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

Let $\varrho \in C^{\mathrm{\infty }}({\mathbb{R}}^d\setminus \{0\})$ be a non-radial homogeneous distance function satisfying $\varrho (t\xi )=t\varrho (\xi )$. For $f\in \mathfrak{S}({\mathbb{R}}^{d+1})$ and $\delta >0$, we consider convolution operator ${\text{Cal}T}^{\delta }$ associated with the smooth cone type multipliers defined by $$\widehat{{\text{Cal}T}^{\delta }f}(\xi ,\tau )=(1-\frac{\varrho (\xi )}{\mathrm{\mid }\tau \mathrm{\mid }}{)}_{+}^{\delta }\widehat{f}(\xi ,\tau ),(\xi ,\tau )\in {\mathbb{R}}^d\times \mathbb{R}\mathrm{.}$$ If the unit sphere $\Sigma_{\varrho}\fallingdotseq{\xi\in {\Bbb R}^d : \varrho(\xi)=1}$isaconvexhypersurfaceoffinitetypeand$\varrho$ is not radial, then we prove that ${\text{Cal}T}^{\delta (p)}$ maps from $H^p({\Bbb R}^{d+1})$,$0<p<1$,intoweak-$L^p(\Gamma_{\gamma})$forthecriticalindex$\delta(p)=d(1/p -1/2)-1/2$,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 $f\in \mathfrak{S}({\mathbb{R}}^{d+1})$ such that $$\underset{\lambda >0}{\sup }{\lambda }^p\mathrm{\mid }\{(x,t)\in \overline{{\mathbb{R}}^{d+1}\setminus {\mathrm{\Gamma }}_{\gamma }}:\mathrm{\mid }{\text{Cal}T}_{\varrho }^{\delta (p)}f(x,t)\mathrm{\mid }>\lambda \}\mathrm{\mid }=\mathrm{\infty }\mathrm{.}$$