Variations on Bochner–Riesz multipliers in the plane
Volume 177 / 2006
Abstract
We consider the multiplier $m_\mu$ defined for $\xi\in{\mathbb R}$ by $$ m_\mu(\xi)\doteq\left(\frac{1-\xi_{1}^{2}-\xi_{2}^{2}} {1-\xi_{1}}\right)^\mu 1_{D}(\xi), $$ ${D}$ denoting the open unit disk in ${\mathbb R}$. Given $p\in\ ]1,\infty[$, we show that the optimal range of $\mu$'s for which $m_\mu$ is a Fourier multiplier on $L^{p}$ is the same as for Bochner–Riesz means. The key ingredient is a lemma about some modifications of Bochner–Riesz means inside convex regions with smooth boundary and non-vanishing curvature, providing a more flexible version of a result by Iosevich et al. [Publ. Mat. 46 (2002)]. As an application, we show that the same characterization also holds true for the multiplier $p_\mu(\xi)\doteq(\xi_{2}-\xi_{1}^{2})_{+}^\mu \xi_{2}^{-\mu}$. Finally, we briefly discuss the $n$-dimensional analogue of these results.
