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1. 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

2. CMB Online first

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

3. CMB Online first

Ostrovskii, Mikhail I.
Connections between metric characterizations of superreflexivity and the Radon-Nikodým 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

4. CMB Online first

Mihai, Adela; Ozgur, Cihan
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

5. CMB Online first

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

6. 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

7. CMB Online first

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

8. 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

9. 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

10. 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

11. 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

12. 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

13. 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

14. 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

15. 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

16. CMB 2012 (vol 57 pp. 166)

Öztop, Serap; Spronk, Nico
On Minimal and Maximal $p$-operator Space Structures
We show that for $p$-operator spaces, there are natural notions of minimal and maximal structures. These are useful for dealing with tensor products.

Keywords:$p$-operator space, min space, max space
Categories:46L07, 47L25, 46G10

17. CMB 2012 (vol 57 pp. 145)

Mustafayev, H. S.
The Essential Spectrum of the Essentially Isometric Operator
Let $T$ be a contraction on a complex, separable, infinite dimensional Hilbert space and let $\sigma \left( T\right) $ (resp. $\sigma _{e}\left( T\right) )$ be its spectrum (resp. essential spectrum). We assume that $T$ is an essentially isometric operator, that is $I_{H}-T^{\ast }T$ is compact. We show that if $D\diagdown \sigma \left( T\right) \neq \emptyset ,$ then for every $f$ from the disc-algebra, \begin{equation*} \sigma _{e}\left( f\left( T\right) \right) =f\left( \sigma _{e}\left( T\right) \right) , \end{equation*} where $D$ is the open unit disc. In addition, if $T$ lies in the class $ C_{0\cdot }\cup C_{\cdot 0},$ then \begin{equation*} \sigma _{e}\left( f\left( T\right) \right) =f\left( \sigma \left( T\right) \cap \Gamma \right) , \end{equation*} where $\Gamma $ is the unit circle. Some related problems are also discussed.

Keywords:Hilbert space, contraction, essentially isometric operator, (essential) spectrum, functional calculus
Categories:47A10, 47A53, 47A60, 47B07

18. CMB 2012 (vol 57 pp. 25)

Bourin, Jean-Christophe; Harada, Tetsuo; Lee, Eun-Young
Subadditivity Inequalities for Compact Operators
Some subadditivity inequalities for matrices and concave functions also hold for Hilbert space operators, but (unfortunately!) with an additional $\varepsilon$ term. It seems not possible to erase this residual term. However, in case of compact operators we show that the $\varepsilon$ term is unnecessary. Further, these inequalities are strict in a certain sense when some natural assumptions are satisfied. The discussion also stresses on matrices and their compressions and several open questions or conjectures are considered, both in the matrix and operator settings.

Keywords:concave or convex function, Hilbert space, unitary orbits, compact operators, compressions, matrix inequalities
Categories:47A63, 15A45

19. CMB 2012 (vol 56 pp. 466)

Aulaskari, Rauno; Rättyä, Jouni
Inclusion Relations for New Function Spaces on Riemann Surfaces
We introduce and study some new function spaces on Riemann surfaces. For certain parameter values these spaces coincide with the classical Dirichlet space, BMOA or the recently defined $Q_p$ space. We establish inclusion relations that generalize earlier known inclusions between the above-mentioned spaces.

Keywords:Bloch space, BMOA, $Q_p$, Green's function, hyperbolic Riemann surface
Categories:30F35, 30H25, 30H30

20. CMB 2012 (vol 56 pp. 551)

Handelman, David
Real Dimension Groups
Dimension groups (not countable) that are also real ordered vector spaces can be obtained as direct limits (over directed sets) of simplicial real vector spaces (finite dimensional vector spaces with the coordinatewise ordering), but the directed set is not as interesting as one would like, i.e., it is not true that a countable-dimensional real vector space that has interpolation can be represented as such a direct limit over the a countable directed set. It turns out this is the case when the group is additionally simple, and it is shown that the latter have an ordered tensor product decomposition. In the Appendix, we provide a huge class of polynomial rings that, with a pointwise ordering, are shown to satisfy interpolation, extending a result outlined by Fuchs.

Keywords:dimension group, simplicial vector space, direct limit, Riesz interpolation
Categories:46A40, 06F20, 13J25, 19K14

21. CMB 2012 (vol 56 pp. 503)

Bu, Qingying
Weak Sequential Completeness of $\mathcal K(X,Y)$
For Banach spaces $X$ and $Y$, we show that if $X^\ast$ and $Y$ are weakly sequentially complete and every weakly compact operator from $X$ to $Y$ is compact then the space of all compact operators from $X$ to $Y$ is weakly sequentially complete. The converse is also true if, in addition, either $X^\ast$ or $Y$ has the bounded compact approximation property.

Keywords:weak sequential completeness, reflexivity, compact operator space
Categories:46B25, 46B28

22. CMB 2011 (vol 56 pp. 306)

Pérez, Juan de Dios; Suh, Young Jin
Real Hypersurfaces in Complex Projective Space Whose Structure Jacobi Operator is Lie $\mathbb{D}$-parallel
We prove the non-existence of real hypersurfaces in complex projective space whose structure Jacobi operator is Lie $\mathbb{D}$-parallel and satisfies a further condition.

Keywords:complex projective space, real hypersurface, structure Jacobi operator
Categories:53C15, 53C40

23. CMB 2011 (vol 56 pp. 225)

Agashe, Amod
On the Notion of Visibility of Torsors
Let $J$ be an abelian variety and $A$ be an abelian subvariety of $J$, both defined over $\mathbf{Q}$. Let $x$ be an element of $H^1(\mathbf{Q},A)$. Then there are at least two definitions of $x$ being visible in $J$: one asks that the torsor corresponding to $x$ be isomorphic over $\mathbf{Q}$ to a subvariety of $J$, and the other asks that $x$ be in the kernel of the natural map $H^1(\mathbf{Q},A) \to H^1(\mathbf{Q},J)$. In this article, we clarify the relation between the two definitions.

Keywords:torsors, principal homogeneous spaces, visibility, Shafarevich-Tate group
Categories:11G35, 14G25

24. CMB 2011 (vol 56 pp. 400)

Prunaru, Bebe
A Factorization Theorem for Multiplier Algebras of Reproducing Kernel Hilbert Spaces
Let $(X,\mathcal B,\mu)$ be a $\sigma$-finite measure space and let $H\subset L^2(X,\mu)$ be a separable reproducing kernel Hilbert space on $X$. We show that the multiplier algebra of $H$ has property $(A_1(1))$.

Keywords:reproducing kernel Hilbert space, Berezin transform, dual algebra
Categories:46E22, 47B32, 47L45

25. CMB 2011 (vol 56 pp. 388)

Mursaleen, M.
Application of Measure of Noncompactness to Infinite Systems of Differential Equations
In this paper we determine the Hausdorff measure of noncompactness on the sequence space $n(\phi)$ of W. L. C. Sargent. Further we apply the technique of measures of noncompactness to the theory of infinite systems of differential equations in the Banach sequence spaces $n(\phi)$ and $m(\phi)$. Our aim is to present some existence results for infinite systems of differential equations formulated with the help of measures of noncompactness.

Keywords:sequence spaces, BK spaces, measure of noncompactness, infinite system of differential equations
Categories:46B15, 46B45, 46B50, 34A34, 34G20
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