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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.

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