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 $\varrho\in C^{\infty}({\Bbb R}^n\setminus{0})$bea$A_t$-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 $${\mathfrak{M}}_{\varrho }^{\delta }f(x)=\underset{t>0}{\sup }\mathrm{\mid }{\text{Cal}F}^{-1}[(1-\varrho \mathrm{/}t{)}_{+}^{\delta }\widehat{f}](x)\mathrm{\mid }\mathrm{.}$$ If $A_t=tI$ and $\{\xi \in {\mathbb{R}}^n\mathrm{\mid }\varrho (\xi )=1\}$ 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 $\delta =n(1\mathrm{/}p-1\mathrm{/}2)-1\mathrm{/}2$ and $0<p<1$; moreover, it is a bounded operator from $H^p({\mathbb{R}}^n)$ into $L^{p,\mathrm{\infty }}({\mathbb{R}}^n)$. If $A_t=tI$ and $\varrho \in C^{\mathrm{\infty }}({\mathbb{R}}^n\setminus \{0\})$, we also prove that ${\mathfrak{M}}_{\varrho }^{\delta }$ is a bounded operator from $H^p({\Bbb R}^n)$into$L^p({\Bbb R}^n)$when$\delta>n(1/p-1/2)-1/2$and$0<p<1$.