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

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

2003-12-10

arxiv.org/abs/math/0312204v6

Mathematics

Classical Analysis and ODEs

13 pages

Scientific paper

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 $H^p({\Bbb R}^{d+1})$, $00} \lambda^p|\{(x,t)\in \bar{{\Bbb R}^{d+1}\setminus\Gamma_{\gamma}} : |{\Cal T}_{\varrho}^{\delta(p)}f(x,t)|>\lambda\}|=\infty.$$

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