The $*$-product of domains in several complex variables

Abstract

In this article we continue the research, carried out in [25], on computing the * -product of domains in C N . Assuming that 0 ∈ G ⊂ C N is an arbitrary Runge domain and 0 ∈ D ⊂ C N is a bounded, smooth and linearly convex domain (or a non-decreasing union of such ones), we establish a geometric relation between D * G and another domain in C N which is 'extremal' (in an appropriate sense) with respect to a special coefficient multiplier dependent only on the dimension N . Next, for N = 2, we derive a characterization of the latter domain expressed in terms of planar geometry. These two results, when combined together, give a formula which allows to calculate D * G for two-dimensional domains D and G satisfying the outlined assumptions.