Weak type estimates on certain Hardy spaces for smooth cone type multipliers
| dc.creator | Hong, Sunggeum | |
| dc.creator | Kim, Yong-Cheol | |
| dc.date | 2003-12-10 | |
| dc.date | 2005-02-03 | |
| dc.date.accessioned | 2026-07-07T05:03:44Z | |
| dc.date.available | 2026-07-07T05:03:44Z | |
| dc.description | 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 $$\sup_{λ>0} λ^p|\{(x,t)\in \bar{{\Bbb R}^{d+1}\setminusΓ_γ} : |{\Cal T}_{\varrho}^{δ(p)}f(x,t)|>λ\}|=\infty.$$ | |
| dc.description | 13 pages | |
| dc.identifier | https://arxiv.org/abs/math/0312204 | |
| dc.identifier | http://arxiv.org/abs/math/0312204 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/69541 | |
| dc.subject | Classical Analysis and ODEs | |
| dc.subject | Analysis of PDEs | |
| dc.subject | 42B15; 42B30 | |
| dc.title | Weak type estimates on certain Hardy spaces for smooth cone type multipliers | |
| dc.type | text |