Peak functions and boundary behaviour of holomorphically invariant distances and metrics on strictly pseudoconvex domains

Abstract

We give a parameter version of Graham-Kerzman approximation theorem for bounded holomorphic functions on strictly pseudoconvex domains. As an application, we present some uniform estimates for the boundary behaviour of the Kobayashi and Carathéodory pseudodistences on such domains.2010 Mathematics Subject Classification. Primary 32T40; Secondary 32T15, 32F45. Key words and phrases. strictly pseudoconvex domains, peak functions, Kobayashi pseudodistance, Carathéodory pseudodistance.Moreover, if m = 1, then N can be chosen to be zero, (C) f t − f t Gt∩B(ζt,ρ) < ε f Gt∩B(ζt,R) .Remark 1.6. Notice that if m = 1, then the estimate in (B) depends in fact only on ε and R.

Parametrix for the localization of the Bergman metric on strictly pseudoconvex domains

“…Remark 1.7. In [9] we proved an analogous result for bounded holomorphic functions, instead of square integrable holomorphic ones (see also [3]). The proof of said assertion is based on stating and solving certain family of subtlē ∂ problems on some deformations of the domains G t , with estimates that do not depend on t ∈ T and ζ t ∈ ∂G t .…”

“…Define ρ := min{ η 2 , η 1 5 }. As in [9], we show that for any s ∈ T we may choose points ζ s 1 , . .…”

“…As in [9], observe that G t ∩ B(ζ 0 , ρ) ⊂⊂ G t 1 ∩ B(ζ 0 , η) and, by (3.2), the distance of the former set to the boundary of the latter one is bounded from below by some positive constant, independent of t and the choice of ζ 0 .…”

“…Proof of Theorem 1.5. The proof of Theorem 1.5 is similar to the proof of Theorem 1.5 from [9]. Therefore, we here focus mainly on the part of it that is new.…”