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 $\varrho(tξ)=t\varrho(ξ)$. For $f\in\frak S ({\Bbb R}^{d+1})$ and $δ>0$, 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 $Σ_{\varrho}\fallingdotseq\{ξ\in {\Bbb R}^d : \varrho(ξ)=1\}$ is a convex hypersurface of finite type and $\varrho$ is not radial, then we prove that ${\Cal T}^{δ(p)}$ maps from $H^p({\Bbb R}^{d+1})$, $0<p<1$, into weak-$L^p(Γ_γ)$ for the critical index $δ(p)=d(1/p -1/2)-1/2$, where $Γ_γ=\{(x,t)\in {\Bbb R}^d\times\Bbb R : |t|\geqγ|x|\}$ for $γ=\max\{\sup_{\varrho(ξ)\leq 1}|ξ|,1\}$. Moreover, we furnish a function $f\in\frak S({\Bbb R}^{d+1})$ 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