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Variations on Bochner-Riesz multipliers in the plane

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Debertol D.

Studia Mathematica

2006

tom 177

nr 1

1-8

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.

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1 Studia Mathematica

1 Debertol D.

1 2006

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Variations on Bochner-Riesz multipliers in the plane