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Weak Type Estimates of the Maximal Quasiradial Bochner-Riesz Operator On Certain Hardy Spaces

  Published:2003-06-01
 Printed: Jun 2003
  • Yong-Cheol Kim
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Abstract

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
MSC Classifications: 42B15, 42B25 show english descriptions Multipliers
Maximal functions, Littlewood-Paley theory
42B15 - Multipliers
42B25 - Maximal functions, Littlewood-Paley theory
 

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