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51. 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 
52. CMB 1999 (vol 42 pp. 344)
Positive Definite Distributions and Subspaces of $L_p$ With Applications to Stable Processes We define embedding of an $n$dimensional normed space into
$L_{p}$, $0

53. CMB 1999 (vol 42 pp. 198)
Commutators and Analytic Dependence of FourierBessel Series on $(0,\infty)$ In this paper we study the boundedness of the commutators $[b,
S_n]$ where $b$ is a $\BMO$ function and $S_n$ denotes the $n$th
partial sum of the FourierBessel series on $(0,\infty)$.
Perturbing the measure by $\exp(2b)$ we obtain that certain
operators related to $S_n$ depend analytically on the functional
parameter $b$.
Keywords:FourierBessel series, commutators, BMO, $A_p$ weights Category:42C10 
54. CMB 1999 (vol 42 pp. 37)
Operators with Closed Range, PseudoInverses, and Perturbation of Frames for a Subspace Recent work of Ding and Huang shows that if we perturb a bounded
operator (between Hilbert spaces) which has closed range, then the
perturbed operator again has closed range. We extend this result by
introducing a weaker perturbation condition, and our result is then
used to prove a theorem about the stability of frames for a subspace.
Category:42C15 
55. CMB 1998 (vol 41 pp. 398)
Bandlimited wavelets with subexponential decay It is well known that the compactly supported wavelets cannot belong to
the class $C^\infty({\bf R})\cap L^2({\bf R})$. This is also true for
wavelets with exponential decay. We show that one can construct
wavelets in the class $C^\infty({\bf R})\cap L^2({\bf R})$ that are
``almost'' of exponential decay and, moreover, they are
bandlimited. We do this by showing that we can adapt the
construction of the Lemari\'eMeyer wavelets \cite{LM} that
is found in \cite{BSW} so that we obtain bandlimited,
$C^\infty$wavelets on $\bf R$ that have subexponential decay,
that is, for every $0<\varepsilon<1$, there exits $C_\varepsilon>0$
such that $\psi(x)\leq C_\varepsilon e^{x^{1\varepsilon}}$,
$x\in\bf R$. Moreover, all of its derivatives have also
subexponential decay. The proof is constructive and uses the
Gevrey classes of functions.
Keywords:Wavelet, Gevrey classes, subexponential decay Category:42C15 
56. 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 
57. 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

58. 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 
59. CMB 1998 (vol 41 pp. 49)
Stability of weighted darma filters We study the stability of linear filters associated with certain types of
linear difference equations with variable coefficients. We show that
stability is determined by the locations of the poles of a rational transfer
function relative to the spectrum of an associated weighted shift operator.
The known theory for filters associated with constantcoefficient difference
equations is a special case.
Keywords:Difference equations, adaptive $\DARMA$ filters, weighted shifts,, stability and boundedness, automatic continuity Categories:47A62, 47B37, 93D25, 42A85, 47N70 
60. CMB 1997 (vol 40 pp. 433)
A uniform $L^{\infty}$ estimate of the smoothing operators related to plane curves In dealing with the spectral synthesis property for a plane curve with
nonzero curvature, a key step is to have a uniform $L^{\infty}$ estimate
for some smoothing operators related to the curve. In this paper, we will
show that the same $L^{\infty}$ estimate holds true for a plane curve
that may have zero curvature.
Categories:42b20, 42b15 
61. 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 
62. 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 