On the analytic continuation of holomorphic mappings

Let $D_1$ and $D_2$ be strictly pseudoconvex domains in $\mathbf C^n$ with real analytic boundaries $\partial D_1$ and $\partial D_2$, and let $\Omega$ be a neighborhood of the point $\zeta\in\partial D_1$ with $\Omega\cap\partial D_1$ connected. Assume that the mapping $f\colon\Omega\cap\overline D_1\to\mathbf C^n$ is holomorphic in $\Omega\cap D_1$, $C_1$ in $\Omega\cap\overline D_1$, and that $f(\Omega\cap\partial D_1)\subset\partial D_2$. The author proves that $f$ can be holomorphically continued to $\Omega\cap\partial D_1$. If the domain $D_2$ is a sphere $\{|z|<1\}$ and $\partial D_1$ is simply connected, then $f$ extends to a biholomorphic mapping from $D_1$ onto $D_2$.
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