1. 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 

2. 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 

3. CMB 2011 (vol 54 pp. 566)
 Zhou, XiangJun; Shi, Lei; Zhou, DingXuan

Nonuniform Randomized Sampling for Multivariate Approximation by High Order Parzen Windows
We consider approximation of multivariate functions in Sobolev
spaces by high order Parzen windows in a nonuniform sampling
setting. Sampling points are neither i.i.d. nor regular, but are
noised from regular grids by nonuniform shifts of a probability
density function. Sample function values at sampling points are
drawn according to probability measures with expected values being
values of the approximated function. The approximation orders are
estimated by means of regularity of the approximated function, the
density function, and the order of the Parzen windows, under
suitable choices of the scaling parameter.
Keywords:multivariate approximation, Sobolev spaces, nonuniform randomized sampling, high order Parzen windows, convergence rates Categories:68T05, 62J02 

4. 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 
