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Search: MSC category 42B15 ( Multipliers )

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1. 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
 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 $0 Categories:42B15, 42B25 3. CMB 2001 (vol 44 pp. 121) Wojciechowski, Michał  A Necessary Condition for Multipliers of Weak Type$(1,1)$Simple necessary conditions for weak type$(1,1)$of invariant operators on$L(\rr^d)$and their applications to rational Fourier multiplier are given. Categories:42B15, 42B20 4. CMB 2000 (vol 43 pp. 17) Bak, Jong-Guk  Multilinear Proofs for Convolution Estimates for Degenerate Plane Curves Suppose that$\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 5. 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