The -product of domains in several complex variables - Zając, Sylwester (Complex variables and elliptic equations, 2024)
In this article, we investigate the problem of computing the $ * $ ∗ -product of domains in $ \mathbb {C}^N $ C N . Assuming that $ 0\in G\subset \mathbb {C}^N $ 0 ∈ G ⊂ C N is an arbitrary Runge domain and $ 0\in D\subset \mathbb {C}^N $ 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 $ D ∗ G and another domain in $ \mathbb {C}^N $ 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 $ D ∗ G for two-dimensional domains D and G satisfying the outlined assumptions.