On the extension of \(L^ 2\) holomorphic functions. III: Negligible weights
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 \textit{K. Takegoshi} and the author [Math. Z. 195, 197-204 (1987; Zbl 0625.32011)]. When \(D\) is strongly pseudoconvex related results were found by \textit{A. Cumenge} [Ann. Inst. Fourier 33, No. 3, 59-97 (1983; Zbl 0519.32012)] and \textit{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)].
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