Weak type estimates for cone type multipliers associated with a convex polygon
Sunggeum HongJoonil KimChan Yang
42B1542B30convex polygonMinkowski functionalcone type multipliersHardy spaces
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 $\delta=2(1/p-1)$.
Indiana University Mathematics Journal
2007
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10.1512/iumj.2007.56.2946
10.1512/iumj.2007.56.2946
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Indiana Univ. Math. J. 56 (2007) 1827 - 1869
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