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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 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

27. 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

28. 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 $1 Keywords:singular integral, rough kernel, Hardy spaceCategory:42B20 29. CMB 1998 (vol 41 pp. 478) Oberlin, Daniel M.  Convolution with measures on curves in$\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$. Categories:42B15, 42B20 30. CMB 1998 (vol 41 pp. 306) Kolasa, Lawrence A.  Oscillatory integrals with nonhomogeneous phase functions related to SchrÃ¶dinger equations In this paper we consider solutions to the free Schr\" odinger equation in$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$, when the initial data is in an appropriate Sobolev space. Categories:42A25, 42B25 31. CMB 1997 (vol 40 pp. 296) Hare, Kathryn E.  A general approach to Littlewood-Paley theorems for orthogonal families A general lacunary Littlewood-Paley type theorem is proved, which applies in a variety of settings including Jacobi polynomials in$[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$multipliers. Categories:42B25, 42C10, 43A80 32. CMB 1997 (vol 40 pp. 169) Cruz-Uribe, David  The class$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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