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26. CMB 2000 (vol 43 pp. 330)
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 
27. CMB 2000 (vol 43 pp. 17)
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$$L^q$ mapping
properties of $T$.
Categories:42A85, 42B15 
28. CMB 2000 (vol 43 pp. 63)
Sharpness Results and Knapp's Homogeneity Argument We prove that the $L^2$ restriction theorem, and $L^p \to L^{p'}$,
$\frac{1}{p}+\frac{1}{p'}=1$, boundedness of the surface averages
imply certain geometric restrictions on the underlying
hypersurface. We deduce that these bounds imply that a certain
number of principal curvatures do not vanish.
Category:42B99 
29. CMB 1999 (vol 42 pp. 463)
A Generalized Characterization of Commutators of Parabolic Singular Integrals Let $x=(x_1, \dots, x_n)\in\rz$ and $\dz_\lz x=(\lz^{\az_1}x_1,
\dots,\lz^{\az_n}x_n)$, where $\lz>0$ and $1\le \az_1\le\cdots
\le\az_n$. Denote $\az=\az_1+\cdots+\az_n$. We characterize those
functions $A(x)$ for which the parabolic Calder\'on commutator
$$
T_{A}f(x)\equiv \pv \int_{\mathbb{R}^n}
K(xy)[A(x)A(y)]f(y)\,dy
$$
is bounded on $L^2(\mathbb{R}^n)$, where $K(\dz_\lz x)=\lz^{\az1}K(x)$,
$K$ is smooth away from the origin and satisfies a certain cancellation
property.
Keywords:parabolic singular integral, commutator, parabolic $\BMO$ sobolev space, homogeneous space, T1theorem, symbol Category:42B20 
30. CMB 1998 (vol 41 pp. 404)
$L^p$boundedness of a singular integral operator Let $b(t)$ be an $L^\infty$ function on $\bR$, $\Omega (\,y')$ be
an $H^1$ function on the unit sphere satisfying the mean zero
property (1) and $Q_m(t)$ be a real polynomial on $\bR$ of degree
$m$ satisfying $Q_m(0)=0$. We prove that the singular integral
operator
$$
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

31. CMB 1998 (vol 41 pp. 478)
Convolution with measures on curves in $\bbd R^3$ We study convolution properties of measures on the curves
$(t^{a_1}, t^{a_2}, t^{a_3})$ in $\hbox{\Bbbvii R}^3$.
Categories:42B15, 42B20 
32. CMB 1998 (vol 41 pp. 306)
Oscillatory integrals with nonhomogeneous phase functions related to SchrÃ¶dinger equations In this paper we consider solutions to the free Schr\" odinger
equation in $n+1$ dimensions. When we restrict the last variable
to be a smooth function of the first $n$ variables we find that the
solution, so restricted, is locally in $L^2$, when the initial data
is in an appropriate Sobolev space.
Categories:42A25, 42B25 
33. CMB 1997 (vol 40 pp. 296)
A general approach to LittlewoodPaley theorems for orthogonal families A general lacunary LittlewoodPaley type theorem is proved, which applies in a
variety of settings including Jacobi polynomials in $[0, 1]$, $\su$, and the
usual classical trigonometric series in $[0, 2 \pi)$. The theorem is used to
derive new results for $\LP$ multipliers on $\su$ and Jacobi $\LP$ multipliers.
Categories:42B25, 42C10, 43A80 
34. CMB 1997 (vol 40 pp. 169)
The class $A^{+}_{\infty}(\lowercase{g})$ and the onesided reverse HÃ¶lder inequality We give a direct proof that $w$ is an $A^{+}_{\infty}(g)$ weight if and only
if $w$ satisfies a onesided, weighted reverse H\"older inequality.
Keywords:onesided maximal operator, onesided $(A_\infty)$, onesided, reverse HÃ¶lder inequality Category:42B25 