The Pompeiu problem on locally symmetric spaces
type: Item
instance of: scholarly article
title: The Pompeiu problem on locally symmetric spaces
zbMATH Open document ID: 0777.43006
DOI: 10.1007/BF03041072
author: Erich Badertscher
published in: Journal d'Analyse Mathématique
publication date: 1993-06-29T00:00:00Z
review text: In [Comment. Math. Helv. 55, 593-621 (1990; Zbl 0452.43012)] \textit{L. Zalcman} and the reviewer stated a very general type of Pompeiu problem as follows. Let \(G\) be a group of transformations of a locally compact space \(X\), \(d\mu\) an invariant measure on \(X\), then an open relatively compact set \(\Omega\subseteq X\) is said to have the Pompeiu property if the map \(P: C(X)\to C(G)\) given by \(Pf(g):=\int_{g(\Omega)} f(x)d\mu(x)\) is injective. In this generality the problem is very hard, but some progress can be made for \(X=G/K\) an irreducible symmetric space of rank one or \(X=\mathbb{R}^ n\), with \(G\) the Euclidean group \(M(n)\). Some of the work of the reviewer on the local Pompeiu property [e.g., \textit{C. A. Berenstein} and \textit{R. Gay}, J. Anal. Math. 52, 133-166 (1989; Zbl 0668.30037)] indicates that the existence of a global transformation group \(G\) might not be necessary, similarly the invariance of the measure might also be disposable [e.g., \textit{C. Berenstein} and \textit{D. Pascuas}, J. Math. (to appear)]. In that case one may just want to consider a Riemannian manifold and preplace the translates of \(\Omega\) by the collection of all balls of a single radius (or a few radii) [e.g., \textit{T. Sunada}, Trans. Am. Math. Soc. 267, 483-501 (1981; Zbl 0514.58037), \textit{E. T. Quinto} (to appear)]. The author proceeds in this direction considering the following idea: Let \(X\) be a Riemannian manifold, \(T_ x\) its tangent space at \(x\in X\). If \(\mu\) is a radial measure with compact support in \(\mathbb{R}^ n\), \(n=\dim X\), then it induces a measure on \(T_ x\) by choosing an orthogonal basis of \(T_ x\) and using it to identify \(T_ x\) to \(\mathbb{R}^ n\), then the Pompeiu transform is \[ P_ \mu(f)(x)= \int_{\mathbb{R}^ n} f(\exp_ x(t))\upsilon_ x(t)d\mu(t), \] where \(\upsilon_ x\) is the Jacobian of \(\exp_ x\). Since \(\mu\) is radial, the definition is independent of the choice of orthonormal basis in \(T_ x\). By taki…
Mathematics Subject Classification ID: 43A85
zbMATH DE Number: 221311
zbMATH Keywords: Pompeiu problem
zbMATH Keywords: group of transformations
zbMATH Keywords: locally compact space
zbMATH Keywords: invariant measure
zbMATH Keywords: irreducible symmetric space
zbMATH Keywords: local Pompeiu property
zbMATH Keywords: Riemannian manifold
zbMATH Keywords: radial measure
zbMATH Keywords: Pompeiu transform
zbMATH Keywords: locally symmetric spaces
reviewed by: Carlos A. Berenstein
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf03041072
OpenAlex ID: W58097721
description: scientific article; zbMATH DE number 221311