Variations on Bochner-Riesz multipliers in the plane

Daniele Debertol. "Variations on Bochner-Riesz multipliers in the plane." Studia Mathematica 177.1 (2006): 1-8. <http://eudml.org/doc/285139>.

@article{DanieleDebertol2006,
abstract = {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.},
author = {Daniele Debertol},
journal = {Studia Mathematica},
keywords = {homogeneous multipliers},
language = {eng},
number = {1},
pages = {1-8},
title = {Variations on Bochner-Riesz multipliers in the plane},
url = {http://eudml.org/doc/285139},
volume = {177},
year = {2006},
}

TY - JOUR
AU - Daniele Debertol
TI - Variations on Bochner-Riesz multipliers in the plane
JO - Studia Mathematica
PY - 2006
VL - 177
IS - 1
SP - 1
EP - 8
AB - 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.
LA - eng
KW - homogeneous multipliers
UR - http://eudml.org/doc/285139
ER -