Fourth International Conference on
Geometry, Integrability and Quantization
June 6–15, 2002, Varna, Bulgaria
Ivaïlo M. Mladenov and Gregory L. Naber, Editors
Coral Press, Sofia 2003, pp 193–205
COMPRESSED PRODUCT OF BALLS AND LOWER
BOUNDARY ESTIMATES ON BERGMAN KERNELS
AZNIV KASPARIAN
Department of Mathematics and Informatics, “St Kliment Ohridski” University
5 James Bourchier Blvd, 1126 Sofia, Bulgaria
Abstract. The image Bpσ ,q of a product of balls Bp × Bq under a
σ
compression cσ (X, V ) = (X, V (1 − t X̄X) 2 ) 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 ∈ Cn ; ρ(z) < 0} with a smooth
boundary, Fefferman [6] and Boutet de Monvel–Sjöstrand [2] have expanded
the diagonal values kD (z) := kD (z, z) of the Bergman kernel in the form
kD (z) = ϕD (ρ)ρ−n−1 + ψD (ρ) log(−ρ), where ϕD (ρ) and ψD (ρ) are power
series in the defining function ρ = ρ(z).
Only few results are known for the boundary behavior of the Bergman kernel
of a weakly pseudoconvexP
domain. For arbitrary m = (m1 , . . . , mn ) ∈ Nn let
n
Em := {z ∈ C ; ρm (z) = nj=1 |zj |2mj − 1 < 0}, z o ∈ ∂Em ,
Pm := {j ∈ N ; zjo = 0 and mj > 1},
Qm := {j ∈ N ; zjo 6= 0 or mj = 1} .
Kamimoto has established in [11] the existence of an open subset U ⊂ Cn with
z o ∈ ∂U and a real analytic
Φm : U → {r ∈ R ; r > 0}, such that
P function
−
m−1
−card
Qm −1
j
j∈Pm
on U . If Pm = ∅ then Φm (z) is
kEm (z) = Φm (z)ρm (z)
o
bounded around z , while limz→zo Φm (z) = ∞ for Pm 6= ∅.
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