## 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.)

### Citations:

Zbl 0625.32011; Zbl 0786.32016
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