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

TL;DR

This paper extends the understanding of the $*$-product of domains in several complex variables, providing geometric relations and explicit formulas for two-dimensional cases under specific conditions.

Contribution

It establishes a geometric relation between $D*G$ and an extremal domain, and derives a planar geometric characterization for the two-dimensional case.

Findings

Derived a formula for $D*G$ in 2D domains.

Established a geometric relation involving extremal domains.

Provided a planar geometric characterization for the case $N=2$.

Abstract

In this article we continue the research, carried out in \cite{zajac}, on computing the $*$-product of domains in $\CC^N$. Assuming that $0\in G\subset\CC^N$ is an arbitrary Runge domain and $0\in D\subset\CC^N$ 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 $\CC^N$ 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.