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Le problème de Pompeiu local. (Local Pompeiu problem). (French) Zbl 0668.30037
This paper extends our work from Isr. J. Math. 55, 267-288 (1986; Zbl 0624.31002). We show here that if a set \(\Omega\) has the Pompeiu property in \({\mathbb{R}}^ n\), generally it also satisfies the same property in balls of radius \(>R_ 0(\Omega)\). In order to avoid complicated definitions we give here two simple corollaries of the theorems proved. Hole Theorem. Let K be a convex closed subset of \({\mathbb{R}}^ n\), \(\Omega\) a cube and f a continuous function in \({\mathbb{R}}^ n\setminus K\). If for every rigid motion \(\sigma\) such that \(\sigma (\Omega)\cap K=\emptyset\) we have \(\int_{\sigma (\Omega)}fdx=0\), then \(f\equiv 0\) in \({\mathbb{R}}^ n\setminus K.\)
Morera Theorem. Let \(T_ 0\) be the boundary of a fixed triangle, \(T_ 0\subseteq \Delta(0,)=\) the open disk of radius \(1/2\) and center 0 in the complex plane. Let \(f\in C(\Delta (0,1))\) be such that \(\int_{T}f(z)dz=0\) for every T congruent (by a Euclidean motion) to \(T_ 0\), \(T\subseteq \Delta (0,1)\). Then f is holomorphic in the unit disk. A follow up paper with R. Gay and A. Yger will also appear in the same journal.
Reviewer: C.A.Berenstein

MSC:
30E99 Miscellaneous topics of analysis in the complex plane
31B05 Harmonic, subharmonic, superharmonic functions in higher dimensions
35R99 Miscellaneous topics in partial differential equations
44A99 Integral transforms, operational calculus
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