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Estimates on the Bergman kernels of convex domains. (English) Zbl 0816.32018
Let $$\Omega$$ be a bounded domain in $$\mathbb{C}^ n$$ with a smooth boundary $$\partial \Omega$$, and let $$K_ \Omega$$ be the Bergman kernel function of $$\Omega$$. Let $$z_ 0 \in \partial \Omega$$ be a boundary point of finite type (in the sense of D’Angelo), near which $$\Omega$$ is convex. Under these assumptions, the author establishes sharp bounds for $$K_ \Omega (z, \zeta)$$ and its derivatives $$D^ k_ z \overline D^ m_ \zeta K_ \Omega (z, \zeta)$$ $$(k,m \in \mathbb{Z}_ +)$$, as $$z$$ and $$\zeta$$ vary independently near $$z_ 0$$. These bounds are described in terms of a noneuclidean pseudometric which arises from consideration of the largest allowable polydisk, centered at an arbitrary point near $$z_ 0$$, which fits inside various level sets of $$\partial \Omega$$. The author also remarks that these bounds imply certain boundedness theorems for the Bergman projector $$B_ \Omega$$ of $$\Omega$$, given by $(B_ \Omega f) (z) = \int_ \Omega K_ \Omega (z, \zeta) f(\zeta) d \zeta \quad (z \in \Omega),$ in all $$L^ P$$-norms and some nonisotropic Hölder norms. In addition to the above main result, the author provides a new proof for a result due to M. E. Fornaess and N. Sibony [Duke Math. J. 58, No. 3, 633-655 (1989; Zbl 0679.32017)] on the sharp subelliptic estimate for the $$\overline \partial$$-Neumann problem. Another interesting result, concerned with the size of the Bergman kernel $$K_ \Omega$$ on the diagonal of $$\Omega \times \Omega$$, is presented. This result was first proved by S.-C. Chen [Ph. D. Diss. Purdue Univ. (1989)] by applying a theorem of D. W. Catlin [Math. Z. 200, No. 3, 429-466 (1989; Zbl 0661.32030)]. The present method of proof of this result is, in fact, a slight modification of Catlin’s argument.

##### MSC:
 32A25 Integral representations; canonical kernels (Szegő, Bergman, etc.) 32T99 Pseudoconvex domains 32W05 $$\overline\partial$$ and $$\overline\partial$$-Neumann operators
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