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On large values of \(L^ 2\) holomorphic functions. (English) Zbl 0865.32009

For a smoothly bounded domain of holomorphy in \(\mathbb{C}^n\), denoted by \(\Omega\), with \(p\in\Omega\) fixed, set \[ M_\Omega(p)= \sup \biggl\{\bigl |f(p) \bigr |^2: f \in {\mathcal O} (\Omega) \cap L^2 (\Omega),\;|f|_{L^2} \leq 1\biggr\}, \] where \({\mathcal O}(\Omega)\) and \(L^2(\Omega)\) denote the holomorphic functions and the square integrable functions on \(\Omega\) respectively. It is known that \(M_\Omega(p)\) gives the value of the Bergman kernel function associated to \(\Omega\). If \(n=1\), it is a classical fact that \(M_\Omega (p)\) is bounded from above and below by a constant factor times \(\text{dist} (p, \partial \Omega)^{-2}\). In higher dimensions, the author gives a lower bound and proves the following theorem. Let \(\Omega= \{z:r(z)<0\}\) be as before and \(z_0\in\partial \Omega\) with a holomorphic support surface \(S\) at \(z_0\). Let \(\nu_{z_0}\) denote the inward unit normal for \(\partial \Omega\) at \(z_0\). The theorem says that there exists a constant \(c\) so that if \(p= z_0+ \delta \nu_{z_0}\), then \[ c\delta^{-2} M_{S_p\cap \Omega}(p)\leq M_{\Omega_\delta}(p), \] where \(S_p=\{z:z-\delta\nu_{z_0}\in S\}\) and \(\Omega_\delta= \{z:r(z) < -{\delta \over 2}\}\). In the above one says that a complex analytic manifold \(S\) with dimension \(n-1\), defined in a neighborhood \(U\) of a boundary point \(q\in\partial\Omega\) is a (weak, local) holomorphic support surface for \(\Omega\) at \(q\) if \(q\in S\) and \(S\cap U\subset\mathbb{C}^n \backslash \Omega\).
The author remarks that for a class of domains whose Levi forms do not necessarily degenerate to finite order, his theorem generalize the result given by T. Ohsawa and K. Takegoshi [Math. Z. 195, 197-204 (1987; Zbl 0625.32011)]. See also the paper by K. Diederich and G. Herbort, J. Geom. Anal. 3, No. 3, 237-267 (1993; Zbl 0786.32016) for some issues behind these results and further references.

MSC:

32T99 Pseudoconvex domains
32D05 Domains of holomorphy
32A25 Integral representations; canonical kernels (Szegő, Bergman, etc.)
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