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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.