Variations on Bochner-Riesz multipliers in the plane

• Autorzy: Daniele Debertol
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 42B15: Multipliers

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2006
• Tom: 177
• Numer: 1
• Strony: 1-8

Daty

wydano

2006

Twórcy

autor

Daniele Debertol

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

Identyfikatory

DOI

10.4064/sm177-1-1