Weak type estimates on certain Hardy spaces for smooth cone type multipliers

Abstract

Let $\varrho\in C^{\infty} ({\Bbb R}^d\setminus\{0\})$ be a non-radial homogeneous distance function satisfying $\varrho(t\xi)=t\varrho(\xi)$. For $f\in\frak S ({\Bbb R}^{d+1})$ and $\delta>0$, we consider convolution operator ${\Cal T}^{\delta}$ associated with the smooth cone type multipliers defined by $$\hat {{\Cal T}^{\delta} f}(\xi,\tau)= (1-\frac{\varrho(\xi)}{|\tau|} )^{\delta}_+\hat f (\xi,\tau), (\xi,\tau)\in {\Bbb R}^d \times \Bbb R.$$ If the unit sphere $\Sigma_{\varrho}\fallingdotseq\{\xi\in {\Bbb R}^d : \varrho(\xi)=1\}$ is a convex hypersurface of finite type and $\varrho$ is not radial, then we prove that ${\Cal T}^{\delta(p)}$ maps from $H^p({\Bbb R}^{d+1})$, $0<p<1$, into weak-$L^p(\Gamma_{\gamma})$ for the critical index $\delta(p)=d(1/p -1/2)-1/2$, where $\Gamma_{\gamma}=\{(x,t)\in {\Bbb R}^d\times\Bbb R : |t|\geq\gamma |x|\}$ for $\gamma=\max\{\sup_{\varrho(\xi)\leq 1}|\xi|,1\}$. Moreover, we furnish a function $f\in\frak S({\Bbb R}^{d+1})$ such that $$\sup_{\lambda>0} \lambda^p|\{(x,t)\in \bar{{\Bbb R}^{d+1}\setminus\Gamma_{\gamma}} : |{\Cal T}_{\varrho}^{\delta(p)}f(x,t)|>\lambda\}|=\infty.$$