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 ${\Bbb R}^n$ generated by the infinitesimal generator $M$ where $A_t=\exp(M\log t)$, and let $\varrho\in C^{\infty}({\Bbb R}^n\setminus\{0\})$ be a $A_t$-homogeneous distance function defined on ${\Bbb R}^n$. For $f\in {\frak S}({\Bbb R}^n)$, we define the maximal quasiradial Bochner-Riesz operator ${\frak M}^{\delta}_{\varrho}$ of index $\delta>0$ by $${\frak M}^{\delta}_{\varrho} f(x)=\sup_{t>0}|{\Cal F}^{-1}[(1-\varrho/t)_+^{\delta}\hat f ](x)|.$$ If $A_t=t I$ and $\{\xi\in {\Bbb R}^n| \varrho(\xi)=1\}$ is a smooth convex hypersurface of finite type, then we prove in an extremely easy way that ${\frak M}^{\delta}_{\varrho}$ is well defined on $H^p({\Bbb R}^n)$ when $\delta=n(1/p-1/2)-1/2$ and $0<p<1$; moreover, it is a bounded operator from $H^p({\Bbb R}^n)$ into $L^{p,\infty}({\Bbb R}^n)$. If $A_t=t I$ and $\varrho\in C^{\infty}({\Bbb R}^n\setminus\{0\})$, we also prove that ${\frak M}^{\delta}_{\varrho}$ 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 $0<p<1$.