26. 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 

27. CMB 2001 (vol 44 pp. 121)
28. 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 

29. 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 

30. 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 

31. 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 

32. 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 

33. CMB 1998 (vol 41 pp. 478)
34. CMB 1998 (vol 41 pp. 306)
35. CMB 1997 (vol 40 pp. 296)
36. CMB 1997 (vol 40 pp. 169)