On the Ohsawa-Takegoshi extension theorem. (English) Zbl 1210.32005

Adachi, Kenzō

Bull. Fac. Educ. Nagasaki Univ., Nat. Sci. 78, 1-16 (2010).

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.

Reviewer: Alberto Saracco (Parma)

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