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The Bergman kernel on uniformly extendable pseudoconvex domains. (English) Zbl 0582.32028
The authors consider the following conjecture of J. J. Kohn [J. Differ. Geom. 6, 523-542 (1972; Zbl 0256.35060)]: Let $$\Omega$$ be a bounded weakly pseudoconvex domain in $${\mathbb{C}}^ n$$ with real-analytic smooth boundaries, and let $$d_{\Omega}(z)$$ denote the distance from a point $$z\in \Omega$$ to $$\partial \Omega$$. Suppose the Levi-form of $$\partial \Omega$$ at the point $$z_ 0\in \partial \Omega$$ has exactly k positive eigenvalues. Then the Bergman kernel $$K_{\Omega}(z)=K_{\Omega}(z,z)$$ grows near $$z_ 0$$ at least like $$K_{\Omega}(z)\geq C d_{\Omega}^{-(k+2+\epsilon)}(z),$$ where $$\epsilon =0$$ if $$k=n-1$$ and $$\epsilon >0$$ if $$k<n-1.$$
This conjecture has been verified by D. Catlin [Several complex variables, Proc. 1981 Hangzhou Conf., 7-12 (1984)] for $$n=2$$. For $$n>2$$, the second author of this paper showed in Manuscr. Math. 45, 69-76 (1983; Zbl 0559.32006) that the principal part of the singularity of $$K_{\Omega}$$ does not have, in general, a rational order and it may contain logarithmic factors. Also, the last author of this paper [Publ. Res. Inst. Math. Sci. 20, 897-902 (1984; Zbl 0569.32013)] has proved the above conjecture for all pseudoconvex $$C^{\infty}$$-smooth domains in $${\mathbb{C}}^ n$$, but with $$\epsilon$$ replaced by -$$\epsilon$$.
In this paper, the authors verify this conjecture for a class of bounded pseudoconvex domains in $${\mathbb{C}}^ n$$, the so-called ”uniformly extendable pseudo-convex domains”, which contains, in particular, all $$\Omega$$ with real-analytic smooth boundaries.
Reviewer: J.Burbea

##### MSC:
 32A25 Integral representations; canonical kernels (Szegő, Bergman, etc.) 32T99 Pseudoconvex domains 32A40 Boundary behavior of holomorphic functions of several complex variables
##### Keywords:
pseudoconvex domain; Levi-form; Bergman kernel
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##### References:
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