The main result of the paper is a new Hartogs type extension theorem for generalized (N, k)-crosses with analytic singularities for separately holomorphic functions and for separately meromorphic functions. Our result is a simultaneous generalization of several known results, from the classical cross theorem, through the extension theorem with analytic singularities for generalized crosses, to the cross theorem with analytic singularities for meromorphic functions.

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.

We prove that given a family (Gt) of strictly pseudoconvex domains varying in C 2 topology on domains, there exists a continuously varying family of exposing maps h t,ζ for all Gt at every ζ ∈ ∂Gt.