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Variation of the Bergman kernel by cutting a hole. (English) Zbl 0595.32033

The author considers the variation of the Bergman kernel of a bounded domain \(\Omega\) in \({\mathbb{C}}^ n\) by cutting a hole, that is, a relatively closed subset \(\omega\) of \(\Omega\) such that \(\Omega\) \(\setminus \omega\) is connected and nonempty. Specifically, the author is concerned with expressing the Bergman kernel \(K_{\Omega \setminus \omega}\) of \(\Omega\) /\(\omega\) in terms of the kernel \(K_{\Omega}\) of \(\Omega\). For \(\omega\) compact, it is proved that for z,w\(\in \Omega \setminus \omega\), \[ (*)\quad K_{\Omega \setminus \omega}(z,w) = K_{\Omega}(z,w) + \sum^{\infty}_{m-1}T^{(m)}_{\Omega \quad \setminus \omega}(z,w), \] where the series converges absolutely and uniformly on compact subsets of \(\Omega\) \(\times \Omega\), and for each m, \(T^{(m)}_{\Omega,\omega}(z,w)\) is given by \[ \int...\int_{\omega^ m}K_{\Omega}(z,\xi_ 1) K_{\Omega}(\xi_ 1,\xi_ 2)... K_{\Omega}(\quad \xi_ m,w) dV(\xi_ 1)... dV(\xi_ m). \] It is also proved that in general, if \(R: L^ 2H(\Omega)\to L^ 2H(\Omega \setminus \omega)\) denotes the restriction mapping, then the range of R is dense if and only if (*) holds for z,w\(\in \Omega \setminus \omega\). The results are applied to show that certain regularity properties of the Bergman projection inherit from \(\Omega\) to \(\Omega\) \(\setminus \omega\).
Reviewer: M.Stoll

MSC:

32A25 Integral representations; canonical kernels (Szegő, Bergman, etc.)

Keywords:

Bergman kernel
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