1. CMB Online first
 Reijonen, Atte

Remark on integral means of derivatives of Blaschke products
If $B$ is the Blachke product with zeros $\{z_n\}$, then $B'(z)\le
\Psi_B(z)$, where
$$\Psi_B(z)=\sum_n \frac{1z_n^2}{1\overline{z}_nz^2}.$$
Moreover, it is a wellknown fact that, for $0\lt p\lt \infty$,
$$M_p(r,B')=
\left(\frac{1}{2\pi}\int_{0}^{2\pi} B'(re^{i\t})^p\,d\t
\right)^{1/p}, \quad 0\le r\lt 1,$$
is bounded if and only if $M_p(r,\Psi_B)$ is bounded.
We find a Blaschke product $B_0$ such that $M_p(r,B_0')$ and
$M_p(r,\Psi_{B_0})$ are not comparable for any $\frac12\lt p\lt \infty$.
In addition, it is shown that, if $0\lt p\lt \infty$, $B$ is a CarlesonNewman
Blaschke product and a weight $\omega$ satisfies a certain regularity
condition, then
$$
\int_\mathbb{D} B'(z)^p\omega(z)\,dA(z)\asymp \int_\mathbb{D} \Psi_B(z)^p\omega(z)\,dA(z),
$$
where $dA(z)$ is the Lebesgue area measure on the unit disc.
Keywords:Bergman space, Blaschke product, Hardy space, integral mean Categories:30J10, 30H10, 30H20 

2. CMB Online first
 Rocha, Pablo Alejandro

A remark on certain integral operators of fractional type
For $m, n \in \mathbb{N}$, $1\lt m \leq n$, we write $n = n_1 +
\dots + n_m$ where $\{ n_1, \dots, n_m \} \subset \mathbb{N}$. Let
$A_1, \dots, A_m$ be $n \times n$ singular real matrices such that
$\bigoplus_{i=1}^{m} \bigcap_{1\leq j \neq i \leq m} \mathcal{N}_j
= \mathbb{R}^{n},$ where
$\mathcal{N}_j = \{ x : A_j x = 0 \}$, $dim(\mathcal{N}_j)=nn_j$
and $A_1+ \dots+ A_m$ is invertible. In this paper we study integral
operators of the form
$T_{r}f(x)= \int_{\mathbb{R}^{n}} \, xA_1 y^{n_1 + \alpha_1}
\cdots xA_m y^{n_m + \alpha_m} f(y) \, dy,$
$n_1 + \dots + n_m = n$, $\frac{\alpha_1}{n_1} = \dots = \frac{\alpha_m}{n_m}=r$,
$0 \lt r \lt 1$, and the matrices $A_i$'s are as above. We obtain
the $H^{p}(\mathbb{R}^{n})L^{q}(\mathbb{R}^{n})$ boundedness
of $T_r$ for $0\lt p\lt \frac{1}{r}$ and $\frac{1}{q}=\frac{1}{p} 
r$.
Keywords:integral operator, Hardy space Categories:42B20, 42B30 

3. CMB Online first
 Cui, Xiaohui; Wang, Chunjie; Zhu, Kehe

Area Integral Means of Analytic Functions in the Unit Disk
For an analytic function $f$ on the unit disk $\mathbb D$ we show that
the $L^2$ integral mean of $f$ on $c\lt z\lt r$ with
respect to the weighted area measure $(1z^2)^\alpha\,dA(z)$
is a logarithmically convex function of $r$ on $(c,1)$,
where $3\le\alpha\le0$ and $c\in[0,1)$. Moreover, the range
$[3,0]$ for $\alpha$ is best possible. When
$c=0$, our arguments here also simplify the proof for several
results we obtained in earlier papers.
Keywords:logarithmic convexity, area integral mean, Bergman space, Hardy space Categories:30H10, 30H20 

4. CMB 2016 (vol 60 pp. 510)
 Haase, Christian; Hofmann, Jan

Convexnormal (Pairs of) Polytopes
In 2012 Gubeladze (Adv. Math. 2012)
introduced the notion of $k$convexnormal polytopes to show
that
integral polytopes all of whose edges are longer than $4d(d+1)$
have
the integer decomposition property.
In the first part of this paper we show that for lattice polytopes
there is no difference between $k$ and $(k+1)$convexnormality
(for
$k\geq 3 $) and improve the bound to $2d(d+1)$. In the second
part we
extend the definition to pairs of polytopes. Given two rational
polytopes $P$ and $Q$, where the normal fan of $P$ is a refinement
of
the normal fan of $Q$.
If every edge $e_P$ of $P$ is at least $d$ times as long as the
corresponding face (edge or vertex) $e_Q$ of $Q$, then $(P+Q)\cap
\mathbb{Z}^d
= (P\cap \mathbb{Z}^d ) + (Q \cap \mathbb{Z}^d)$.
Keywords:integer decomposition property, integrally closed, projectively normal, lattice polytopes Categories:52B20, 14M25, 90C10 

5. CMB 2016 (vol 60 pp. 131)
6. CMB Online first
 Xu, Xu; Zhu, Laiyi

Rational function operators from Poisson integrals
In this paper, we construct two classes of rational function
operators by using the Poisson integrals of the function on the
whole real
axis. The convergence rates of the uniform and mean approximation
of such rational function operators on the whole real axis are
studied.
Keywords:rational function operators, Poisson integrals, convergence rate, uniform approximation, mean approximation Categories:41A20, 41A25, 41A35 

7. CMB 2015 (vol 58 pp. 486)
 Duc, Dinh Thanh; Nhan, Nguyen Du Vi; Xuan, Nguyen Tong

Inequalities for Partial Derivatives and their Applications
We present various weighted integral inequalities for partial
derivatives acting on products and compositions of functions
which are applied to establish some new Opialtype inequalities
involving functions of several independent variables. We also
demonstrate the usefulness of our results in the field of partial
differential equations.
Keywords:inequality for integral, Opialtype inequality, HÃ¶lder's inequality, partial differential operator, partial differential equation Categories:26D10, 35A23 

8. CMB 2014 (vol 58 pp. 196)
9. CMB 2014 (vol 58 pp. 19)
 Chen, Jiecheng; Hu, Guoen

Compact Commutators of Rough Singular Integral Operators
Let $b\in \mathrm{BMO}(\mathbb{R}^n)$ and $T_{\Omega}$ be the singular
integral operator with kernel $\frac{\Omega(x)}{x^n}$, where
$\Omega$ is homogeneous of degree zero, integrable and has mean
value zero on the unit sphere $S^{n1}$. In this paper, by Fourier
transform estimates and approximation to the operator $T_{\Omega}$
by integral operators with smooth kernels, it is proved that if
$b\in \mathrm{CMO}(\mathbb{R}^n)$ and $\Omega$ satisfies a certain
minimal size condition, then the commutator generated by $b$ and
$T_{\Omega}$ is a compact operator on $L^p(\mathbb{R}^n)$ for
appropriate index $p$. The associated maximal operator is also
considered.
Keywords:commutator,singular integral operator, compact operator, maximal operator Category:42B20 

10. CMB 2013 (vol 57 pp. 9)
11. CMB 2011 (vol 56 pp. 593)
12. CMB 2011 (vol 55 pp. 646)
 Zhou, Jiang; Ma, Bolin

Marcinkiewicz Commutators with Lipschitz Functions in Nonhomogeneous Spaces
Under the assumption that $\mu$ is a nondoubling
measure, we study certain commutators generated by the
Lipschitz function and the Marcinkiewicz integral whose kernel
satisfies a HÃ¶rmandertype condition. We establish the boundedness
of these commutators on the Lebesgue spaces, Lipschitz spaces, and
Hardy spaces. Our results are extensions of known theorems in the
doubling case.
Keywords:non doubling measure, Marcinkiewicz integral, commutator, ${\rm Lip}_{\beta}(\mu)$, $H^1(\mu)$ Categories:42B25, 47B47, 42B20, 47A30 

13. CMB 2011 (vol 56 pp. 80)
 Islam, Muhammad N.

Three Fixed Point Theorems: Periodic Solutions of a Volterra Type Integral Equation with Infinite Heredity
In this paper we study the existence of periodic solutions of a Volterra type integral equation with infinite heredity. Banach fixed point theorem, Krasnosel'skii's fixed point theorem, and a combination of Krasnosel'skii's
and Schaefer's fixed point theorems are employed in the analysis.
The combination theorem of Krasnosel'skii and Schaefer requires an a priori bound on all solutions.
We employ Liapunov's direct method to obtain such an a priori bound.
In the process, we compare these theorems in terms of assumptions and outcomes.
Keywords:Volterra integral equation, periodic solutions, Liapunov's method, Krasnosel'skii's fixed point theorem, Schaefer's fixed point theorem Categories:45D05, 45J05 

14. CMB 2011 (vol 55 pp. 176)
 Spirn, Daniel; Wright, J. Douglas

Linear Dispersive Decay Estimates for the 3+1 Dimensional Water Wave Equation with Surface Tension
We consider the linearization of the threedimensional water waves
equation with surface tension about a flat interface. Using
oscillatory integral methods, we prove that solutions of this equation
demonstrate dispersive decay at the somewhat surprising rate of
$t^{5/6}$. This rate is due to competition between surface tension
and gravitation at $O(1)$ wave numbers and is connected to the fact
that, in the presence of surface tension, there is a socalled "slowest
wave". Additionally, we combine our dispersive estimates with $L^2$
type energy bounds to prove a family of Strichartz estimates.
Keywords:oscillatory integrals, water waves, surface tension, Strichartz estimates Categories:76B07, 76B15, 76B45 

15. CMB 2010 (vol 54 pp. 207)
 Chen, Jiecheng; Fan, Dashan

A Bilinear Fractional Integral on Compact Lie Groups
As an analog of a wellknown theorem on the bilinear
fractional integral on $\mathbb{R}^{n}$ by Kenig and Stein,
we establish the similar boundedness
property for a bilinear fractional integral on a compact Lie group. Our
result is also a generalization of our recent theorem
about the
bilinear fractional integral on torus.
Keywords:bilinear fractional integral, $L^p$ spaces, Heat kernel Categories:43A22, 43A32, 43B25 

16. CMB 2010 (vol 54 pp. 100)
 Fan, Dashan; Wu, Huoxiong

On the Generalized Marcinkiewicz Integral Operators with Rough Kernels
A class of generalized Marcinkiewicz
integral operators is introduced, and, under rather weak conditions
on the integral kernels, the boundedness of such operators on $L^p$
and TriebelLizorkin spaces is established.
Keywords: Marcinkiewicz integral, LittlewoodPaley theory, TriebelLizorkin space, rough kernel, product domain Categories:42B20, , , , , 42B25, 42B30, 42B99 

17. CMB 2010 (vol 53 pp. 639)
 Coykendall, Jim; Dutta, Tridib

A Generalization of Integrality
In this paper, we explore a generalization of the notion of
integrality. In particular, we study a nearintegrality condition that is
intermediate between the concepts of integral and almost integral.
This property (referred to as the $\Omega$almost integral
property) is a representative independent specialization of the
standard notion of almost integrality. Some of the properties of
this generalization are explored in this paper, and these properties
are compared with the notion of pseudointegrality introduced by
Anderson, Houston, and Zafrullah. Additionally, it is
shown that the $\Omega$almost integral property serves to
characterize the survival/lying over pairs of Dobbs and Coykendall
Keywords:integral closure, complete integral closure Categories:13B22, 13G05, 13B21 

18. CMB 2010 (vol 53 pp. 503)
 Kurenok, V. P.

The Time Change Method and SDEs with Nonnegative Drift
Using the time change method we show how to construct a solution to the stochastic equation $dX_t=b(X_{t})dZ_t+a(X_t)dt$ with a nonnegative drift $a$ provided there exists a solution to the auxililary equation $dL_t=[a^{1/\alpha}b](L_{t})d\bar Z_t+dt$ where $Z, \bar Z$ are two symmetric stable processes of the same index $\alpha\in(0,2]$. This approach allows us to prove the existence of solutions for both stochastic equations for the values $0<\alpha<1$ and only measurable coefficients $a$ and $b$ satisfying some conditions of boundedness. The existence proof for the auxililary equation uses the method of integral estimates in the sense of Krylov.
Keywords:Onedimensional SDEs, symmetric stable processes, nonnegative drift, time change, integral estimates, weak convergence Categories:60H10, 60J60, 60J65, 60G44 

19. CMB 2010 (vol 53 pp. 526)
 Milian, Anna

On Some Stochastic Perturbations of Semilinear Evolution Equations
We consider semilinear evolution equations with some locally Lipschitz nonlinearities, perturbed by Banach space valued, continuous, and adapted stochastic process. We show that under some assumptions there exists a solution to the equation. Using the result we show that there exists a mild, continuous, global solution to a semilinear ItÃ´ equation with locally Lipschitz nonlinearites. An example of the equation is given.
Keywords:evolution equation, mild solution, nonLipschitz drift, Ito integral Category:60H20 

20. CMB 2009 (vol 53 pp. 263)
 Feuto, Justin; Fofana, Ibrahim; Koua, Konin

Weighted Norm Inequalities for a Maximal Operator in Some Subspace of Amalgams
We give weighted norm inequalities for the maximal fractional operator $ \mathcal M_{q,\beta }$ of HardyÂLittlewood and the fractional integral $I_{\gamma}$. These inequalities are established between $(L^{q},L^{p}) ^{\alpha }(X,d,\mu )$ spaces (which are superspaces of Lebesgue spaces $L^{\alpha}(X,d,\mu)$ and subspaces of amalgams $(L^{q},L^{p})(X,d,\mu)$) and in the setting of space of homogeneous type $(X,d,\mu)$. The conditions on the weights are stated in terms of Orlicz norm.
Keywords:fractional maximal operator, fractional integral, space of homogeneous type Categories:42B35, 42B20, 42B25 

21. CMB 2008 (vol 51 pp. 618)
 Valmorin, V.

Vanishing Theorems in Colombeau Algebras of Generalized Functions
Using a canonical linear embedding of the algebra
${\mathcal G}^{\infty}(\Omega)$ of Colombeau generalized functions in the space of
$\overline{\C}$valued $\C$linear maps on the space
${\mathcal D}(\Omega)$ of smooth functions with compact support, we give vanishing
conditions for functions and linear integral operators of class
${\mathcal G}^\infty$. These results are then applied to the zeros of holomorphic
generalized functions in dimension greater than one.
Keywords:Colombeau generalized functions, linear integral operators, generalized holomorphic functions Categories:32A60, 45P05, 46F30 

22. CMB 2008 (vol 51 pp. 386)
 Lan, K. Q.; Yang, G. C.

Positive Solutions of the FalknerSkan Equation Arising in the Boundary Layer Theory
The wellknown FalknerSkan equation is one of the most important
equations in laminar boundary layer theory and is used to describe
the steady twodimensional flow of a slightly viscous
incompressible fluid past wedge shaped bodies of angles related to
$\lambda\pi/2$, where $\lambda\in \mathbb R$ is a parameter
involved in the equation. It is known that there exists
$\lambda^{*}<0$ such that the equation with suitable boundary
conditions has at least one positive solution for each $\lambda\ge
\lambda^{*}$ and has no positive solutions for
$\lambda<\lambda^{*}$. The known numerical result shows
$\lambda^{*}=0.1988$. In this paper, $\lambda^{*}\in
[0.4,0.12]$ is proved analytically by establishing a singular
integral equation which is equivalent to the FalknerSkan
equation. The equivalence result
provides new techniques to study properties and existence of solutions of
the FalknerSkan equation.
Keywords:FalknerSkan equation, boundary layer problems, singular integral equation, positive solutions Categories:34B16, 34B18, 34B40, 76D10 

23. CMB 2008 (vol 51 pp. 372)
24. CMB 2008 (vol 51 pp. 161)
 Agarwal, Ravi P.; OteroEspinar, Victoria; Perera, Kanishka; Vivero, Dolores R.

Wirtinger's Inequalities on Time Scales
This paper is devoted to the study of Wirtingertype
inequalities for the Lebesgue $\Delta$integral on an arbitrary time scale $\T$.
We prove a general inequality for a class of absolutely continuous
functions on closed subintervals of an adequate subset of $\T$.
By using this expression and by assuming that $\T$ is bounded,
we deduce that
a general inequality is valid for every absolutely continuous function on $\T$
such that its $\Delta$derivative belongs to $L_\Delta^2([a,b)\cap\T)$ and at most it vanishes
on the boundary of $\T$.
Keywords:time scales calculus, $\Delta$integral, Wirtinger's inequality Category:39A10 

25. CMB 2007 (vol 50 pp. 284)
 McIntosh, Richard J.

Second Order Mock Theta Functions
In his last letter to Hardy, Ramanujan
defined 17 functions $F(q)$, where $q<1$. He called them mock theta
functions, because as $q$ radially approaches any point $e^{2\pi ir}$
($r$ rational), there is a theta function $F_r(q)$ with $F(q)F_r(q)=O(1)$.
In this paper we establish the relationship between two families of mock
theta functions.
Keywords:$q$series, mock theta function, Mordell integral Categories:11B65, 33D15 
