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51. 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 $$T_{A}f(x)\equiv \pv \int_{\mathbb{R}^n} K(x-y)[A(x)-A(y)]f(y)\,dy$$ is bounded on $L^2(\mathbb{R}^n)$, where $K(\dz_\lz x)=\lz^{-|\az|-1}K(x)$, $K$ is smooth away from the origin and satisfies a certain cancellation property. Keywords:parabolic singular integral, commutator, parabolic $\BMO$ sobolev space, homogeneous space, T1-theorem, symbolCategory:42B20

52. CMB 1999 (vol 42 pp. 344)

Koldobsky, Alexander
 Positive Definite Distributions and Subspaces of $L_p$ With Applications to Stable Processes We define embedding of an $n$-dimensional normed space into $L_{-p}$, $0 Categories:42A82, 46B04, 46F12, 60E07 53. CMB 1999 (vol 42 pp. 198) Guadalupe, José J.; Pérez, Mario; Varona, Juan L.  Commutators and Analytic Dependence of Fourier-Bessel Series on$(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$-th partial sum of the Fourier-Bessel series on$(0,\infty)$. Perturbing the measure by$\exp(2b)$we obtain that certain operators related to$S_n$depend analytically on the functional parameter$b$. Keywords:Fourier-Bessel series, commutators, BMO,$A_p$weightsCategory:42C10 54. CMB 1999 (vol 42 pp. 37) Christensen, Ole  Operators with Closed Range, Pseudo-Inverses, and Perturbation of Frames for a Subspace Recent work of Ding and Huang shows that if we perturb a bounded operator (between Hilbert spaces) which has closed range, then the perturbed operator again has closed range. We extend this result by introducing a weaker perturbation condition, and our result is then used to prove a theorem about the stability of frames for a subspace. Category:42C15 55. CMB 1998 (vol 41 pp. 398) Dziubański, Jacek; Hernández, Eugenio  Band-limited wavelets with subexponential decay It is well known that the compactly supported wavelets cannot belong to the class$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})$that are almost'' of exponential decay and, moreover, they are band-limited. We do this by showing that we can adapt the construction of the Lemari\'e-Meyer wavelets \cite{LM} that is found in \cite{BSW} so that we obtain band-limited,$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 decayCategory:42C15 56. CMB 1998 (vol 41 pp. 478) Oberlin, Daniel M.  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) Al-Hasan, 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 $$T_{Q_m,b} (\,f) (x)=p.v. \int_\bR^n b(|y|) \Omega(\,y) |y|^{-n} f \left( x-Q_m (|y|) y' \right) \,dy$$ is bounded in$L^p (\bR^n)$for$1 Keywords:singular integral, rough kernel, Hardy spaceCategory:42B20

58. CMB 1998 (vol 41 pp. 306)

Kolasa, Lawrence A.
 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)

Harrison, K. J.; Ward, J. A.; Eaton, L-J.
 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 constant-coefficient difference equations is a special case. Keywords:Difference equations, adaptive $\DARMA$ filters, weighted shifts,, stability and boundedness, automatic continuityCategories:47A62, 47B37, 93D25, 42A85, 47N70

60. CMB 1997 (vol 40 pp. 433)

Guo, Kanghui
 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)

Hare, Kathryn E.
 A general approach to Littlewood-Paley theorems for orthogonal families A general lacunary Littlewood-Paley 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)

Cruz-Uribe, David
 The class $A^{+}_{\infty}(\lowercase{g})$ and the one-sided reverse HÃ¶lder inequality We give a direct proof that $w$ is an $A^{+}_{\infty}(g)$ weight if and only if $w$ satisfies a one-sided, weighted reverse H\"older inequality. Keywords:one-sided maximal operator, one-sided $(A_\infty)$, one-sided, reverse HÃ¶lder inequalityCategory:42B25
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