Boundary uniqueness theorems in $\textbf {C}^ n$

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.