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1. CMB Online first

He, Ziyi; Yang, Dachun; Yuan, Wen
Littlewood-Paley Characterizations of Second-Order Sobolev Spaces via Averages on Balls
In this paper, the authors characterize second-order Sobolev spaces $W^{2,p}({\mathbb R}^n)$, with $p\in [2,\infty)$ and $n\in\mathbb N$ or $p\in (1,2)$ and $n\in\{1,2,3\}$, via the Lusin area function and the Littlewood-Paley $g_\lambda^\ast$-function in terms of ball means.

Keywords:Sobolev space, ball means, Lusin-area function, $g_\lambda^*$-function
Categories:46E35, 42B25, 42B20, 42B35

2. CMB Online first

Ongaro, Jared; Shapiro, Boris
A note on planarity stratification of Hurwitz spaces
One can easily show that any meromorphic function on a complex closed Riemann surface can be represented as a composition of a birational map of this surface to $\mathbb{CP}^2$ and a projection of the image curve from an appropriate point $p\in \mathbb{CP}^2$ to the pencil of lines through $p$. We introduce a natural stratification of Hurwitz spaces according to the minimal degree of a plane curve such that a given meromorphic function can be represented in the above way and calculate the dimensions of these strata. We observe that they are closely related to a family of Severi varieties studied earlier by J. Harris, Z. Ran and I. Tyomkin.

Keywords:Hurwitz spaces, meromorphic functions, Severi varieties

3. CMB 2015 (vol 58 pp. 271)

Jafari, Sayyed Heidar; Jafari Rad, Nader
On Domination of Zero-divisor Graphs of Matrix Rings
We study domination in zero-divisor graphs of matrix rings over a commutative ring with $1$.

Keywords:vector space, linear transformation, zero-divisor graph, domination, local ring

4. CMB Online first

Alfuraidan, Monther Rashed
The Contraction Principle for Multivalued Mappings on a Modular Metric Space with a Graph
We study the existence of fixed points for contraction multivalued mappings in modular metric spaces endowed with a graph. The notion of a modular metric on an arbitrary set and the corresponding modular spaces, generalizing classical modulars over linear spaces like Orlicz spaces, were recently introduced. This paper can be seen as a generalization of Nadler's and Edelstein's fixed point theorems to modular metric spaces endowed with a graph.

Keywords:fixed point theory, modular metric spaces, multivalued contraction mapping, connected digraph.
Categories:47H09, 46B20, 47H10, 47E10

5. CMB Online first

Han, Yanchang
Embedding theorem for inhomogeneous Besov and Triebel-Lizorkin spaces on RD-spaces
In this article we prove the embedding theorem for inhomogeneous Besov and Triebel-Lizorkin spaces on RD-spaces. The crucial idea is to use the geometric density condition on the measure.

Keywords:spaces of homogeneous type, test function space, distributions, Calderón reproducing formula, Besov and Triebel-Lizorkin spaces, embedding
Categories:42B25, 46F05, 46E35

6. CMB 2015 (vol 58 pp. 350)

Merino-Cruz, Héctor; Wawrzynczyk, Antoni
On Closed Ideals in a Certain Class of Algebras of Holomorphic Functions
We recently introduced a weighted Banach algebra $\mathfrak{A}_G^n$ of functions which are holomorphic on the unit disc $\mathbb{D}$, continuous up to the boundary and of the class $C^{(n)}$ at all points where the function $G$ does not vanish. Here, $G$ refers to a function of the disc algebra without zeros on $\mathbb{D}$. Then we proved that all closed ideals in $\mathfrak{A}_G^n$ with at most countable hull are standard. In the present paper, on the assumption that $G$ is an outer function in $C^{(n)}(\overline{\mathbb{D}})$ having infinite roots in $\mathfrak{A}_G^n$ and countable zero set $h(G)$, we show that all the closed ideals $I$ with hull containing $h(G)$ are standard.

Keywords:Banach algebra, disc algebra, holomorphic spaces, standard ideal
Categories:46J15, 46J20, 30H50

7. CMB Online first

Hsu, Ming-Hsiu; Lee, Ming-Yi
VMO space associated with parabolic sections and its application
In this paper we define $VMO_\mathcal{P}$ space associated with a family $\mathcal{P}$ of parabolic sections and show that the dual of $VMO_\mathcal{P}$ is the Hardy space $H^1_\mathcal{P}$. As an application, we prove that almost everywhere convergence of a bounded sequence in $H^1_\mathcal{P}$ implies weak* convergence.

Keywords:Monge-Ampere equation, parabolic section, Hardy space, BMO, VMO

8. CMB Online first

Casini, Emanuele; Miglierina, Enrico; Piasecki, Lukasz
Hyperplanes in the space of convergent sequences and preduals of $\ell_1$
The main aim of the present paper is to investigate various structural properties of hyperplanes of $c$, the Banach space of the convergent sequences. In particular, we give an explicit formula for the projection constants and we prove that an hyperplane of $c$ is isometric to the whole space if and only if it is $1$-complemented. Moreover, we obtain the classification of those hyperplanes for which their duals are isometric to $\ell_{1}$ and we give a complete description of the preduals of $\ell_{1}$ under the assumption that the standard basis of $\ell_{1}$ is weak$^{*}$-convergent.

Keywords:space of convergent sequences, projection, $\ell_1$-predual, hyperplane
Categories:46B45, 46B04

9. CMB Online first

Anona, F. M.; Randriambololondrantomalala, Princy; Ravelonirina, H. S. G.
Sur les algèbres de Lie associées à une connexion
Let $\Gamma$ be a connection on a smooth manifold $M$, in this paper we give some properties of $\Gamma$ by studying the corresponding Lie algebras. In particular, we compute the first Chevalley-Eilenberg cohomology space of the horizontal vector fields Lie algebra on the tangent bundle of $M$, whose the corresponding Lie derivative of $\Gamma$ is null, and of the horizontal nullity curvature space.

Keywords:algèbre de Lie, connexion, cohomologie de Chevalley-Eilenberg, champs dont la dérivée de Lie correspondante à une connexion est nulle, espace de nullité de la courbure
Categories:17B66, 53B15

10. CMB Online first

Liu, Feng; Wu, Huoxiong
On the Regularity of the Multisublinear Maximal Functions
This paper is concerned with the study of the regularity for the multisublinear maximal operator. It is proved that the multisublinear maximal operator is bounded on first-order Sobolev spaces. Moreover, two key point-wise inequalities for the partial derivatives of the multisublinear maximal functions are established. As an application, the quasi-continuity on the multisublinear maximal function is also obtained.

Keywords:regularity, multisublinear maximal operator, Sobolev spaces, partial deviative, quasicontinuity
Categories:42B25, 46E35

11. CMB 2015 (vol 58 pp. 241)

Botelho, Fernanda
Isometries and Hermitian Operators on Zygmund Spaces
In this paper we characterize the isometries of subspaces of the little Zygmund space. We show that the isometries of these spaces are surjective and represented as integral operators. We also show that all hermitian operators on these settings are bounded.

Keywords:Zygmund spaces, the little Zygmund space, Hermitian operators, surjective linear isometries, generators of one-parameter groups of surjective isometries
Categories:46E15, 47B15, 47B38

12. CMB 2014 (vol 58 pp. 432)

Yang, Dachun; Yang, Sibei
Second-order Riesz Transforms and Maximal Inequalities Associated with Magnetic Schrödinger Operators
Let $A:=-(\nabla-i\vec{a})\cdot(\nabla-i\vec{a})+V$ be a magnetic Schrödinger operator on $\mathbb{R}^n$, where $\vec{a}:=(a_1,\dots, a_n)\in L^2_{\mathrm{loc}}(\mathbb{R}^n,\mathbb{R}^n)$ and $0\le V\in L^1_{\mathrm{loc}}(\mathbb{R}^n)$ satisfy some reverse Hölder conditions. Let $\varphi\colon \mathbb{R}^n\times[0,\infty)\to[0,\infty)$ be such that $\varphi(x,\cdot)$ for any given $x\in\mathbb{R}^n$ is an Orlicz function, $\varphi(\cdot,t)\in {\mathbb A}_{\infty}(\mathbb{R}^n)$ for all $t\in (0,\infty)$ (the class of uniformly Muckenhoupt weights) and its uniformly critical upper type index $I(\varphi)\in(0,1]$. In this article, the authors prove that second-order Riesz transforms $VA^{-1}$ and $(\nabla-i\vec{a})^2A^{-1}$ are bounded from the Musielak-Orlicz-Hardy space $H_{\varphi,\,A}(\mathbb{R}^n)$, associated with $A$, to the Musielak-Orlicz space $L^{\varphi}(\mathbb{R}^n)$. Moreover, the authors establish the boundedness of $VA^{-1}$ on $H_{\varphi, A}(\mathbb{R}^n)$. As applications, some maximal inequalities associated with $A$ in the scale of $H_{\varphi, A}(\mathbb{R}^n)$ are obtained.

Keywords:Musielak-Orlicz-Hardy space, magnetic Schrödinger operator, atom, second-order Riesz transform, maximal inequality
Categories:42B30, 42B35, 42B25, 35J10, 42B37, 46E30

13. CMB 2014 (vol 58 pp. 150)

Ostrovskii, Mikhail I.
Connections Between Metric Characterizations of Superreflexivity and the Radon-Nikodý Property for Dual Banach Spaces
Johnson and Schechtman (2009) characterized superreflexivity in terms of finite diamond graphs. The present author characterized the Radon-Nikodým property (RNP) for dual spaces in terms of the infinite diamond. This paper is devoted to further study of relations between metric characterizations of superreflexivity and the RNP for dual spaces. The main result is that finite subsets of any set $M$ whose embeddability characterizes the RNP for dual spaces, characterize superreflexivity. It is also observed that the converse statement does not hold, and that $M=\ell_2$ is a counterexample.

Keywords:Banach space, diamond graph, finite representability, metric characterization, Radon-Nikodým property, superreflexivity
Categories:46B85, 46B07, 46B22

14. CMB 2014 (vol 58 pp. 158)

Özgür, Cihan; Mihai, Adela
Corrigendum to "Chen Inequalities for Submanifolds of Real Space Forms with a Semi-symmetric Non-metric Connection"
We fix the coefficients in the inequality (4.1) in the Theorem 4.1(i) from A. Mihai and C. Özgür, "Chen inequalities for submanifolds of real space forms with a semi-symmetric non-metric connection" Canad. Math. Bull. 55 (2012), no. 3, 611-622.

Keywords:real space form, semi-symmetric non-metric connection, Ricci curvature
Categories:53C40, 53B05, 53B15

15. CMB 2014 (vol 58 pp. 128)

Marković, Marijan
A Sharp Constant for the Bergman Projection
For the Bergman projection operator $P$ we prove that \begin{equation*} \|P\colon L^1(B,d\lambda)\rightarrow B_1\| = \frac {(2n+1)!}{n!}. \end{equation*} Here $\lambda$ stands for the hyperbolic metric in the unit ball $B$ of $\mathbb{C}^n$, and $B_1$ denotes the Besov space with an adequate semi--norm. We also consider a generalization of this result. This generalizes some recent results due to Perälä.

Keywords:Bergman projections, Besov spaces
Categories:45P05, 47B35

16. CMB 2014 (vol 57 pp. 749)

Cavalieri, Renzo; Marcus, Steffen
Geometric Perspective on Piecewise Polynomiality of Double Hurwitz Numbers
We describe double Hurwitz numbers as intersection numbers on the moduli space of curves $\overline{\mathcal{M}}_{g,n}$. Using a result on the polynomiality of intersection numbers of psi classes with the Double Ramification Cycle, our formula explains the polynomiality in chambers of double Hurwitz numbers, and the wall crossing phenomenon in terms of a variation of correction terms to the $\psi$ classes. We interpret this as suggestive evidence for polynomiality of the Double Ramification Cycle (which is only known in genera $0$ and $1$).

Keywords:double Hurwitz numbers, wall crossings, moduli spaces, ELSV formula

17. CMB 2014 (vol 58 pp. 297)

Khamsi, M. A.
Approximate Fixed Point Sequences of Nonlinear Semigroup in Metric Spaces
In this paper, we investigate the common approximate fixed point sequences of nonexpansive semigroups of nonlinear mappings $\{T_t\}_{t \geq 0}$, i.e., a family such that $T_0(x)=x$, $T_{s+t}=T_s(T_t(x))$, where the domain is a metric space $(M,d)$. In particular we prove that under suitable conditions, the common approximate fixed point sequences set is the same as the common approximate fixed point sequences set of two mappings from the family. Then we use the Ishikawa iteration to construct a common approximate fixed point sequence of nonexpansive semigroups of nonlinear mappings.

Keywords:approximate fixed point, fixed point, hyperbolic metric space, Ishikawa iterations, nonexpansive mapping, semigroup of mappings, uniformly convex hyperbolic space
Categories:47H09, 46B20, 47H10, 47E10

18. CMB 2014 (vol 57 pp. 803)

Gabriyelyan, S. S.
Free Locally Convex Spaces and the $k$-space Property
Let $L(X)$ be the free locally convex space over a Tychonoff space $X$. Then $L(X)$ is a $k$-space if and only if $X$ is a countable discrete space. We prove also that $L(D)$ has uncountable tightness for every uncountable discrete space $D$.

Keywords:free locally convex space, $k$-space, countable tightness
Categories:46A03, 54D50, 54A25

19. CMB 2014 (vol 57 pp. 780)

Erzakova, Nina A.
Measures of Noncompactness in Regular Spaces
Previous results by the author on the connection between three of measures of non-compactness obtained for $L_p$, are extended to regular spaces of measurable functions. An example of advantage in some cases one of them in comparison with another is given. Geometric characteristics of regular spaces are determined. New theorems for $(k,\beta)$-boundedness of partially additive operators are proved.

Keywords:measure of non-compactness, condensing map, partially additive operator, regular space, ideal space
Categories:47H08, 46E30, 47H99, 47G10

20. CMB 2014 (vol 57 pp. 683)

Aurichi, Leandro F.; Dias, Rodrigo R.
Topological Games and Alster Spaces
In this paper we study connections between topological games such as Rothberger, Menger and compact-open, and relate these games to properties involving covers by $G_\delta$ subsets. The results include: (1) If Two has a winning strategy in the Menger game on a regular space $X$, then $X$ is an Alster space. (2) If Two has a winning strategy in the Rothberger game on a topological space $X$, then the $G_\delta$-topology on $X$ is Lindelöf. (3) The Menger game and the compact-open game are (consistently) not dual.

Keywords:topological games, selection principles, Alster spaces, Menger spaces, Rothberger spaces, Menger game, Rothberger game, compact-open game, $G_\delta$-topology
Categories:54D20, 54G99, 54A10

21. CMB 2014 (vol 57 pp. 765)

da Silva, Rosângela Maria; Tenenblat, Keti
Helicoidal Minimal Surfaces in a Finsler Space of Randers Type
We consider the Finsler space $(\bar{M}^3, \bar{F})$ obtained by perturbing the Euclidean metric of $\mathbb{R}^3$ by a rotation. It is the open region of $\mathbb{R}^3$ bounded by a cylinder with a Randers metric. Using the Busemann-Hausdorff volume form, we obtain the differential equation that characterizes the helicoidal minimal surfaces in $\bar{M}^3$. We prove that the helicoid is a minimal surface in $\bar{M}^3$, only if the axis of the helicoid is the axis of the cylinder. Moreover, we prove that, in the Randers space $(\bar{M}^3, \bar{F})$, the only minimal surfaces in the Bonnet family, with fixed axis $O\bar{x}^3$, are the catenoids and the helicoids.

Keywords:minimal surfaces, helicoidal surfaces, Finsler space, Randers space
Categories:53A10, 53B40

22. CMB 2013 (vol 57 pp. 794)

Fang, Zhong-Shan; Zhou, Ze-Hua
New Characterizations of the Weighted Composition Operators Between Bloch Type Spaces in the Polydisk
We give some new characterizations for compactness of weighted composition operators $uC_\varphi$ acting on Bloch-type spaces in terms of the power of the components of $\varphi,$ where $\varphi$ is a holomorphic self-map of the polydisk $\mathbb{D}^n,$ thus generalizing the results obtained by Hyvärinen and Lindström in 2012.

Keywords:weighted composition operator, compactness, Bloch type spaces, polydisk, several complex variables
Categories:47B38, 47B33, 32A37, 45P05, 47G10

23. CMB 2013 (vol 57 pp. 598)

Lu, Yufeng; Yang, Dachun; Yuan, Wen
Interpolation of Morrey Spaces on Metric Measure Spaces
In this article, via the classical complex interpolation method and some interpolation methods traced to Gagliardo, the authors obtain an interpolation theorem for Morrey spaces on quasi-metric measure spaces, which generalizes some known results on ${\mathbb R}^n$.

Keywords:complex interpolation, Morrey space, Gagliardo interpolation, Calderón product, quasi-metric measure space
Categories:46B70, 46E30

24. CMB 2012 (vol 57 pp. 90)

Lazar, Aldo J.
Compact Subsets of the Glimm Space of a $C^*$-algebra
If $A$ is a $\sigma$-unital $C^*$-algebra and $a$ is a strictly positive element of $A$ then for every compact subset $K$ of the complete regularization $\mathrm{Glimm}(A)$ of $\mathrm{Prim}(A)$ there exists $\alpha \gt 0$ such that $K\subset \{G\in \mathrm{Glimm}(A) \mid \Vert a + G\Vert \geq \alpha\}$. This extends a result of J. Dauns to all $\sigma$-unital $C^*$-algebras. However, there are a $C^*$-algebra $A$ and a compact subset of $\mathrm{Glimm}(A)$ that is not contained in any set of the form $\{G\in \mathrm{Glimm}(A) \mid \Vert a + G\Vert \geq \alpha\}$, $a\in A$ and $\alpha \gt 0$.

Keywords:primitive ideal space, complete regularization

25. CMB 2012 (vol 57 pp. 42)

Fonf, Vladimir P.; Zanco, Clemente
Covering the Unit Sphere of Certain Banach Spaces by Sequences of Slices and Balls
e prove that, given any covering of any infinite-dimensional Hilbert space $H$ by countably many closed balls, some point exists in $H$ which belongs to infinitely many balls. We do that by characterizing isomorphically polyhedral separable Banach spaces as those whose unit sphere admits a point-finite covering by the union of countably many slices of the unit ball.

Keywords:point finite coverings, slices, polyhedral spaces, Hilbert spaces
Categories:46B20, 46C05, 52C17
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