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Search: MSC category 42B ( Harmonic analysis in several variables {For automorphic theory, see mainly 11F30} )

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26. 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 $0 Categories:42B15, 42B25 27. CMB 2002 (vol 45 pp. 46) Dafni, Galia  Local$\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$is proved. Categories:42B30, 46E99 28. CMB 2002 (vol 45 pp. 25) Bloom, Steven; Kerman, Ron  Extrapolation of$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)$as well. Category:42B25 29. CMB 2001 (vol 44 pp. 121) Wojciechowski, Michał  A Necessary Condition for Multipliers of Weak Type$(1,1)$Simple necessary conditions for weak type$(1,1)$of invariant operators on$L(\rr^d)$and their applications to rational Fourier multiplier are given. Categories:42B15, 42B20 30. CMB 2000 (vol 43 pp. 330) Hare, Kathryn E.  Maximal Operators and Cantor Sets We consider maximal operators in the plane, defined by Cantor sets of directions, and show such operators are not bounded on$L^2$if the Cantor set has positive Hausdorff dimension. Keywords:maximal functions, Cantor set, lacunary setCategories:42B25, 43A46 31. CMB 2000 (vol 43 pp. 17) Bak, Jong-Guk  Multilinear Proofs for Convolution Estimates for Degenerate Plane Curves Suppose that$\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 32. 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 33. 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 34. 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

35. 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

36. 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

37. 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

38. 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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