Littlewood maximal function associated to F, Monge-Amp`ere singular integral operators and fractional integrals on M_F~(p,q)(R~n). We also prove that the Morrey spaces M_F~(p,q) (R~n)and the Campanato spaces E_F~(p,q) (R~n) are equivalent with 1 ≤ q ≤ p ∞.     

DOMAIN OF THE DOUBLE SEQUENTIAL BAND MATRIX B（r,s） ON SOME MADDOX＇S SPACES

The domain of generalized difference matrix B(r, s) in the classical spaces l∞,c, and c0 was recently studied by Kirisci and Bassar in [16]. The main goal of this article is to introduce the paranormed sequence spaces l∞( B, p), c( B, p), and c0( B, p), which are more general and comprehensive than the corresponding consequences of the matrix domain of B(r, s), as well as other studies in literature. Besides this, the alpha-, beta-, and gamma-duals of the spaces l∞( B, p), c( B, p), and c0( B, p) are computed and the bases of the spaces c( B, p)and c0( B, p) are constructed. The final section of this article is devoted to the characterization of the classes(λ( B, p) :) and( : λ( B, p)), where λ∈ {c, c0, l∞}and is any given sequence space. Additionally, the characterization of some other classes which are related to the space of almost convergent sequences is obtained by means of a given lemma.     

The hyperspace of the regions below continuous maps with the fell topology

For a Tychonoff space X,we use ↓USC F(X) and ↓C F(X) to denote the families of the hypographs of all semi-continuous maps and of all continuous maps from X to I = [0,1] with the subspace topologies of the hyperspace Cld F(X × I) consisting of all non-empty closed sets in X × I endowed with the Fell topology.In this paper,we shall show that there exists a homeomorphism h:↓USC F(X) → Q = [1,1] ω such that h(↓CF(X))=c0 = {(xn)∈Q| lim n→∞ x n = 0} if and only if X is a locally compact separable metrizable space and the set of isolated points is not dense in X.     

实值函数与某些覆盖性质

In the paper [properties defined with semi-continuous functions and some related spaces',Houston J.Math.,2015,41(3):1097–1106] properties(U L)~(wl)_m,(U L)~K_m and(U L)_m were defined and it was shown that spaces having these properties coincide with countably paracompact spaces,countably mesocompact spaces and countably metacompact spaces,respectively.In this paper,we continue with the study on the relationship between properties defined with real-valued functions and some covering properties.Some characterizations of countably compact spaces and pseudo-compact spaces in terms of real-valued functions are obtained.