1. CMB Online first
2. CMB Online first
 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 

3. CMB Online first
 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 

4. 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 

5. 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 

6. 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 

7. CMB 2011 (vol 55 pp. 708)
8. CMB 2011 (vol 55 pp. 303)
 Han, Yongsheng; Lee, MingYi; Lin, ChinCheng

Atomic Decomposition and Boundedness of Operators on Weighted Hardy Spaces
In this article, we establish a new atomic decomposition for $f\in L^2_w\cap H^p_w$,
where the decomposition converges in $L^2_w$norm rather than in the distribution sense.
As applications of this decomposition, assuming that $T$ is a linear
operator bounded on $L^2_w$ and $0
Keywords:$A_p$ weights, atomic decomposition, CalderÃ³n reproducing formula, weighted Hardy spaces Categories:42B25, 42B30 

9. 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 

10. CMB 2010 (vol 53 pp. 491)
 Jizheng, Huang; Liu, Heping

The Weak Type (1,1) Estimates of Maximal Functions on the Laguerre Hypergroup
In this paper, we discuss various maximal functions on the Laguerre hypergroup $\mathbf{K}$ including the heat maximal function, the Poisson maximal function, and the HardyLittlewood maximal function which is consistent with the structure of hypergroup of $\mathbf{K}$. We shall establish the weak type $(1,1)$ estimates for these maximal functions. The $L^p$ estimates for $p>1$ follow from the interpolation. Some applications are included.
Keywords:Laguerre hypergroup, maximal function, heat kernel, Poisson kernel Categories:42B25, 43A62 

11. 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 

12. CMB 2009 (vol 52 pp. 521)
13. 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 

14. CMB 2003 (vol 46 pp. 191)
 Kim, YongCheol

Weak Type Estimates of the Maximal Quasiradial BochnerRiesz Operator On Certain Hardy Spaces
Let $\{A_t\}_{t>0}$ be the dilation group in $\mathbb{R}^n$ generated
by the infinitesimal generator $M$ where $A_t=\exp(M\log t)$, and let
$\varrho\in C^{\infty}(\mathbb{R}^n\setminus\{0\})$ be a
$A_t$homogeneous distance function defined on $\mathbb{R}^n$. For
$f\in \mathfrak{S}(\mathbb{R}^n)$, we define the maximal quasiradial
BochnerRiesz operator $\mathfrak{M}^{\delta}_{\varrho}$ of index
$\delta>0$ by
$$
\mathfrak{M}^{\delta}_{\varrho} f(x)=\sup_{t>0}\mathcal{F}^{1}
[(1\varrho/t)_+^{\delta}\hat f ](x).
$$
If $A_t=t I$ and $\{\xi\in \mathbb{R}^n\mid \varrho(\xi)=1\}$ is a
smooth convex hypersurface of finite type, then we prove in an
extremely easy way that $\mathfrak{M}^{\delta}_{\varrho}$ is well
defined on $H^p(\mathbb{R}^n)$ when $\delta=n(1/p1/2)1/2$ and
$0 n(1/p1/2)1/2$ and $0
Categories:42B15, 42B25 

15. CMB 2002 (vol 45 pp. 25)
 Bloom, Steven; Kerman, Ron

Extrapolation of $L^p$ Data from a Modular Inequality
If an operator $T$ satisfies a modular inequality on a rearrangement
invariant space $L^\rho (\Omega,\mu)$, and if $p$ is strictly between
the indices of the space, then the Lebesgue inequality $\int Tf^p
\leq C \int f^p$ holds. This extrapolation result is a partial
converse to the usual interpolation results. A modular inequality for
Orlicz spaces takes the form $\int \Phi (Tf) \leq \int \Phi (C
f)$, and here, one can extrapolate to the (finite) indices $i(\Phi)$
and $I(\Phi)$ as well.
Category:42B25 

16. CMB 2000 (vol 43 pp. 330)
 Hare, Kathryn E.

Maximal Operators and Cantor Sets
We consider maximal operators in the plane, defined by Cantor sets of
directions, and show such operators are not bounded on $L^2$ if the
Cantor set has positive Hausdorff dimension.
Keywords:maximal functions, Cantor set, lacunary set Categories:42B25, 43A46 

17. CMB 1998 (vol 41 pp. 306)
18. CMB 1997 (vol 40 pp. 296)
19. CMB 1997 (vol 40 pp. 169)