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Search: MSC category 42B15 ( Multipliers )

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1. CMB 2006 (vol 49 pp. 3)

Al-Salman, Ahmad
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 spaces
Categories:42B20, 42B15, 42B25

2. 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

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

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