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51. CMB 2005 (vol 48 pp. 260)

Oberlin, Daniel M.
 A Restriction Theorem for a \\$k$-Surface in $\mathbb R ^n$ We establish a sharp Fourier restriction estimate for a measure on a $k$-surface in $\mathbb R ^n$, where $n=k(k+3)/2$. Keywords:Fourier restrictionCategory:42B10

52. CMB 2004 (vol 47 pp. 475)

 Uniqueness of Almost Everywhere Convergent Vilenkin Series D. J. Grubb [3] has shown that uniqueness holds, under a mild growth condition, for Vilenkin series which converge almost everywhere to zero. We show that, under even less restrictive growth conditions, one can replace the limit function 0 by an arbitrary $f\in L^q$, when $q>1$. Categories:43A75, 42C10

53. CMB 2004 (vol 47 pp. 3)

 Singular Integrals With Rough Kernels In this paper we establish the $L^p$ boundedness of a class of singular integrals with rough kernels associated to polynomial mappings. Category:42B20

54. CMB 2003 (vol 46 pp. 191)

Kim, Yong-Cheol
 Weak Type Estimates of the Maximal Quasiradial Bochner-Riesz 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 Bochner-Riesz 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 $\delta=n(1/p-1/2)-1/2$ and $0n(1/p-1/2)-1/2$ and $0 Categories:42B15, 42B25 55. CMB 2002 (vol 45 pp. 25) Bloom, Steven; Kerman, Ron  Extrapolation of$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)$as well. Category:42B25 56. CMB 2002 (vol 45 pp. 46) Dafni, Galia  Local$\VMO$and Weak Convergence in$\hone$A local version of$\VMO$is defined, and the local Hardy space$\hone$is shown to be its dual. An application to weak-$*$convergence in$\hone$is proved. Categories:42B30, 46E99 57. CMB 2001 (vol 44 pp. 121) Wojciechowski, Michał  A Necessary Condition for Multipliers of Weak Type$(1,1)$Simple necessary conditions for weak type$(1,1)$of invariant operators on$L(\rr^d)$and their applications to rational Fourier multiplier are given. Categories:42B15, 42B20 58. CMB 2000 (vol 43 pp. 355) Kelly, Brian P.  A Dimension-Free Weak-Type Estimate for Operators on UMD-Valued Functions Let$\T$denote the unit circle in the complex plane, and let$X$be a Banach space that satisfies\break Burkholder's UMD condition. Fix a natural number,$N \in \N$. Let$\od$denote the reverse lexicographical order on$\Z^N$. For each$f \in L^1 (\T^N,X)$, there exists a strongly measurable function$\wt{f}$such that formally, for all$\bfn \in \Z^N$,$\Dual{{\wt{f}}} (\bfn) = -i \sgn_\od (\bfn) \Dual{f} (\bfn)$. In this paper, we present a summation method for this conjugate function directly analogous to the martingale methods developed by Asmar and Montgomery-Smith for scalar-valued functions. Using a stochastic integral representation and an application of Garling's characterization of UMD spaces, we prove that the associated maximal operator satisfies a weak-type$(1,1)$inequality with a constant independent of the dimension~$N$. Category:42A61 59. CMB 2000 (vol 43 pp. 330) Hare, Kathryn E.  Maximal Operators and Cantor Sets We consider maximal operators in the plane, defined by Cantor sets of directions, and show such operators are not bounded on$L^2$if the Cantor set has positive Hausdorff dimension. Keywords:maximal functions, Cantor set, lacunary setCategories:42B25, 43A46 60. CMB 2000 (vol 43 pp. 63) Iosevich, Alex; Lu, Guozhen  Sharpness Results and Knapp's Homogeneity Argument We prove that the$L^2$restriction theorem, and$L^p \to L^{p'}$,$\frac{1}{p}+\frac{1}{p'}=1$, boundedness of the surface averages imply certain geometric restrictions on the underlying hypersurface. We deduce that these bounds imply that a certain number of principal curvatures do not vanish. Category:42B99 61. CMB 2000 (vol 43 pp. 17) Bak, Jong-Guk  Multilinear Proofs for Convolution Estimates for Degenerate Plane Curves Suppose that$\g \in C^2\bigl([0,\infty)\bigr)$is a real-valued function such that$\g(0)=\g'(0)=0$, and$\g''(t)\approx t^{m-2}$, for some integer$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 62. 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 63. 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

64. 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

65. 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

66. 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

67. 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

68. 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 $1 Keywords:singular integral, rough kernel, Hardy spaceCategory:42B20 69. CMB 1998 (vol 41 pp. 306) Kolasa, Lawrence A.  Oscillatory integrals with nonhomogeneous phase functions related to SchrÃ¶dinger equations In this paper we consider solutions to the free Schr\" odinger equation in$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$, when the initial data is in an appropriate Sobolev space. Categories:42A25, 42B25 70. CMB 1998 (vol 41 pp. 49) Harrison, K. J.; Ward, J. A.; Eaton, L-J.  Stability of weighted darma filters We study the stability of linear filters associated with certain types of linear difference equations with variable coefficients. We show that stability is determined by the locations of the poles of a rational transfer function relative to the spectrum of an associated weighted shift operator. The known theory for filters associated with constant-coefficient difference equations is a special case. Keywords:Difference equations, adaptive$\DARMA$filters, weighted shifts,, stability and boundedness, automatic continuityCategories:47A62, 47B37, 93D25, 42A85, 47N70 71. CMB 1997 (vol 40 pp. 433) Guo, Kanghui  A uniform$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}$estimate holds true for a plane curve that may have zero curvature. Categories:42b20, 42b15 72. CMB 1997 (vol 40 pp. 296) Hare, Kathryn E.  A general approach to Littlewood-Paley theorems for orthogonal families A general lacunary Littlewood-Paley type theorem is proved, which applies in a variety of settings including Jacobi polynomials in$[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$multipliers. Categories:42B25, 42C10, 43A80 73. CMB 1997 (vol 40 pp. 169) Cruz-Uribe, David  The class$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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