1. CMB Online first
 Awonusika, Richard; Taheri, Ali

A spectral identity on Jacobi polynomials and its analytic implications
The Jacobi coefficients $c^{\ell}_{j}(\alpha,\beta)$ ($1\leq
j\leq \ell$, $\alpha,\beta\gt 1$) are linked to the Maclaurin
spectral expansion of the Schwartz kernel of functions of the
Laplacian on a compact rank one symmetric space. It
is proved that these coefficients can be computed by transforming
the even derivatives of the the Jacobi polynomials $P_{k}^{(\alpha,\beta)}$ ($k\geq 0, \alpha,\beta\gt 1$) into a spectral sum associated with
the Jacobi operator. The first few coefficients are explicitly
computed and a direct trace
interpretation of the Maclaurin coefficients is presented.
Keywords:Jacobi coefficient, LaplaceBeltrami operator, symmetric space, Maclaurin expansion, Jacobi polynomial Categories:33C05, 33C45, 35A08, 35C05, 35C10, 35C15 

2. CMB Online first
 Lee, Hao

Irregular Weight one points with $D_{4}$ Image
Darmon, Lauder and Rotger conjectured that the relative tangent space of the eigencurve at a classical, ordinary, irregular weight one point is of dimension two. This space can be identified with the space of normalized overconvergent generalized eigenforms, whose Fourier coefficients can be conjecturally described explicitly in terms of $p$adic logarithms of algebraic numbers. This article presents the proof of this conjecture in the case where the weight one point is the intersection of two Hida families of Hecke theta series.
Keywords:weight one points, irregular, dihedral image, generalized eigenform, eigencurve, tangent space, Categories:11F33, 11F80 

3. 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 

4. CMB Online first
 Marković, Marijan

Differentialfree characterisation of smooth mappings with controlled growth
In this paper we give some generalizations and
improvements of the PavloviÄ result on the
HollandWalsh type characterization of the Bloch space of
continuously differentiable (smooth) functions in
the unit ball in $\mathbf{R}^m$.
Keywords:Bloch type space, Lipschitz type space, HollandWalsh characterisation, hyperbolic distance, analytic function, Mobius transform Categories:32A18, 30D45 

5. 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 

6. CMB Online first
 Bénéteau, Catherine Anne; Fleeman, Matthew C.; Khavinson, Dmitry S.; Seco, Daniel; Sola, Alan

Remarks on inner functions and optimal approximants
We discuss the concept of inner function in reproducing kernel Hilbert spaces with an orthogonal basis of monomials and examine connections between inner functions and optimal polynomial approximants to $1/f$, where $f$ is a function in the space. We revisit some classical examples from this perspective, and show how a construction of Shapiro and Shields can be modified to produce inner functions.
Keywords:inner function, reproducing Kernel Hilbert Space, operatortheoretic function theory Categories:46E22, 30J05 

7. CMB 2017 (vol 60 pp. 673)
 Abtahi, Fatemeh; Azizi, Mohsen; Rejali, Ali

Character Amenability of the Intersection of Lipschitz Algebras
Let $(X,d)$ be a metric space and $J\subseteq [0,\infty)$ be
nonempty. We study the structure of the arbitrary intersections
of
Lipschitz algebras, and define a special Banach subalgebra of
$\bigcap_{\gamma\in J}\operatorname{Lip}_\gamma X$, denoted by
$\operatorname{ILip}_J X$. Mainly,
we investigate $C$character amenability of $\operatorname{ILip}_J X$, in
particular Lipschitz algebras. We address a gap in the proof
of a
recent result in this field. Then we remove this gap, and obtain
a
necessary and sufficient condition for $C$character amenability
of $\operatorname{ILip}_J X$, specially Lipschitz algebras, under an additional
assumption.
Keywords:amenability, character amenability, Lipschitz algebra, metric space Categories:46H05, 46J10, 11J83 

8. CMB Online first
 Bu, Shangquan; Cai, Gang

Periodic solutions of second order degenerate differential equations with delay in Banach spaces
We give necessary and sufficient
conditions of the $L^p$wellposedness (resp. $B_{p,q}^s$wellposedness) for the second order degenerate
differential equation with finite delays:
$(Mu)''(t)+Bu'(t)+Au(t)=Gu'_t+Fu_t+f(t),(t\in [0,2\pi])$ with periodic
boundary conditions $(Mu)(0)=(Mu)(2\pi)$, $(Mu)'(0)=(Mu)'(2\pi)$, where
$A, B, M$ are closed linear operators on a complex Banach space $X$ satisfying
$D(A)\cap D(B)\subset D(M)$, $F$ and $G$ are bounded linear operators from
$L^p([2\pi,0];X)$ (resp. $B_{p,q}^s([2\pi,0];X)$) into $X$.
Keywords:second order degenerate differential equation, Fourier multiplier theorem, wellposedness, LebesgueBochner space, Besov space Categories:34G10, 34K30, 43A15, 47D06 

9. 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 

10. CMB Online first
 Figiel, Tadeusz; Johnson, William

Quotients of Essentially Euclidean Spaces
A precise quantitative version of the following qualitative statement
is proved: If a finite dimensional normed space contains approximately
Euclidean subspaces of all proportional dimensions, then every
proportional dimensional quotient space has the same property.
Keywords:essentially euclidean space Categories:46B20, 46B07, 46B99 

11. CMB 2017 (vol 60 pp. 571)
12. CMB Online first
 Wang, Lian Daniel

A Multiplier Theorem on Anisotropic Hardy Spaces
We present a multiplier theorem on anisotropic
Hardy spaces. When $m$ satisfies the anisotropic, pointwise Mihlin
condition, we obtain boundedness of the multiplier operator $T_m
: H_A^p (\mathbb R^n) \rightarrow H_A^p (\mathbb R^n)$, for the range of $p$
that depends on the eccentricities of the dilation $A$ and the
level of regularity of a multiplier symbol $m$. This extends
the classical multiplier theorem of Taibleson and Weiss.
Keywords:anisotropic Hardy space, multiplier, Fourier transform Categories:42B30, 42B25, 42B35 

13. CMB Online first
 Bu, Shangquan; Cai, Gang

HÃ¶lder continuous solutions of degenerate differential equations with finite delay
Using known operatorvalued Fourier multiplier results on vectorvalued
HÃ¶lder continuous function spaces $C^\alpha (\mathbb R; X)$, we completely
characterize the $C^\alpha$wellposedness of the first order
degenerate differential equations with finite delay $(Mu)'(t)
= Au(t) + Fu_t + f(t)$ for $t\in\mathbb R$
by the boundedness of the $(M, F)$resolvent of $A$ under suitable
assumption on the delay operator $F$, where $A, M$ are closed
linear
operators on a Banach space $X$ satisfying $D(A)\cap D(M) \not=\{0\}$,
the delay operator $F$ is a bounded linear operator
from $C([r, 0]; X)$ to $X$ and $r \gt 0$ is fixed.
Keywords:wellposedness, degenerate differential equation, $\dot{C}^\alpha$multiplier, HÃ¶lder continuous function space Categories:34N05, 34G10, 47D06, 47A10, 34K30 

14. CMB 2017 (vol 60 pp. 690)
 Bao, Guanlong; Göğüş, Nıhat Gökhan; Pouliasis, Stamatis

$\mathcal{Q}_p$ Spaces and Dirichlet Type Spaces
In this paper, we show that the MÃ¶bius invariant
function space $\mathcal {Q}_p$ can be generated by variant
Dirichlet type spaces
$\mathcal{D}_{\mu, p}$ induced by finite positive Borel measures
$\mu$ on the open unit disk. A criterion for the equality between
the space $\mathcal{D}_{\mu, p}$ and the usual Dirichlet type
space $\mathcal {D}_p$ is given. We obtain a sufficient condition
to construct different $\mathcal{D}_{\mu, p}$ spaces
and we provide examples.
We establish decomposition theorems for $\mathcal{D}_{\mu,
p}$ spaces, and prove that the nonHilbert space $\mathcal
{Q}_p$ is equal to the intersection of Hilbert spaces $\mathcal{D}_{\mu,
p}$. As an application of the relation between $\mathcal {Q}_p$
and $\mathcal{D}_{\mu, p}$ spaces, we also obtain that there
exist different $\mathcal{D}_{\mu, p}$ spaces; this is a trick
to prove the existence without constructing examples.
Keywords:$\mathcal {Q}_p$ space, Dirichlet type space, MÃ¶bius invariant function space Categories:30H25, 31C25, 46E15 

15. CMB 2016 (vol 60 pp. 655)
 Zhuo, Ciqiang; Sickel, Winfried; Yang, Dachun; Yuan, Wen

Characterizations of BesovType and TriebelLizorkinType Spaces via Averages on Balls
Let $\ell\in\mathbb N$ and $\alpha\in (0,2\ell)$. In this article,
the authors establish
equivalent characterizations
of Besovtype spaces, TriebelLizorkintype
spaces and BesovMorrey spaces via the sequence
$\{fB_{\ell,2^{k}}f\}_{k}$ consisting of the difference between
$f$ and
the ball average $B_{\ell,2^{k}}f$. These results give a way
to introduce Besovtype spaces,
TriebelLizorkintype spaces and BesovMorrey spaces with any
smoothness order
on metric measure spaces. As special cases, the authors obtain
a new characterization of MorreySobolev spaces
and $Q_\alpha$ spaces with $\alpha\in(0,1)$, which are of independent
interest.
Keywords:Besov space, TriebelLizorkin space, ball average, CalderÃ³n reproducing formula Categories:42B25, 46E35, 42B35 

16. CMB 2016 (vol 60 pp. 712)
 Chen, ChungChuan

Disjoint Hypercyclicity and Weighted Translations on Discrete Groups
Let $1\leq p\lt \infty$, and let $G$ be a discrete group. We give
a sufficient and necessary condition
for weighted translation operators on the Lebesgue space $\ell^p(G)$
to be densely disjoint hypercyclic.
The characterization for the dual of a weighted translation to
be densely disjoint hypercyclic is also obtained.
Keywords:disjoint hypercyclicity, topological transitivity, weighted translation, $\ell^p$space Categories:47A16, 47B38, 43A15 

17. CMB 2016 (vol 60 pp. 131)
18. CMB 2016 (vol 60 pp. 217)
 Wang, Yuanyi

Condition $C'_{\wedge}$ of Operator Spaces
In this paper, we study condition $C'_{\wedge}$ which is a
projective tensor product analogue of condition $C'$. We show
that
the finitedimensional OLLP operator spaces have condition
$C'_{\wedge}$ and $M_{n}$ $(n\gt 2)$ does not have that property.
Keywords:operator space, local theory, tensor product Category:46L07 

19. CMB 2016 (vol 60 pp. 104)
20. CMB 2016 (vol 60 pp. 522)
 Iena, Oleksandr; Leytem, Alain

On the Singular Sheaves in the Fine Simpson Moduli Spaces of $1$dimensional Sheaves
In the Simpson moduli space $M$ of semistable sheaves with
Hilbert polynomial $dm1$ on a projective plane we study the
closed subvariety $M'$ of sheaves that are not locally free on
their support. We show that for $d\ge 4$ it is a singular subvariety
of codimension $2$ in $M$. The blow up of $M$ along $M'$ is interpreted
as a (partial) modification of $M\setminus M'$ by line bundles
(on support).
Keywords:Simpson moduli spaces, coherent sheaves, vector bundles on curves, singular sheaves Category:14D20 

21. CMB 2016 (vol 60 pp. 586)
 Liu, Feng; Wu, Huoxiong

Endpoint Regularity of Multisublinear Fractional Maximal Functions
In this paper we investigate
the endpoint regularity properties of the multisublinear
fractional maximal operators, which include the multisublinear
HardyLittlewood maximal operator. We obtain some new bounds
for the derivative of the onedimensional multisublinear
fractional maximal operators acting on vectorvalued function
$\vec{f}=(f_1,\dots,f_m)$ with all $f_j$ being $BV$functions.
Keywords:multisublinear fractional maximal operators, Sobolev spaces, bounded variation Categories:42B25, 46E35 

22. CMB 2016 (vol 59 pp. 813)
23. CMB 2016 (vol 60 pp. 77)
 Christ, Michael; Rieffel, Marc A.

Nilpotent Group C*algebras as Compact Quantum Metric Spaces
Let $\mathbb{L}$ be a length function on a group $G$, and let $M_\mathbb{L}$
denote the
operator of pointwise multiplication by $\mathbb{L}$ on $\lt(G)$.
Following Connes,
$M_\mathbb{L}$ can be used as a ``Dirac'' operator for the reduced
group C*algebra $C_r^*(G)$. It defines a
Lipschitz seminorm on $C_r^*(G)$, which defines a metric on the
state space of
$C_r^*(G)$. We show that
for any length function satisfying a strong form of polynomial
growth on a discrete group,
the topology from this metric
coincides with the
weak$*$ topology (a key property for the
definition of a ``compact quantum metric
space''). In particular, this holds for all wordlength functions
on finitely generated nilpotentbyfinite groups.
Keywords:group C*algebra, Dirac operator, quantum metric space, discrete nilpotent group, polynomial growth Categories:46L87, 20F65, 22D15, 53C23, 58B34 

24. CMB 2016 (vol 60 pp. 546)
25. CMB 2016 (vol 59 pp. 834)
 Liao, Fanghui; Liu, Zongguang

Some Properties of TriebelLizorkin and Besov Spaces Associated with Zygmund Dilations
In this paper, using CalderÃ³n's
reproducing formula and almost orthogonality estimates, we
prove the lifting property and the embedding theorem of the TriebelLizorkin
and Besov spaces associated with Zygmund dilations.
Keywords:TriebelLizorkin and Besov spaces, Riesz potential, CalderÃ³n's reproducing formula, almost orthogonality estimate, Zygmund dilation, embedding theorem Categories:42B20, 42B35 
