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26. CMB 2008 (vol 51 pp. 348)

Casazza, Peter G.; Christensen, Ole
 The Reconstruction Property in Banach Spaces and a Perturbation Theorem Perturbation theory is a fundamental tool in Banach space theory. However, the applications of the classical results are limited by the fact that they force the perturbed sequence to be equivalent to the given sequence. We will develop a more general perturbation theory that does not force equivalence of the sequences. Category:42C15

27. CMB 2007 (vol 50 pp. 85)

Han, Deguang
 Classification of Finite Group-Frames and Super-Frames Given a finite group $G$, we examine the classification of all frame representations of $G$ and the classification of all $G$-frames, \emph{i.e.,} frames induced by group representations of $G$. We show that the exact number of equivalence classes of $G$-frames and the exact number of frame representations can be explicitly calculated. We also discuss how to calculate the largest number $L$ such that there exists an $L$-tuple of strongly disjoint $G$-frames. Keywords:frames, group-frames, frame representations, disjoint framesCategories:42C15, 46C05, 47B10

28. CMB 2006 (vol 49 pp. 438)

Mercer, Idris David
 Unimodular Roots of\\ Special Littlewood Polynomials We call $\alpha(z) = a_0 + a_1 z + \dots + a_{n-1} z^{n-1}$ a Littlewood polynomial if $a_j = \pm 1$ for all $j$. We call $\alpha(z)$ self-reciprocal if $\alpha(z) = z^{n-1}\alpha(1/z)$, and call $\alpha(z)$ skewsymmetric if $n = 2m+1$ and $a_{m+j} = (-1)^j a_{m-j}$ for all $j$. It has been observed that Littlewood polynomials with particularly high minimum modulus on the unit circle in $\bC$ tend to be skewsymmetric. In this paper, we prove that a skewsymmetric Littlewood polynomial cannot have any zeros on the unit circle, as well as providing a new proof of the known result that a self-reciprocal Littlewood polynomial must have a zero on the unit circle. Categories:26C10, 30C15, 42A05

29. CMB 2006 (vol 49 pp. 414)

Jiang, Liya; Jia, Houyu; Xu, Han
 Commutators Estimates on Triebel--Lizorkin Spaces In this paper, we consider the behavior of the commutators of convolution operators on the Triebel--Lizorkin spaces $\dot{F}^{s, q} _p$. Keywords:commutators, Triebel--Lizorkin spaces, paraproductCategories:42B, 46F

30. CMB 2006 (vol 49 pp. 3)

 On a Class of Singular Integral Operators With Rough Kernels In this paper, we study the $L^p$ mapping properties of a class of singular integral operators with rough kernels belonging to certain block spaces. We prove that our operators are bounded on $L^p$ provided that their kernels satisfy a size condition much weaker than that for the classical Calder\'{o}n--Zygmund singular integral operators. Moreover, we present an example showing that our size condition is optimal. As a consequence of our results, we substantially improve a previously known result on certain maximal functions. Keywords:Singular integrals, Rough kernels, Square functions,, Maximal functions, Block spacesCategories:42B20, 42B15, 42B25

31. CMB 2005 (vol 48 pp. 382)

De Carli, Laura
 Uniform Estimates of Ultraspherical Polynomials of Large Order In this paper we prove the sharp inequality $$|P_n^{(s)}(x)|\leq P_n^{(s)}(1)\bigl(|x|^n +\frac{n-1}{2 s+1}(1-|x|^n)\bigr),$$ where $P_n^{(s)}(x)$ is the classical ultraspherical polynomial of degree $n$ and order $s\ge n\frac{1+\sqrt 5}{4}$. This inequality can be refined in $[0,z_n^s]$ and $[z_n^s,1]$, where $z_n^s$ denotes the largest zero of $P_n^{(s)}(x)$. Categories:42C05, 33C47

32. CMB 2005 (vol 48 pp. 370)

Daly, J. E.; Fridli, S.
 Trigonometric Multipliers on $H_{2\pi}$ In this paper we consider multipliers on the real Hardy space $H_{2\pi}$. It is known that the Marcinkiewicz and the H\"ormander--Mihlin conditions are sufficient for the corresponding trigonometric multiplier to be bounded on $L_{2\pi}^p$, $1 Keywords:Multipliers, Hardy spaceCategories:42A45, 42A50, 42A85 33. CMB 2005 (vol 48 pp. 260) Oberlin, Daniel M.  A Restriction Theorem for a \\$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$. Keywords:Fourier restrictionCategory:42B10 34. CMB 2004 (vol 47 pp. 475) Wade, W. R.  Uniqueness of Almost Everywhere Convergent Vilenkin Series D. J. Grubb [3] has shown that uniqueness holds, under a mild growth condition, for Vilenkin series which converge almost everywhere to zero. We show that, under even less restrictive growth conditions, one can replace the limit function 0 by an arbitrary$f\in L^q$, when$q>1$. Categories:43A75, 42C10 35. CMB 2004 (vol 47 pp. 3) Al-Salman, Ahmad; Pan, Yibiao  Singular Integrals With Rough Kernels In this paper we establish the$L^p$boundedness of a class of singular integrals with rough kernels associated to polynomial mappings. Category:42B20 36. CMB 2003 (vol 46 pp. 191) Kim, Yong-Cheol  Weak Type Estimates of the Maximal Quasiradial Bochner-Riesz Operator On Certain Hardy Spaces Let$\{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

37. 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

38. 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

39. 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

40. 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

41. 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

42. 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

43. 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

44. 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

45. 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 $L_{-p}$, $0 Categories:42A82, 46B04, 46F12, 60E07 46. CMB 1999 (vol 42 pp. 198) Guadalupe, José J.; Pérez, Mario; Varona, Juan L.  Commutators and Analytic Dependence of Fourier-Bessel Series on$(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$weightsCategory:42C10 47. CMB 1999 (vol 42 pp. 37) Christensen, Ole  Operators with Closed Range, Pseudo-Inverses, and Perturbation of Frames for a Subspace Recent work of Ding and Huang shows that if we perturb a bounded operator (between Hilbert spaces) which has closed range, then the perturbed operator again has closed range. We extend this result by introducing a weaker perturbation condition, and our result is then used to prove a theorem about the stability of frames for a subspace. Category:42C15 48. CMB 1998 (vol 41 pp. 398) Dziubański, Jacek; Hernández, Eugenio  Band-limited wavelets with subexponential decay It is well known that the compactly supported wavelets cannot belong to the class$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 49. 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 50. 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
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