article

Title: Weak type estimates for cone type multipliers associated with a convex polygon

Authors: Sunggeum Hong, Joonil Kim and Chan Woo Yang

Issue: Volume 56 (2007), Issue 4, 1827-1869

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 $\rho(\xi) = \inf \{ \varepsilon > 0 : \varepsilon^{-1}\xi \in \mathcal{P} \}$, $\xi \neq 0$. We consider the family of convolution operators $T^{\delta}$ associated with cone type multipliers \[ \left(1 - \frac{\rho(\xi)^2}{\tau^2} \right)_{+}^{\delta}, \quad (\xi,\tau) \in \mathbb{R}^2 \times \mathbb{R}, \] and show that $T^{\delta}$ is of weak type $(p,p)$ on $H^p(\mathbb{R}^3)$, $\frac{1}{2} < p < 1$ for the critical value $\delta=2(1/p-1)$.$