Sharp bounds for multiplier operators of negative indices associated with degenerate curves
type: Item
instance of: scholarly article
title: Sharp bounds for multiplier operators of negative indices associated with degenerate curves
zbMATH Open document ID: 1228.42015
author: Sanghyuk Lee
author: Ihyeok Seo
published in: Mathematische Zeitschrift
publication date: 2011-03-02T00:00:00Z
review text: Studied are the Bochner-Riesz type means associated with the plane curve \(C=\{(t,\gamma(t)):t\in[-1,1]\}\), where \(\gamma:[-1,1] \to\mathbb R\) is a smooth function. They are multiplier operators and are defined as \[ \widehat{S^\alpha f}(x)=\frac{(x_2-\gamma(x_1))^\alpha_+}{\Gamma(\alpha+1)} \chi(x) \widehat{f}(x), \] where \(x=(x_1,x_2)\in\mathbb R^2\) and \(\chi\) is a smooth function compactly supported in a small neighborhood of \((0,\gamma(0))\). When \(\text{Re}(\alpha)\leq-1\), \(S^\alpha\) is defined by analytic continuation. Also considered are related multiplier operators. When \(C\) is a part of the circle, one gets usual Bochner-Riesz means for which classical results on \(L^p\)-\(L^q\) boundedness are known. In the paper under review these results are generalized to multiplier operators of negative indices associated with degenerate curves. The proofs are quite sophisticated and the obtained results are sharp.
reviewed by: Elijah Liflyand
Mathematics Subject Classification ID: 42B15
Mathematics Subject Classification ID: 42B25
zbMATH DE Number: 5859316
zbMATH Keywords: multiplier
zbMATH Keywords: negative index
zbMATH Keywords: Bochner-Riesz means
MaRDI profile type: Publication
full work available at URL: s00209-009-0620-x
OpenAlex ID: W2024154456
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DOI: 10.1007/S00209-009-0620-X
description: scientific article; zbMATH DE number 5859316