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On the Ohsawa-Takegoshi extension theorem. (English) Zbl 1210.32005
This paper gives a simple proof of the Ohsawa-Takegoshi extension theorem [for other proofs see T. Ohsawa and K. Takegoshi, in: Math. Z. 195, 197–204 (1987; Zbl 0625.32011) and Y.-T. Siu, Geometric complex analysis. Singapore: World Scientific. 577–592 (1996; Zbl 0941.32021)].
Let $$\Omega\subset\mathbb C^n$$ be a bounded pseudoconvex domain and let $$H\subset\mathbb C^n$$ be the hyperplane of the equation $$z_n=0$$. For any plurisubharmonic function $$\varphi$$ on $$\Omega$$, there exists a positive constant $$C$$ such that every holomorphic function $$f$$ of $$\Omega\cap H$$ can be extended to a holomorphic function $$F$$ on $$\Omega$$ satisfying the following $$L^2$$-estimate
$\int_\Omega |F|^2e^{-\varphi}\,dV_n\,\leq\,C\int_{H\cap \Omega}|f|^2e^{-\varphi}\,dV_{n-1}\,.$ The original proof of Ohsawa and Takegoshi was obtained via Carleman estimates for the $$\overline\partial$$-operator, while in this paper Hörmander $$L^2$$ estimates are used.
##### MSC:
 32A70 Functional analysis techniques applied to functions of several complex variables 46E20 Hilbert spaces of continuous, differentiable or analytic functions
##### Keywords:
holomorphic extension; Hörmander $$L^2$$ estimates