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Weak type estimates for cone type mutipliers associated with convex polygons


  Sunggeum 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.


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