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Compressed product of balls and lower boundary estimates of Bergman kernels. (English) Zbl 1057.32003
Mladenov, Ivaïlo M. (ed.) et al., Proceedings of the 4th international conference on geometry, integrability and quantization, Sts. Constantine and Elena, Bulgaria, June 6–15, 2002. Sofia: Coral Press Scientific Publishing (ISBN 954-90618-4-1/pbk). 193-205 (2003).
The image $$B_{p^\sigma,q}$$ of a product of balls $$B_p\times B_q$$ under a compression $c_\sigma(X,Y)=(X, V(1-\,^t\bar XX)^{\sigma/2})$ is called a compressed product of balls of exponent $$\sigma\in {\mathbb R}$$.
The author shows that the group $$\operatorname{Aut}(B_{p^\sigma,q})$$ of the holomorphic automorphisms and the $$\operatorname{Aut}(B_{p^\sigma,q})$$-orbit structure of $$B_{p^\sigma,q}$$ is verified by an explicit calculation of the Bergman kernel. As a consequence, local lower boundary estimates of the Bergman kernels of the bounded pseudoconvex domains are obtained, which are locally inscribed in $$B_{p^\sigma,q}$$ at a common boundary point.
For the entire collection see [Zbl 1008.00022].
##### MSC:
 32A25 Integral representations; canonical kernels (Szegő, Bergman, etc.) 32M05 Complex Lie groups, group actions on complex spaces
##### Keywords:
Bergman kernel; holomorphic automorphisms; estimates