1. CMB Online first
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 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 

5. CMB 2017 (vol 60 pp. 571)
6. CMB 2016 (vol 60 pp. 131)
7. 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 

8. 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 

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 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 

11. CMB 2011 (vol 55 pp. 555)
 Michalowski, Nicholas; Rule, David J.; Staubach, Wolfgang

Weighted $L^p$ Boundedness of Pseudodifferential Operators and Applications
In this paper we prove weighted norm inequalities with weights in
the $A_p$ classes, for pseudodifferential operators with symbols in
the class ${S^{n(\rho 1)}_{\rho, \delta}}$ that fall outside the
scope of CalderÃ³nZygmund theory. This is accomplished by
controlling the sharp function of the pseudodifferential operator by
HardyLittlewood type maximal functions. Our weighted norm
inequalities also yield $L^{p}$ boundedness of commutators of
functions of bounded mean oscillation with a wide class of operators
in $\mathrm{OP}S^{m}_{\rho, \delta}$.
Keywords:weighted norm inequality, pseudodifferential operator, commutator estimates Categories:42B20, 42B25, 35S05, 47G30 

12. CMB 2010 (vol 54 pp. 113)
 Hytönen, Tuomas P.

On the Norm of the BeurlingAhlfors Operator in Several Dimensions
The generalized BeurlingAhlfors operator $S$ on
$L^p(\mathbb{R}^n;\Lambda)$, where $\Lambda:=\Lambda(\mathbb{R}^n)$ is the
exterior algebra with its natural Hilbert space norm, satisfies the
estimate
$$\S\_{\mathcal{L}(L^p(\mathbb{R}^n;\Lambda))}\leq(n/2+1)(p^*1),\quad
p^*:=\max\{p,p'\}$$
This improves on earlier results in all dimensions $n\geq 3$. The
proof is based on the heat extension and relies at the bottom on
Burkholder's sharp inequality for martingale transforms.
Categories:42B20, 60G46 

13. 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 

14. 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 

15. CMB 2009 (vol 52 pp. 521)
16. CMB 2006 (vol 49 pp. 3)
 AlSalman, Ahmad

On a Class of Singular Integral Operators With Rough Kernels
In this paper, we study the $L^p$ mapping properties of a class of singular
integral operators with rough kernels belonging to certain block spaces. We
prove that our operators are bounded on $L^p$ provided that their kernels
satisfy a size condition much weaker than that for the classical
Calder\'{o}nZygmund singular integral operators. Moreover, we present an
example showing that our size condition is optimal. As a consequence of our
results, we substantially improve a previously known result on certain maximal
functions.
Keywords:Singular integrals, Rough kernels, Square functions,, Maximal functions, Block spaces Categories:42B20, 42B15, 42B25 

17. CMB 2004 (vol 47 pp. 3)
18. CMB 2001 (vol 44 pp. 121)
19. CMB 1999 (vol 42 pp. 463)
 Hofmann, Steve; Li, Xinwei; Yang, Dachun

A Generalized Characterization of Commutators of Parabolic Singular Integrals
Let $x=(x_1, \dots, x_n)\in\rz$ and $\dz_\lz x=(\lz^{\az_1}x_1,
\dots,\lz^{\az_n}x_n)$, where $\lz>0$ and $1\le \az_1\le\cdots
\le\az_n$. Denote $\az=\az_1+\cdots+\az_n$. We characterize those
functions $A(x)$ for which the parabolic Calder\'on commutator
$$
T_{A}f(x)\equiv \pv \int_{\mathbb{R}^n}
K(xy)[A(x)A(y)]f(y)\,dy
$$
is bounded on $L^2(\mathbb{R}^n)$, where $K(\dz_\lz x)=\lz^{\az1}K(x)$,
$K$ is smooth away from the origin and satisfies a certain cancellation
property.
Keywords:parabolic singular integral, commutator, parabolic $\BMO$ sobolev space, homogeneous space, T1theorem, symbol Category:42B20 

20. CMB 1998 (vol 41 pp. 478)
21. CMB 1998 (vol 41 pp. 404)
 AlHasan, Abdelnaser J.; Fan, Dashan

$L^p$boundedness of a singular integral operator
Let $b(t)$ be an $L^\infty$ function on $\bR$, $\Omega (\,y')$ be
an $H^1$ function on the unit sphere satisfying the mean zero
property (1) and $Q_m(t)$ be a real polynomial on $\bR$ of degree
$m$ satisfying $Q_m(0)=0$. We prove that the singular integral
operator
$$
T_{Q_m,b} (\,f) (x)=p.v. \int_\bR^n b(y) \Omega(\,y) y^{n} f
\left( xQ_m (y) y' \right) \,dy
$$
is bounded in $L^p (\bR^n)$ for $1
Keywords:singular integral, rough kernel, Hardy space Category:42B20 
