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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.
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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