1. CMB Online first
 Dang, Pei; Liu, Hua; Qian, Tao

Hilbert Transformation and Representation of the $ax+b$ Group
In this paper we study the Hilbert transformations over
$L^2(\mathbb{R})$
and $L^2(\mathbb{T})$ from
the viewpoint of symmetry. For a linear operator over $L^2(\mathbb{R})$
commutative with the ax+b group we show that the operator is
of the form
$
\lambda I+\eta H,
$
where $I$ and $H$ are the identity operator and Hilbert transformation
respectively, and $\lambda,\eta$ are complex numbers. In the
related literature this result was proved through first invoking
the boundedness result of the operator, proved though a big
machinery.
In our setting the boundedness is a consequence of the boundedness
of the Hilbert transformation. The methodology that we use is
GelfandNaimark's representation of the ax+b group. Furthermore
we prove a similar result on the unit circle. Although there
does not exist a group like ax+b on the unit circle, we construct
a semigroup to play the same symmetry role for the Hilbert transformations
over the circle $L^2(\mathbb{T}).$
Keywords:singular integral, Hilbert transform, the $ax+b$ group Categories:30E25, 44A15, 42A50 

2. CMB Online first
 Li, Junfeng; Yu, Haixia

Oscillatory HyperHilbert Transform Associated with Plane Curves
In this paper, the bounded properties of oscillatory hyperHilbert
transform along certain plane curves $\gamma(t)$
$$T_{\alpha,\beta}f(x,y)=\int_{0}^1f(xt,y\gamma(t))e^{ i t^{\beta}}\frac{\textrm{d}t}{t^{1+\alpha}}$$
were studied. For a general curves, these operators are bounded
in ${L^2(\mathbb{R}^{2})}$, if $\beta\geq 3\alpha$. And their
boundedness in $L^p(\mathbb{R}^{2})$
were also obtained, whenever $\beta\gt 3\alpha$, $\frac{2\beta}{2\beta3\alpha}\lt p\lt \frac{2\beta}{3\alpha}$.
Keywords:oscillatory hyperHilbert transform, oscillatory integral Categories:42B20, 42B35 

3. 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 

4. CMB Online first
 Ding, Yong; Lai, Xudong

On a singular integral of ChristJournÃ© type with homogeneous kernel
In this paper, we prove that the following singular integral
defined by
$$T_{\Omega,a}f(x)=\operatorname{p.v.}\int_{\mathbb{R}^{d}}\frac{\Omega(xy)}{xy^d}\cdot m_{x,y}a\cdot
f(y)dy$$
is bounded on $L^p(\mathbb{R}^d)$ for $1\lt p\lt \infty$ and is of weak type
(1,1), where $\Omega\in L\log^+L(\mathbb{S}^{d1})$ and
$m_{x,y}a=:\int_0^1a(sx+(1s)y)ds$
with $a\in L^\infty(\mathbb{R}^d)$ satisfying some restricted conditions.
Keywords:CalderÃ³n commutator, rough kernel, weak type (1,1) Category:42B20 

5. CMB 2017 (vol 60 pp. 571)
6. 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 

7. CMB Online first
8. 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 

9. CMB 2016 (vol 60 pp. 131)
10. 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 

11. 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 

12. CMB 2016 (vol 59 pp. 528)
 Jahan, Qaiser

Characterization of Lowpass Filters on Local Fields of Positive Characteristic
In this article, we give necessary and sufficient conditions
on a function to be a lowpass filter on a local field $K$ of
positive characteristic associated to the scaling function for
multiresolution analysis of $L^2(K)$. We use probability and
martingale methods to provide such a characterization.
Keywords:multiresolution analysis, local field, lowpass filter, scaling function, probability, conditional probability and martingales Categories:42C40, 42C15, 43A70, 11S85 

13. CMB 2016 (vol 59 pp. 497)
14. CMB 2016 (vol 59 pp. 521)
 Hare, Kathryn; Ramsey, L. Thomas

The Relationship Between $\epsilon$Kronecker Sets and Sidon Sets
A subset $E$ of a discrete abelian group is called $\epsilon
$Kronecker if
all $E$functions of modulus one can be approximated to within
$\epsilon $
by characters. $E$ is called a Sidon set if all bounded $E$functions
can be
interpolated by the Fourier transform of measures on the dual
group. As $%
\epsilon $Kronecker sets with $\epsilon \lt 2$ possess the same
arithmetic
properties as Sidon sets, it is natural to ask if they are Sidon.
We use the
Pisier net characterization of Sidonicity to prove this is true.
Keywords:Kronecker set, Sidon set Categories:43A46, 42A15, 42A55 

15. CMB 2015 (vol 59 pp. 62)
 Feng, Han

Uncertainty Principles on Weighted Spheres, Balls and Simplexes
This paper studies the uncertainty principle for spherical
$h$harmonic expansions on the unit sphere of $\mathbb{R}^d$ associated
with a weight function invariant under a general finite reflection
group, which
is in full analogy with the classical Heisenberg inequality.
Our proof is motivated by a new decomposition of the DunklLaplaceBeltrami
operator on the weighted sphere.
Keywords:uncertainty principle, Dunkl theory Categories:42C10, 42B10 

16. CMB 2015 (vol 59 pp. 104)
 He, Ziyi; Yang, Dachun; Yuan, Wen

LittlewoodPaley Characterizations of SecondOrder Sobolev Spaces via Averages on Balls
In this paper, the authors characterize secondorder 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 LittlewoodPaley $g_\lambda^\ast$function in
terms of ball means.
Keywords:Sobolev space, ball means, Lusinarea function, $g_\lambda^*$function Categories:46E35, 42B25, 42B20, 42B35 

17. CMB 2015 (vol 58 pp. 877)
 Zaatra, Mohamed

Generating Some Symmetric Semiclassical Orthogonal Polynomials
We show that if $v$ is a regular semiclassical form
(linear functional), then the symmetric form $u$ defined by the
relation
$x^{2}\sigma u = \lambda v$,
where $(\sigma f)(x)=f(x^{2})$ and the odd
moments of $u$ are $0$, is also
regular and semiclassical form for every
complex $\lambda $ except for a discrete set of numbers depending
on $v$. We give explicitly the threeterm recurrence relation
and the
structure relation coefficients of the orthogonal polynomials
sequence associated with $u$ and the class of the form $u$ knowing
that of $v$. We conclude with an illustrative example.
Keywords:orthogonal polynomials, quadratic decomposition, semiclassical forms, structure relation Categories:33C45, 42C05 

18. CMB 2015 (vol 59 pp. 211)
 Totik, Vilmos

Universality Under SzegÅ's Condition
This paper presents a
theorem on universality on orthogonal polynomials/random matrices
under a weak local condition on the weight function $w$.
With a new inequality for
polynomials and with the use of fast decreasing polynomials,
it is shown that an approach of
D. S. Lubinsky is applicable. The proof works
at all points which are Lebesguepoints both
for the weight function $w$ and for $\log w$.
Keywords:universality, random matrices, Christoffel functions, asymptotics, potential theory Categories:42C05, 60B20, 30C85, 31A15 

19. CMB 2015 (vol 58 pp. 757)
20. CMB 2015 (vol 58 pp. 507)
 Hsu, MingHsiu; Lee, MingYi

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:MongeAmpere equation, parabolic section, Hardy space, BMO, VMO Category:42B30 

21. CMB 2015 (vol 58 pp. 808)
 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
firstorder Sobolev spaces. Moreover, two key pointwise
inequalities for the partial derivatives of the multisublinear
maximal functions are established. As an application, the
quasicontinuity on the multisublinear maximal function is also
obtained.
Keywords:regularity, multisublinear maximal operator, Sobolev spaces, partial deviative, quasicontinuity Categories:42B25, 46E35 

22. CMB 2014 (vol 58 pp. 432)
 Yang, Dachun; Yang, Sibei

Secondorder Riesz Transforms and Maximal Inequalities Associated with Magnetic SchrÃ¶dinger Operators
Let $A:=(\nablai\vec{a})\cdot(\nablai\vec{a})+V$ be a
magnetic SchrÃ¶dinger 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
HÃ¶lder 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
secondorder Riesz transforms $VA^{1}$ and
$(\nablai\vec{a})^2A^{1}$ are bounded from the
MusielakOrliczHardy space $H_{\varphi,\,A}(\mathbb{R}^n)$, associated with $A$,
to the MusielakOrlicz 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:MusielakOrliczHardy space, magnetic SchrÃ¶dinger operator, atom, secondorder Riesz transform, maximal inequality Categories:42B30, 42B35, 42B25, 35J10, 42B37, 46E30 

23. 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 

24. CMB 2014 (vol 58 pp. 144)
25. CMB 2014 (vol 57 pp. 834)
 Koh, Doowon

Restriction Operators Acting on Radial Functions on Vector Spaces Over Finite Fields
We study $L^pL^r$ restriction estimates for
algebraic varieties $V$ in the case when restriction operators act on
radial functions in the finite field setting.
We show that if the varieties $V$ lie in odd dimensional vector
spaces over finite fields, then the conjectured restriction estimates
are possible for all radial test functions.
In addition, assuming that the varieties $V$ are defined in even
dimensional spaces and have few intersection points with the sphere
of zero radius, we also obtain the conjectured exponents for all
radial test functions.
Keywords:finite fields, radial functions, restriction operators Categories:42B05, 43A32, 43A15 
