A restriction theorem for the Heisenberg group
type: Item
instance of: scholarly article
title: A restriction theorem for the Heisenberg group
zbMATH Open document ID: 0731.43003
DOI: 10.2307/1971471
author: Detlef Müller
published in: Annals of Mathematics. Second Series
publication date: 1990-01-01T00:00:00Z
review text: If p is close enough to one, then the Fourier transform of an \(L^ p\)- function on \({\mathbb{R}}^ n\) has a restriction to the unit sphere which is square integrable. It follows that the convolution operator \({\mathcal P}f=f*K\), where K denotes the Fourier transform of the surface measure of the unit sphere, is bounded from \(L^ p\) to \(L^{p'}\). In this paper one proves an analogue for the Heisenberg group \(H_ n\). One considers the spectral resolution E(\(\alpha\)) of the Kohn-Laplacian L on \(H_ n\), \(L=\int^{\infty}_{0}\lambda dE(\lambda)\), and one defines \[ {\mathcal P}_{\alpha}=\lim_{\beta \to \alpha}(E(\beta)-E(\alpha))/(\beta - \alpha). \] Then \({\mathcal P}_{\alpha}\) is a convolution operator, \({\mathcal P}_{\alpha}f=f*G_{\alpha}\), where \(G_{\alpha}\) is computed explicitly. The main result is: if \(1\leq p<2\), then \(\| {\mathcal P}_ 1f\|_{(p',\infty)}\leq C_ p\| f\|_{(p,1)}\). One proves also estimates for the operators \(E_{\alpha,\beta}=E(\beta)-E(\alpha)\). The proofs use the Gelfand transform for the commutative algebra \(L^ 1_ r\) of polyradial integrable functions on \(H_ n\).
Mathematics Subject Classification ID: 43A30
Mathematics Subject Classification ID: 43A15
Mathematics Subject Classification ID: 22E30
Mathematics Subject Classification ID: 22E25
zbMATH DE Number: 4208790
zbMATH Keywords: Fourier transform
zbMATH Keywords: square integrable
zbMATH Keywords: convolution operator
zbMATH Keywords: Heisenberg group
zbMATH Keywords: spectral resolution
zbMATH Keywords: Kohn-Laplacian
zbMATH Keywords: Gelfand transform
zbMATH Keywords: polyradial integrable functions
reviewed by: Jacques Faraut
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2307/1971471
OpenAlex ID: W2318368089
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description: scientific article; zbMATH DE number 4208790