Capacities, Green function, and Bergman functions. (English) Zbl 1539.31001
S. Fu [Proc. Am. Math. Soc. 121, No. 3, 979–980 (1994; Zbl 0808.32024)] inductively applied the famed Ohsawa-Takegoshi extension theorem [T. Ohsawa and K. Takegoshi, Math. Z. 195, 197–204 (1987; Zbl 0625.32011)] to show that on a bounded pseudoconvex domain in \(\mathbb{C}^n\) with boundary of class \(C^2\), the product of the Bergman kernel function on the diagonal with the square of the distance to the boundary is bounded below by a positive constant. The author generalizes this technique by proving and applying some quantitative estimates in one-dimensional potential theory.
One of the main results states that the indicated lower bound on the Bergman kernel still holds when the hypothesis of class \(C^2\) boundary is replaced by the hypothesis that the boundary is locally the graph of a continuous function. Another of the striking theorems in the article is that the estimate on the Bergman kernel holds for every bounded pseudoconvex Runge domain.
One of the main results states that the indicated lower bound on the Bergman kernel still holds when the hypothesis of class \(C^2\) boundary is replaced by the hypothesis that the boundary is locally the graph of a continuous function. Another of the striking theorems in the article is that the estimate on the Bergman kernel holds for every bounded pseudoconvex Runge domain.
Reviewer: Harold P. Boas (College Station)
MSC:
| 31A15 | Potentials and capacity, harmonic measure, extremal length and related notions in two dimensions |
| 32A25 | Integral representations; canonical kernels (Szegő, Bergman, etc.) |
| 32T27 | Geometric and analytic invariants on weakly pseudoconvex boundaries |
| 32U35 | Plurisubharmonic extremal functions, pluricomplex Green functions |
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