Spherical means and CR functions on the Heisenberg group
type: Item
instance of: scholarly article
title: Spherical means and CR functions on the Heisenberg group
zbMATH Open document ID: 0822.43001
DOI: 10.1007/BF03008426
author: S. Thangavelu
published in: Journal d'Analyse Mathématique
publication date: 1995-10-05T00:00:00Z
review text: The injectivity of the spherical mean value operator is studied. The main concern of this paper is the case of the spherical mean on the Heisenberg group. Let \(\mu_ r\) denote the normalized surface measure on \(\{(z, 0): | z| = r\}\) in the Heisenberg group \(\mathbb{H}^ n = \mathbb{C}^ n \times \mathbb{R}\). It is proved that the spherical mean value operator \(f \to f * \mu_ r\) on the Heisenberg group is injective when \(f \in L^ p(\mathbb{H}^ n)\), \(1 \leq p < \infty\). When \(f(z, \cdot) \in L^ p (\mathbb{R})\), \(1 \leq p \leq 2\), the same result is proved under much weaker conditions such that the partial Fourier transform \(f^ \lambda (z) = \int^ \infty_{-\infty} e^{i \lambda t} f(z,t) dt\) is of tempered growth on \(\mathbb{C}^ n\) for a.e. \(\lambda\). Moreover, some results regarding CR functions on the Heisenberg group are proved. Those results are extensions of recent ones obtained by \textit{M. Agranovsky, C. Berenstein, D. Chang} and \textit{D. Pascuas} [J. Anal. Math. 57, 282-296 (1991; Zbl 0773.32012)].
Mathematics Subject Classification ID: 43A80
Mathematics Subject Classification ID: 43A07
zbMATH DE Number: 569423
zbMATH Keywords: injectivity of spherical mean value
zbMATH Keywords: Heisenberg group
zbMATH Keywords: partial Fourier transforms
zbMATH Keywords: CR functions
reviewed by: Koichi Saka
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf03008426
OpenAlex ID: W2065903332
description: scientific article; zbMATH DE number 569423