Weak type estimates of the maximal quasiradial Bochner-Riesz operator on certain Hardy spaces

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 ${\mathfrak{M}}_{\varrho }^{\delta }$ of index $\delta >0$ by $${\frak M}^δ_{\varrho} f(x)=\sup_{t>0}|{\Cal F}^{-1}[(1-\varrho/t)_+^δ\hat f ](x)|.$$ If $A_t=tI$ and $\{\xi \in {\mathbb{R}}^n|\varrho (\xi )=1\}$ 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 $H^p({\mathbb{R}}^n)$ when $\delta =n(1/p-1/2)-1/2$ and $0<p<1$; moreover, it is a bounded operator from $H^p({\mathbb{R}}^n)$ into $L^{p,\infty }({\mathbb{R}}^n)$. If $A_t=tI$ and $\varrho \in C^{\infty }({\mathbb{R}}^n\setminus \{0\})$, we also prove that ${\mathfrak{M}}_{\varrho }^{\delta }$ is a bounded operator from $H^p({\mathbb{R}}^n)$ into $L^p({\mathbb{R}}^n)$ when $\delta >n(1/p-1/2)-1/2$ and $0<p<1$.

11 pages, to appear in Canadian Mathematical Bulletin