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Weak Type Estimates of the Maximal Quasiradial Bochner-Riesz Operator On Certain Hardy Spaces
Published online by Cambridge University Press: 20 November 2018
Abstract
Let ${{\left\{ {{A}_{t}} \right\}}_{t>0}}$ be the dilation group in ${{\mathbb{R}}^{n}}$ generated by the infinitesimal generator $M$ where ${{A}_{t}}\,=\,\exp \left( M\,\log \,t \right)$, and let $\varrho \,\in \,{{C}^{\infty }}\left( \mathbb{R}{{}^{n\,}}\backslash \,\left\{ 0 \right\} \right)$ be a ${{A}_{t}}$-homogeneous distance function defined on ${{\mathbb{R}}^{n}}$. For $f\,\in \,\mathfrak{S}\left( \mathbb{R}{{}^{n}} \right)$, we define the maximal quasiradial Bochner-Riesz operator $\mathfrak{M}_{\varrho }^{\delta }$ of index $\delta \,>\,0$ by
If ${{A}_{t\,}}=\,tI$ and $\left\{ \xi \,\in \,{{\mathbb{R}}^{n\,}}\,|\,\varrho \left( \xi \right)\,=\,1 \right\}$ is a smooth convex hypersurface of finite type, then we prove in an extremely easy way that $\mathfrak{M}_{\varrho }^{\delta }$ is well defined on ${{H}^{p}}\left( {{\mathbb{R}}^{n}} \right)$ when $\delta \,=\,n(1/p\,-\,1/2)\,-\,1/2$ and $0\,<\,p\,<\,1$; moreover, it is a bounded operator from ${{H}^{p}}\left( {{\mathbb{R}}^{n}} \right)$ into ${{L}^{p,\infty }}\left( {{\mathbb{R}}^{n}} \right)$ .
If ${{A}_{t}}\,=\,tI$ and $\varrho \,\in \,{{C}^{\infty }}\left( \mathbb{R}{{}^{n\,}}\backslash \,\left\{ 0 \right\} \right)$ , we also prove that $\mathfrak{M}_{\varrho }^{\delta }$ is a bounded operator from ${{H}^{p}}\left( {{\mathbb{R}}^{n}} \right)$ into ${{L}^{p}}\,\left( {{\mathbb{R}}^{n}} \right)$ when $\delta \,>\,n(1/p\,-\,1/2)\,-\,1/2$ and $0\,<\,p\,<\,1$.
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- Copyright © Canadian Mathematical Society 2003
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