Compressed Product of Balls and Lower Boundary Estimates on Bergman Kernels

Abstract

The image Bpσ ,q of a product of balls Bp × Bq under a compression cσ(X,V ) = (X,V (1 − XX) σ2 ) is called a compressed product of balls of exponent σ ∈ R. The present note obtains the group Aut(Bpσ ,q) of the holomorphic automorphisms and the Aut(Bpσ ,q)orbit structure of Bpσ ,q and its boundary ∂Bpσ ,q for σ > 1. The Bergman completeness of Bpσ ,q is verified by an explicit calculation of the Bergman kernel. As a consequence, local lower boundary estimates on the Bergman kernels of the bounded pseudoconvex domains are obtained, which are locally inscribed in Bpσ ,q at a common boundary point. For a strictly pseudoconvex domain D = {z ∈ C ; ρ(z) 1}, Qm := {j ∈ N ; z j 6= 0 or mj = 1} . Kamimoto has established in [11] the existence of an open subset U ⊂ C with z ∈ ∂U and a real analytic function Φm : U → {r ∈ R ; r > 0}, such that kEm (z) = Φm(z)ρm(z) − ∑ j∈Pm m−1 j −cardQm−1 on U . If Pm = ∅ then Φm(z) is bounded around z, while limz→zo Φm(z) =∞ for Pm 6= ∅.