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Transactions of the American Mathematical Society

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Boundary uniqueness theorems in $ {\bf C}\sp n$


Authors: Joseph A. Cima and Emil J. Straube
Journal: Trans. Amer. Math. Soc. 294 (1986), 333-339
MSC: Primary 32A40; Secondary 32F25
DOI: https://doi.org/10.1090/S0002-9947-1986-0819951-6
MathSciNet review: 819951
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Abstract: Let $ n$-dimensional manifolds $ {\Gamma _k},\,k = 1,2, \ldots $, be given in a smoothly bounded domain $ \Omega \subset {{\mathbf{C}}^n}$. Assume that the $ {\Gamma _k}$ "converge" to an $ n$-dimensional, totally real manifold $ \Gamma \subseteq \partial \Omega $ and that a function $ f$ analytic in $ \Omega $ has the property that its traces $ {f_k}$ on $ {\Gamma _k}$ have distributional limit zero as $ k \to \infty $ (or assume that $ {f_k} \to 0$ pointwise). Then under the assumption that $ f$ is polynomially bounded near $ P \in \Gamma $ by $ {(\operatorname{dist} (z,\partial \Omega ))^{ - 1}}$ we conclude that $ f$ is identically zero.


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DOI: https://doi.org/10.1090/S0002-9947-1986-0819951-6
Article copyright: © Copyright 1986 American Mathematical Society