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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
##### Keywords:
Pompeiu transform; Gauss theorem; Morera Theorem
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##### References:
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