关于某些广义可数紧空间

We first give alternative expressions of some generalized countably compact spaces such as quasi-γ spaces, quasi-Nagata spaces, M#-spaces and wM-spaces with g-functions. Then by means of these expressions, we present some characterizations of the corresponding spaces with real-valued functions.     

取值于序拓扑向量空间的映射与层空间

In the paper [Monotone countable paracompactness and maps to ordered topological vector spaces, Top. Appl., 2014, 169(3): 51–70], Yamazaki initiated the study on maps with values into ordered topological vector spaces. Characterizations of monotonically countably paracompact spaces and some other spaces in terms of maps to ordered topological vector spaces were obtained. In this paper, following Yamazaki's method, we present some characterizations of stratifiable spaces and k-semi-stratifiable spaces in terms of maps with values into ordered topological vector spaces.     

CALCULUS ON CANTOR TRIADIC SET （I）——DERIVATIVE

For real-valued functions defined on Cantor triadic ,set. a derivative with corresponding formula of Newton-Leihniz‘s type is given In particular, for the self-simltar functions and alter-nately jumping functions defined in this paper, their derivative and exceptional sets are studied ac-curately by using ergodic theory on Е2 and Duffin-Scbaeffer‘s theorem coneerning metric diophan-tine approximation. In addition, Haar basis of L2(Е2) is constructed and Flaar expansion of stan-drd self-similar function is given.     

A linear perturbed Palais-Smale condition for lower semicontinuous functions on Banach spaces

This paper introduces a notion of linear perturbed Palais-Smale condition for real-valued functions on Banach spaces. In terms of strongly exposed points, it presents a characterization which guarantees linear perturbed Palais-Smale condition holds for lower semicontinuous functions with bounded effective domains defined on a Banach space with the Radon-Nikody＇m property; and gives an example showing that linear perturbed P-S condition is strictly weaker than the P-S condition.     

COMPOSITION OPERATORS BETWEEN SOME WEIGHTED HARDY SPACES

Let D be the unit disc and H(D) be the set of all analytic functions on D. In [2], C. Cowen defined a space H = f ∈ H(D) : f(z) =sum from k=o to ∞ ak(z + 1)k, z∈ D, ‖f‖2 = sum from k=o to ∞ |ak|24k < ∞In this article, the authors consider the similar Hardy spaces with arbitrary weights and discuss some properties of them. Boundedness and compactness of composition operators between such spaces are also studied.     

SOME CLASSES OF DIFFERENCE SEQUENCES OF FUZZY NUMBERS DEFINED BY A SEQUENCE OF MODULI

In this article, we introduce and examine some properties of new difference sequence spaces of fuzzy numbers defined using a sequence of modulus functions.     

Remarks on g-functions on H^n

In this paper, the authors study some properties of Littlewood-Paley g-functions gψ（f）,Lusin area functions Sψ,α（f） and Littlewood-Paley gψ^＊,λ（f） functions defined on H^n, where α,λ 〉 0 and ψ, f are suitable functions. They are the generalization of the corresponding operators on R^n.     

Subordination and superordination properties for a class of integral operators

In the present paper, we obtain some subordination- and superordinatiompreserving properties of certain integral operators defined on the space of normalized analytic functions in the open unit disk. The sandwich-type theorems for these integral operators are also considered.     

Function spaces on local fields

We study the function spaces on local fields in this paper, such as Triebel B-type and F-type spaces, Holder type spaces, Sobolev type spaces, and so on, moreover, study the relationship between the p-type derivatives and the Holder type spaces. Our obtained results show that there exists quite difference between the functions defined on Euclidean spaces and local fields, respectively. Furthermore, many properties of functions defined on local fields motivate the new idea of solving some important topics on fractal analysis.     

CONSTRUCTION OF SOME KIESSWETTER-LIKE FUNCTIONS-THE CONTINUOUS BUT NON-DIFFERENTIABLE FUNCTION DEFINED BY QUINARY DECIMAL

In this paper, we construct some continuous but non-differentiable functions defined by quinary decimal, that are Kiesswetter-like functions. We discuss their properties, then investigate the Hausdorff dimensions of graphs of these functions and give a detailed proof.     

关于某些广义可数紧空间II

文[Erguang YANG. On some generalized countably compact spaces. J. Math. Res. Appl., 2019, 39(5): 540-550]中给出了对某些广义可数紧空间,如拟-$\gamma$空间、拟-Nagata空间、$wN$-空间及$wM$-空间等的实值函数刻画.本文继续这一研究,给出了上述空间类的其它形式的实值函数刻画.     

Convolutions,Tensor Products and Multipliers of the Orlicz-Lorentz Spaces

In this paper, the authors first give the properties of the convolutions of OrliczLorentz spaces Λ1,wand Λ2,won the locally compact abelian group. Secondly, the authors obtain the concrete representation as function spaces for the tensor products of Orlicz-Lorentz spaces Λ1,wand Λ2,w, and get the space of multipliers from the spaceΛ1,wto the space M2*,w. Finally, the authors discuss the homogeneous properties for the Orlicz-Lorentz space Λp,q,w.     

拟线性次椭圆方程组在Morrey空间上的部分正则性

证明了拟线性次椭圆方程组-X_α~*(a_(ij)~(αβ)(x,u)X_βu~j)=-X_α~*f_i~α+g_i,i=1,2,…,N,x∈Ω的弱解广义梯度Xu在Morrey空间L_x~(p,λ)(Ω,R~(mN))(p2)上的部分正则性,其中光滑实向量场族X=(X_1,X_2,…,X_m)满足H(o|¨)rmander有限秩条件,X_α~*是X_α的共轭;而且主项系数a_(ij)~(αβ)(x,u)关于x一致VMO(Vanishing Mean Oscillation的缩写,消失平均震荡)间断,且关于u为一致连续.     

New Sobolev-Weinstein Spaces and Applications

In this paper, we consider the generalized Weinstein operator $\Delta_{W}^{d,\alpha,n}$, we introduce new Sobolev-Weinstein spaces denoted $\mathscr H_{\alpha,d,n}^{s}(\mathbb{R}_{+}^{d+1}),$ $s\in\mathbb{R},$ associated with the generalized Weinstein operator and we investigate their properties. Next, as application, we study the extremal functions on the spaces $\mathscr H_{\alpha,d,n}^{s}(\mathbb{R}_{+}^{d+1})$ using the theory of reproducing kernels.     

变指标Herz-Morrey空间上的Marcinkiewicz积分算子高阶交换子

In the case of Ω∈ Lipγ(Sn-1)(0 γ≤ 1), we prove the boundedness of the Marcinkiewicz integral operator μΩon the variable exponent Herz-Morrey spaces. Also, we prove the boundedness of the higher order commutators μmΩ,bwith b ∈ BMO(Rn) on both variable exponent Herz spaces and Herz-Morrey spaces, and extend some known results.     

Boundedness of Commutators of θ-Type Calderόn-Zygmund Operators on Non-homogeneous Metric Measure Spaces

Let(X, d, μ) be a metric measure space satisfying both the upper doubling and the geometrically doubling conditions in the sense of Hyt?nen. In this paper, the authors obtain the boundedness of the commutators of θ-type Calderón-Zygmund operators with RBMO functions from L~∞(μ) into RBMO(μ) and from H_(at)~(1,∞)(μ) into L~1(μ), respectively.As a consequence of these results, they establish the L~p(μ) boundedness of the commutators on the non-homogeneous metric spaces.     

Tractability of Approximation for Weighted Korobov Spaces on Classical and Quantum Computers

We study the approximation problem (or problem of optimal recovery in the $L_2$-norm) for weighted Korobov spaces with smoothness parameter $\a$. The weights $\gamma_j$ of the Korobov spaces moderate the behavior of periodic functions with respect to successive variables. The nonnegative smoothness parameter $\a$ measures the decay of Fourier coefficients. For $\a=0$, the Korobov space is the $L_2$ space, whereas for positive $\a$, the Korobov space is a space of periodic functions with some smoothness and the approximation problem corresponds to a compact operator. The periodic functions are defined on $[0,1]^d$ and our main interest is when the dimension $d$ varies and may be large. We consider algorithms using two different classes of information. The first class $\lall$ consists of arbitrary linear functionals. The second class $\lstd$ consists of only function values and this class is more realistic in practical computations. We want to know when the approximation problem is tractable. Tractability means that there exists an algorithm whose error is at most $\e$ and whose information cost is bounded by a polynomial in the dimension $d$ and in $\e^{-1}$. Strong tractability means that the bound does not depend on $d$ and is polynomial in $\e^{-1}$. In this paper we consider the worst case, randomized, and quantum settings. In each setting, the concepts of error and cost are defined differently and, therefore, tractability and strong tractability depend on the setting and on the class of information. In the worst case setting, we apply known results to prove that strong tractability and tractability in the class $\lall$ are equivalent. This holds if and only if $\a>0$ and the sum-exponent $s_{\g}$ of weights is finite, where $s_{\g}= \inf\{s>0 : \xxsum_{j=1}^\infty\g_j^s\,<\,\infty\}$. In the worst case setting for the class $\lstd$ we must assume that $\a>1$ to guarantee that functionals from $\lstd$ are continuous. The notions of strong tractability and tractability are not equivalent. In particular, strong tractability holds if and only if $\a>1$ and $\xxsum_{j=1}^\infty\g_j<\infty$. In the randomized setting, it is known that randomization does not help over the worst case setting in the class $\lall$. For the class $\lstd$, we prove that strong tractability and tractability are equivalent and this holds under the same assumption as for the class $\lall$ in the worst case setting, that is, if and only if $\a>0$ and $s_{\g} < \infty$. In the quantum setting, we consider only upper bounds for the class $\lstd$ with $\a>1$. We prove that $s_{\g}<\infty$ implies strong tractability. Hence for $s_{\g}>1$, the randomized and quantum settings both break worst case intractability of approximation for the class $\lstd$. We indicate cost bounds on algorithms with error at most $\e$. Let $\cc(d)$ denote the cost of computing $L(f)$ for $L\in \lall$ or $L\in \lstd$, and let the cost of one arithmetic operation be taken as unity. The information cost bound in the worst case setting for the class $\lall$ is of order $\cc (d) \cdot \e^{-p}$ with $p$ being roughly equal to $2\max(s_\g,\a^{-1})$. Then for the class $\lstd$ in the randomized setting, we present an algorithm with error at most $\e$ and whose total cost is of order $\cc(d)\e^{-p-2} + d\e^{-2p-2}$, which for small $\e$ is roughly $$ d\e^{-2p-2}. $$ In the quantum setting, we present a quantum algorithm with error at most $\e$ that uses about only $d + \log \e^{-1}$ qubits and whose total cost is of order $$ (\cc(d) +d) \e^{-1-3p/2}. $$ The ratio of the costs of the algorithms in the quantum setting and the randomized setting is of order $$ \frac{d}{\cc(d)+d}\,\left(\frac1{\e}\right)^{1+p/2}. $$ Hence, we have a polynomial speedup of order $\e^{-(1+p/2)}$. We stress that $p$ can be arbitrarily large, and in this case the speedup is huge.