1. CJM Online first
 Guo, Xiaoli; Hu, Guoen

On the commutators of singular integral operators with rough convolution kernels
Let $T_{\Omega}$ be the singular integral operator with kernel
$\frac{\Omega(x)}{x^n}$, where $\Omega$ is homogeneous of degree
zero, has mean value zero and belongs to $L^q(S^{n1})$ for
some
$q\in (1,\,\infty]$. In this paper, the authors establish the
compactness on weighted $L^p$ spaces, and the Morrey spaces,
for the commutator generated by $\operatorname{CMO}(\mathbb{R}^n)$ function
and $T_{\Omega}$. The associated maximal operator and the discrete
maximal operator are also considered.
Keywords:commutator, singular integral operator, compact operator, completely continuous operator, maximal operator, Morrey space Categories:42B20, 47B07 

2. CJM 2014 (vol 66 pp. 1358)
 Osėkowski, Adam

Sharp Localized Inequalities for Fourier Multipliers
In the paper we study sharp localized $L^q\colon L^p$ estimates for
Fourier multipliers resulting from modulation of the jumps of
LÃ©vy
processes.
The proofs of these estimates rest on probabilistic methods and
exploit related sharp bounds for differentially subordinated
martingales, which are of independent interest. The lower bounds
for
the constants involve the analysis of laminates, a family of
certain
special probability measures on $2\times 2$ matrices. As an
application, we obtain new sharp bounds for the real and imaginary
parts of the BeurlingAhlfors operator .
Keywords:Fourier multiplier, martingale, laminate Categories:42B15, 60G44, 42B20 

3. CJM 2013 (vol 66 pp. 1382)
 Wu, Xinfeng

Weighted Carleson Measure Spaces Associated with Different Homogeneities
In this paper, we introduce weighted Carleson measure spaces associated
with different homogeneities and prove that these spaces are the dual spaces
of weighted Hardy spaces studied in a forthcoming paper.
As an application, we establish
the boundedness of composition of two CalderÃ³nZygmund operators with
different homogeneities on the weighted Carleson measure spaces; this,
in particular, provides the weighted endpoint estimates for the operators
studied by PhongStein.
Keywords:composition of operators, weighted Carleson measure spaces, duality Categories:42B20, 42B35 

4. CJM 2013 (vol 65 pp. 1217)
 Cruz, Victor; Mateu, Joan; Orobitg, Joan

Beltrami Equation with Coefficient in Sobolev and Besov Spaces
Our goal in this work is to present some function spaces on the
complex plane $\mathbb C$, $X(\mathbb C)$, for which the quasiregular solutions of
the Beltrami equation, $\overline\partial f (z) = \mu(z) \partial f
(z)$, have first derivatives locally in $X(\mathbb C)$, provided that the
Beltrami coefficient $\mu$ belongs to $X(\mathbb C)$.
Keywords:quasiregular mappings, Beltrami equation, Sobolev spaces, CalderÃ³nZygmund operators Categories:30C62, 35J99, 42B20 

5. CJM 2012 (vol 65 pp. 299)
 Grafakos, Loukas; Miyachi, Akihiko; Tomita, Naohito

On Multilinear Fourier Multipliers of Limited Smoothness
In this paper,
we prove certain $L^2$estimate
for multilinear Fourier multiplier operators
with multipliers of limited smoothness.
As a result,
we extend the result of CalderÃ³n and Torchinsky
in the linear theory to the multilinear case.
The sharpness of our results and some
related estimates in Hardy spaces
are also discussed.
Keywords:multilinear Fourier multipliers, HÃ¶rmander multiplier theorem, Hardy spaces Categories:42B15, 42B20 

6. CJM 2011 (vol 64 pp. 892)
 Hytönen, Tuomas; Liu, Suile; Yang, Dachun; Yang, Dongyong

Boundedness of CalderÃ³nZygmund Operators on Nonhomogeneous Metric Measure Spaces
Let $({\mathcal X}, d, \mu)$ be a
separable metric measure space satisfying the known upper
doubling condition, the geometrical doubling condition, and the
nonatomic condition that $\mu(\{x\})=0$ for all $x\in{\mathcal X}$.
In this paper, we show that the boundedness of a CalderÃ³nZygmund
operator $T$ on $L^2(\mu)$ is equivalent to that of $T$ on
$L^p(\mu)$ for some $p\in (1, \infty)$, and that of $T$ from $L^1(\mu)$
to $L^{1,\,\infty}(\mu).$ As an application, we prove that if $T$ is a
CalderÃ³nZygmund operator bounded on $L^2(\mu)$,
then its maximal operator is bounded on $L^p(\mu)$
for all $p\in (1, \infty)$ and from
the space of all complexvalued Borel measures on
${\mathcal X}$ to $L^{1,\,\infty}(\mu)$.
All these results generalize the corresponding results of Nazarov et al.
on metric spaces with
measures satisfying the socalled polynomial growth condition.
Keywords:upper doubling, geometrical doubling, dominating function, weak type $(1,1)$ estimate, CalderÃ³nZygmund operator, maximal operator Categories:42B20, 42B25, 30L99 

7. CJM 2011 (vol 64 pp. 257)
 Chen, Yanping; Ding, Yong; Wang, Xinxia

Compactness of Commutators for Singular Integrals on Morrey Spaces
In this paper we characterize the
compactness of the commutator $[b,T]$ for the singular integral
operator on the Morrey spaces $L^{p,\lambda}(\mathbb R^n)$. More
precisely, we prove that if
$b\in \operatorname{VMO}(\mathbb R^n)$, the $\operatorname {BMO}
(\mathbb R^n)$closure of $C_c^\infty(\mathbb R^n)$,
then $[b,T]$ is a compact operator on the
Morrey spaces $L^{p,\lambda}(\mathbb R^n)$ for $1\lt p\lt \infty$ and
$0\lt \lambda\lt n$. Conversely, if $b\in \operatorname{BMO}(\mathbb R^n)$ and
$[b,T]$ is a compact operator on the $L^{p,\,\lambda}(\mathbb R^n)$
for some $p\ (1\lt p\lt \infty)$, then $b\in \operatorname {VMO}(\mathbb R^n)$.
Moreover, the boundedness of a rough singular integral operator $T$
and its commutator $[b,T]$ on $L^{p,\,\lambda}(\mathbb R^n)$ are also
given. We obtain a sufficient condition for a
subset in Morrey space to be a strongly precompact set,
which has interest in its own right.
Keywords:singular integral, commutators, compactness, VMO, Morrey space Categories:42B20, 42B99 

8. CJM 2009 (vol 61 pp. 807)
9. CJM 2006 (vol 58 pp. 154)
10. CJM 2004 (vol 56 pp. 655)
 Tao, Xiangxing; Wang, Henggeng

On the Neumann Problem for the SchrÃ¶dinger Equations with Singular Potentials in Lipschitz Domains
We consider the Neumann problem for the Schr\"odinger equations $\Delta u+Vu=0$,
with singular nonnegative potentials $V$ belonging to the reverse H\"older class
$\B_n$, in a connected Lipschitz domain $\Omega\subset\mathbf{R}^n$. Given
boundary data $g$ in $H^p$ or $L^p$ for $1\epsilon
Keywords:Neumann problem, SchrÃ¶dinger equation, Lipschitz, domain, reverse HÃ¶lder class, $H^p$ space Categories:42B20, 35J10 

11. CJM 2003 (vol 55 pp. 504)
 Chen, Jiecheng; Fan, Dashan; Ying, Yiming

Certain Operators with Rough Singular Kernels
We study the singular integral operator
$$
T_{\Omega,\alpha}f(x) = \pv \int_{R^n} b(y) \Omega(y')
y^{n\alpha} f(xy)\,dy,
$$
defined on all test functions $f$,where $b$ is a bounded function, $\alpha\geq 0$,
$\Omega(y')$ is an integrable function on the unit sphere $S^{n1}$ satisfying
certain cancellation conditions. We prove that, for $1
Categories:42B20, 42B25, 42B15 

12. CJM 2001 (vol 53 pp. 1031)
 Sampson, G.; Szeptycki, P.

The Complete $(L^p,L^p)$ Mapping Properties of Some Oscillatory Integrals in Several Dimensions
We prove that the operators $\int_{\mathbb{R}_+^2} e^{ix^a \cdot
y^b} \varphi (x,y) f(y)\, dy$ map $L^p(\mathbb{R}^2)$ into itself
for $p \in J =\bigl[\frac{a_l+b_l}{a_l+(\frac{b_l r}{2})},\frac{a_l+b_l}
{a_l(1\frac{r}{2})}\bigr]$ if $a_l,b_l\ge 1$ and $\varphi(x,y)=xy^{r}$,
$0\le r <2$, the result is sharp. Generalizations to dimensions $d>2$
are indicated.
Categories:42B20, 46B70, 47G10 

13. CJM 2000 (vol 52 pp. 381)
 Miyachi, Akihiko

Hardy Space Estimate for the Product of Singular Integrals
$H^p$ estimate for the multilinear operators which are finite sums
of pointwise products of singular integrals and fractional
integrals is given. An application to Sobolev space and some
examples are also given.
Keywords:$H^p$ space, multilinear operator, singular integral, fractional integration, Sobolev space Category:42B20 

14. CJM 1998 (vol 50 pp. 29)