On the restriction conjecture
type: Item
instance of: scholarly article
title: On the restriction conjecture
zbMATH Open document ID: 1103.42007
author: Loukas Grafakos
author: Laura De Carli
published in: Michigan Mathematical Journal
publication date: 2004-11-11T00:00:00Z
review text: The restriction problem in harmonic analysis reads as follows: for which sets \(S\subset\mathbb R^d\) and which \(p,\) \(1\leq p\leq2,\) can the Fourier transform of an \(L^p(\mathbb R^d)\) function be meaningfully restricted. Even for \(S\) to be the unit sphere \(\mathbb S^{d-1}\) in \(\mathbb R^d\) the problem is not completely solved for \(d>2.\) By this, the study of other spaces where the restriction problem holds true makes sense. It is known that this is the case for radial functions. In the paper under review the authors consider the space of radial function multiplied by spherical harmonics. The restriction estimate is true there for familiar parameters \(1\leq p<2d/(d=1)\) and \(q=(d-1)p/(d+1)(p-1),\) where the Fourier transform from \(L^p(\mathbb R^d)\) is restricted to \(L^q(\mathbb S^{d-1}).\) Since the publication of this paper new results appeared in this area. We mention, for example, a nice survey paper by \textit{T. Tao} [''Some recent progress on the restriction conjecture'', Fourier analysis and convexity. Boston, MA: Birkhäuser. (2004; Zbl 1083.42008)]; and \textit{D. Oberlin}, [Can. Math. Bull. 48, No. 2, 260--266 (2005; Zbl 1083.42007)].
reviewed by: Elijah Liflyand
Mathematics Subject Classification ID: 42B10
Mathematics Subject Classification ID: 42B15
zbMATH DE Number: 2114462
zbMATH Keywords: Fourier restriction
zbMATH Keywords: Fourier transform
zbMATH Keywords: radial function
zbMATH Keywords: spherical harmonics
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1307/mmj/1080837741
OpenAlex ID: W2079389413
Wikidata QID: Q123234660
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DOI: 10.1307/MMJ/1080837741
description: scientific article; zbMATH DE number 2114462