Weak type estimates for cone type mutipliers associated with convex polygonsSunggeum Hong Department of Mathematics, Chosun University KOREA, REPUBLIC OF given at strobl07 (17.06.07) id: 813 length: min status: accepted type: poster LINK-Preprint: http://www.iumj.indiana.edu/IUMJ/nextissue.php LINK-Presentation: ABSTRACT: Let $\mathcal{P}$ be a convex polygon in $\mathbb{R}^{2}$ which contains the origin in its interior. Let $\rho$ be the associated Minkowski functional defined by $\rho(\xi) = \inf \{ \epsilon > 0 : {\epsilon}^{-1}{\xi} \in \mathcal{P}\}, \ \ \xi \neq 0$. We consider the family of convolution operators ${T}^{\delta}$ associated with cone type multipliers \begin{equation*} {\big (} 1 - \frac{{\rho}({\xi})^2}{\tau^2} {\big )}_{+}^{\delta}, \ \ \ (\xi,\tau) \in \mathbb{R}^{2} \times \mathbb{R}, \end{equation*} and show that ${T}^{\delta}$ is of weak type $(p,p)$ on $H^{p}(\mathbb R^{3})$, $1/2 < p < 1$ for the critical value $\delta = 2(1/p-1)$. This is a joint work of Sunggeum Hong, Joonil Kim and Chan Woo Yang. |