A Forelli–Rudin Construction and Asymptotics of Weighted Bergman Kernels
✍ Scribed by Miroslav Engliš
- Publisher
- Elsevier Science
- Year
- 2000
- Tongue
- English
- Weight
- 203 KB
- Volume
- 177
- Category
- Article
- ISSN
- 0022-1236
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✦ Synopsis
Let 0 be a pseudoconvex domain in C N with smooth boundary, &,, & two smooth defining functions for 0=[,>0] such that &log , &log , are plurisubharmonic, z # 0 a point at which &log , is strictly plurisubharmonic, and M 0 an integer. We show that as k Ä , the Bergman kernels with respect to the weights , k M have the asymptotic expansion
For 0 strongly pseudoconvex with real-analytic boundary, ,, real analytic and &log ,, &log strictly plurisubharmonic on 0, we obtain also the analogous result for K , k M (x, y) for (x, y) near the diagonal and discuss consequences for the asymptotics of the Berezin transform and for the Berezin quantization. The proofs rely on Fefferman's expansion for the Bergman kernel in a certain Forelli Rudin type domain over 0; as another application, they also yield a generalization of the cited Fefferman's expansion to a class of weighted Bergman kernels.
2000