1. CMB Online first
2. CMB Online first
 Hou, Xianming; Wu, Huoxiong

On the limiting weaktype behaviors for maximal operators associated with power weighted measure
Let $\beta\ge 0$ and $e_1=(1,0,\ldots,0)$ is a unit vector on
$\mathbb{R}^{n}$, $d\mu(x)=x^\beta dx$ is a power weighted
measure on $\mathbb{R}^n$. For $0\le \alpha\lt n$, let $M_\mu^\alpha$
be the centered HardyLittlewood maximal function and fractional
maximal functions associated to measure $\mu$. This paper shows
that for $q=n/(n\alpha)$, $f\in L^1(\mathbb{R}^n,d\mu)$,
$$\lim\limits_{\lambda\to 0+}\lambda^q \mu(\{x\in\mathbb{R}^n:M_\mu^\alpha
f(x)\gt \lambda\})=\frac{\omega_{n1}}{(n+\beta)\mu(B(e_1,1))}\f\_{L^1(\mathbb{R}^n,
d\mu)}^q,$$
and
$$\lim_{\lambda\to 0+}\lambda^q \mu\Big(\Big\{x\in\mathbb{R}^n:\BigM_\mu^\alpha
f(x)\frac{\f\_{L^1(\mathbb{R}^n, d\mu)}}{\mu(B(x,x))^{1\alpha/n}}\Big\gt \lambda\Big\}\Big)=0,$$
which is new and stronger than the previous result even if $\beta=0$.
Meanwhile, the corresponding results for the uncentered maximal
functions as well as the fractional integral operators with respect
to measure $\mu$ are also obtained.
Keywords:limiting weak type behavior, power weight, HardyLittlewood maximal operator, fractional maximal operator, fractional integral Categories:42B20, 42B25 

3. CMB Online first
 de Carli, Laura; Mizrahi, Alberto; Tepper, Alexander

Three problems on exponential bases
We consider three special and significant cases of the following
problem. Let $D\subset\mathbb{R}^d$ be a (possibly unbounded) set
of finite Lebesgue measure.
Let $E( \mathbb{Z}^d)=\{e^{2\pi i x\cdot n}\}_{n\in\mathbb{Z}^d}$ be the standard
exponential basis on the unit cube of $\mathbb{R}^d$.
Find conditions on $D$ for which $E(\mathbb{Z}^d)$ is a frame, a
Riesz sequence, or a Riesz basis for $L^2(D)$.
Keywords:exponential basis, frame, Riesz sequence, lattice Categories:42C15, 42C30 

4. CMB Online first
 CruzUribe, David; Rodney, Scott; Rosta, Emily

PoincarÃ© Inequalities and Neumann Problems for the $p$Laplacian
We prove an equivalence between weighted PoincarÃ© inequalities
and
the existence of weak solutions to a Neumann problem related
to a
degenerate $p$Laplacian. The PoincarÃ© inequalities are
formulated in the context of degenerate Sobolev spaces defined
in
terms of a quadratic form, and the associated matrix is the
source of
the degeneracy in the $p$Laplacian.
Keywords:degenerate Sobolev space, $p$Laplacian, PoincarÃ© inequalities Categories:30C65, 35B65, 35J70, 42B35, 42B37, 46E35 

5. CMB Online first
6. 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 

7. CMB 2017 (vol 61 pp. 70)
 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 

8. CMB 2017 (vol 61 pp. 370)
 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 

9. CMB 2017 (vol 61 pp. 97)
 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 

10. CMB 2017 (vol 60 pp. 571)
11. CMB 2017 (vol 61 pp. 390)
 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 

12. CMB 2017 (vol 61 pp. 191)
13. 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 

14. CMB 2016 (vol 60 pp. 131)
15. 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 

16. 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 

17. 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 

18. CMB 2016 (vol 59 pp. 497)
19. 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 

20. 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 

21. 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 

22. 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 

23. 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 

24. CMB 2015 (vol 58 pp. 757)
25. 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 
