# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
  Catlin, D.: Invariant metrics on pseudoconvex domains. Several complex variables. Proceedings of the 1981 Hangzhou Conference. Boston 1984 · Zbl 0588.32028  Diederich, K.: Das Randverhalten der Bergmanschen Kernfunktion und Metrik in streng pseudokonvexen Gebieten. Math. Ann.189, 9-36 (1970) · Zbl 0189.36702 · doi:10.1007/BF01368157  Diederich, K., Lieb, I.: Konvexität in der komplexen Analysis. Neue Ergebnisse und Methoden. DMV-Seminar, Bd. 2. Basel, Boston, Stuttgart: Birkhäuser 1981 · Zbl 0473.32015  Diederich, K., Fornaess, J.E.: Proper holomorphic maps onto pseudoconvex domains with real-analytic boundary. Ann. Math.110, 575-592 (1979) · Zbl 0417.32010 · doi:10.2307/1971240  Fefferman, Ch.: The Bergman kernel and biholomorphic mappings of pseudoconvex domains. Invent. Math.26, 1-65 (1974) · Zbl 0289.32012 · doi:10.1007/BF01406845  Herbort, G.: Logarithmic growth of the Bergman kernel for weakly pseudoconvex domains in 478-1 of finite type. Manuscr. Math.45, 69-76 (1983) · Zbl 0559.32006 · doi:10.1007/BF01168581  Hörmander, L.:L 2-estimates and existence theorems for the 478-2. Acta Math.113, 89-152 (1965) · Zbl 0158.11002 · doi:10.1007/BF02391775  Kohn, J.J.: Boundary behavior of 478-3 on weakly pseudoconvex manifolds of dimension two. J. Differ. Geom.6, 523-542 (1972) · Zbl 0256.35060  Ohsawa, T.: On complete Kähler domains withC 1-boundary. Publ. RIMS Kyoto Univ.16, 929-940 (1980) · Zbl 0458.32010 · doi:10.2977/prims/1195186937  Ohsawa, T.: Boundary behavior of the Bergman kernel function on pseudoconvex domains. Publ. RIMS Kyoto Univ.20, 897-902 (1984) · Zbl 0569.32013 · doi:10.2977/prims/1195180870  Ohsawa, T., Takegoshi, K.:L 2 cohomology with weights for complete Kähler domains (in preparation) · Zbl 0625.32011
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.