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Boundary uniqueness theorems in $${\mathbb{C}}^ n$$. (English) Zbl 0588.32007
Let $$\Gamma_ k$$, $$k=1,2,...$$, be n-dimensional manifolds which converge (near a point $$P\in \Gamma)$$ to an n-dimensional totally real manifold $$\Gamma$$ $$\subset \partial \Omega$$, the boundary of a $$C^{\infty}$$- smooth bounded domain $$\Omega \subset {\mathbb{C}}^ n$$. Suppose that the function f, analytic in $$\Omega$$, is such that its traces $$f_ k$$ on $$\Gamma_ k$$ have distributional limit 0 as $$k\to \infty$$ (or as $$f_ k\to 0$$ pointwise). Then the authors obtain the following Theorem 2.2 and Corollary 2.3. If f is polynomially bounded near P by $$(dist(z,\partial \Omega))^{-1}$$, then f is identically 0.
Theorem 2.2 generalizes to polynomially bounded functions a result of S. I. Pinĉuk [Math. Notes 15, 116-120 (1974; Zbl 0292.32002)] for continuous functions. If f is known a priori to omit a (finite) value, then the growth condition of f is relaxable to obtain Corollary 2.7 which applies to the Nevanlinna class (holomorphic functions f satisfying $$\sup_{\epsilon >0}\int_{\partial \Omega \epsilon}\log^+| f| d\sigma_{\epsilon}<\infty).$$
The authors’ technique allows for tangential approach (without restrictions) of the manifolds $$\Gamma_ k$$ to $$\partial \Omega$$ for $$n>1$$. This permits in Theorem 3.1, an extension of the result of A. Sadullaev [Math. USSR, Sb. 30(1976), 501-514 (1978; Zbl 0385.32007)] on bounded functions, to include tangential limits.
Reviewer: J.A.Adepoju
##### MSC:
 32A40 Boundary behavior of holomorphic functions of several complex variables 32A17 Special families of functions of several complex variables 32A05 Power series, series of functions of several complex variables 32A22 Nevanlinna theory; growth estimates; other inequalities of several complex variables
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