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On the extension of \(L^ 2\) holomorphic functions. III: Negligible weights. (English) Zbl 0823.32006
Let \(D \subset \mathbb{C}^ n\) be a bounded pseudoconvex domain, let \(H \subset \mathbb{C}^ n\) be a complex hyperplane, and let \(d(z)\) be the euclidean distance from \(z\) to \(H\). Let \(\psi : D \to [\xi - \infty, \infty)\) be any plurisubharmonic function on \(D\) such that \(\psi (z) + 2 \log d(z)\) is bounded from above.
The following extension theorem is proved in this article.
Theorem. There exists a constant \(C\) depending only on \(\sup (\psi + 2 \log d(z))\) such that, for any plurisubharmonic function \(\nu^ D\) on \(D\) and for any holomorphic function \(f\) on \(D \cap H\) satisfying \[ \int_{D \cap H} e^{-\nu (z) - \psi (z)} \bigl | f(z) \bigr |^ 2 < \infty, \] there exists a holomorphic function \(F\) on \(D\) such that \(F | D \cap H = f\) and \[ \int_ D e^{-\nu (z)} \bigl | F(z) \bigr |^ 2 < C \int_{D \cap H} e^{-\nu (z) - \psi (z)} \bigl | f(z) \bigr |^ 2. \] The special case \(\psi = 0\) has been proved earlier in part I of this paper by K. Takegoshi and the author [Math. Z. 195, 197-204 (1987; Zbl 0625.32011)]. When \(D\) is strongly pseudoconvex related results were found by A. Cumenge [Ann. Inst. Fourier 33, No. 3, 59-97 (1983; Zbl 0519.32012)] and F. Beatrous jun. [Mich. Math. J. 32, 361-380 (1985; Zbl 0584.32024)].
[For part II see Publ. Res. Inst. Math. Sci. 24, No. 2, 265-275 (1988; Zbl 0653.32012)].
Reviewer: T.Ohsawa (Nagoya)

MSC:
32E30 Holomorphic, polynomial and rational approximation, and interpolation in several complex variables; Runge pairs
32W05 \(\overline\partial\) and \(\overline\partial\)-Neumann operators
32D15 Continuation of analytic objects in several complex variables
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