Variations on Bochner–Riesz multipliers in the plane

Daniele Debertol¹

¹Scuola Normale Superiore Piazza dei Cavalieri 7 56126 Pisa (PI), Italy

Studia Mathematica, Tome 177 (2006) no. 1, pp. 1-8

Résumé

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.

Détail

DOI : 10.4064/sm177-1-1

Keywords: consider multiplier defined mathbb doteq frac right denoting unit disk mathbb given infty optimal range mus which fourier multiplier bochner riesz means key ingredient lemma about modifications bochner riesz means inside convex regions smooth boundary non vanishing curvature providing flexible version result iosevich publ mat application characterization holds multiplier doteq finally briefly discuss n dimensional analogue these results