Variations on Bochner-Riesz multipliers in the plane

Daniele Debertol

Studia Mathematica, Tome 173 (2006), p. 1-8 / Harvested from The Polish Digital Mathematics Library

Résumé

We consider the multiplier $m_{μ}$ defined for ξ ∈ ℝ by $m_{μ}(ξ) ≐ ((1-ξ₁²-ξ₂²)/(1-ξ₁))^{μ} 1_{D}(ξ)$, D denoting the open unit disk in ℝ. Given p ∈ ]1,∞[, we show that the optimal range of μ’s for which $m_{μ}$ 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 $p_{μ}(ξ) ≐ (ξ₂ -ξ₁²)₊^{μ} ξ₂^{-μ}$. Finally, we briefly discuss the n-dimensional analogue of these results.

Détail

Publié le : 2006-01-01

Zbl 1133.42015

EUDML-ID : urn:eudml:doc:285139