26. 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 

27. CMB 2005 (vol 48 pp. 260)
28. CMB 2004 (vol 47 pp. 3)
29. 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 

30. CMB 2002 (vol 45 pp. 46)
31. 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 

32. CMB 2001 (vol 44 pp. 121)
33. 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 

34. CMB 2000 (vol 43 pp. 17)
 Bak, JongGuk

Multilinear Proofs for Convolution Estimates for Degenerate Plane Curves
Suppose that $\g \in C^2\bigl([0,\infty)\bigr)$ is a realvalued function
such that $\g(0)=\g'(0)=0$, and $\g''(t)\approx t^{m2}$, for some integer
$m\geq 2$. Let $\Gamma (t)=\bigl(t,\g(t)\bigr)$, $t>0$, be a curve in the
plane, and let $d \lambda =dt$ be a measure on this curve. For a
function $f$
on $\bR^2$, let
$$
Tf(x)=(\lambda *f)(x)=\int_0^{\infty} f\bigl(x\Gamma(t)\bigr)\,dt,
\quad x\in\bR^2 .
$$
An elementary proof is given for the optimal $L^p$$L^q$ mapping
properties of $T$.
Categories:42A85, 42B15 

35. CMB 2000 (vol 43 pp. 63)
 Iosevich, Alex; Lu, Guozhen

Sharpness Results and Knapp's Homogeneity Argument
We prove that the $L^2$ restriction theorem, and $L^p \to L^{p'}$,
$\frac{1}{p}+\frac{1}{p'}=1$, boundedness of the surface averages
imply certain geometric restrictions on the underlying
hypersurface. We deduce that these bounds imply that a certain
number of principal curvatures do not vanish.
Category:42B99 

36. 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 

37. 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 

38. CMB 1998 (vol 41 pp. 478)
39. CMB 1998 (vol 41 pp. 306)
40. CMB 1997 (vol 40 pp. 296)
41. CMB 1997 (vol 40 pp. 169)