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

Abstract

Let $\varrho \in C^{\infty }({\mathbb{R}}^d\setminus \{0\})$ 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 ${\Cal T}^δ$ associated with the smooth cone type multipliers defined by $$\hat {{\Cal T}^δ f}(ξ,τ)= (1-\frac{\varrho(ξ)}{|τ|} )^δ_+\hat f (ξ,τ), (ξ,τ)\in {\Bbb R}^d \times \Bbb R.$$ If the unit sphere ${\Sigma }_{\varrho }\fallingdotseq \{\xi \in {\mathbb{R}}^d:\varrho (\xi )=1\}$ is a convex hypersurface of finite type and $\varrho$ is not radial, then we prove that ${\Cal T}^{δ(p)}$ maps from $H^p({\mathbb{R}}^{d+1})$, $0<p<1$, into weak-$L^p({\Gamma }_{\gamma })$ for the critical index $\delta (p)=d(1/p-1/2)-1/2$, where ${\Gamma }_{\gamma }=\{(x,t)\in {\mathbb{R}}^d\times \mathbb{R}:|t|\ge \gamma |x|\}$ for $\gamma =\max \{{\sup }_{\varrho (\xi )\le 1}|\xi |,1\}$. Moreover, we furnish a function $f\in \mathfrak{S}({\mathbb{R}}^{d+1})$ such that $$\sup_{λ>0} λ^p|\{(x,t)\in \bar{{\Bbb R}^{d+1}\setminusΓ_γ} : |{\Cal T}_{\varrho}^{δ(p)}f(x,t)|>λ\}|=\infty.$$

13 pages