The *-product of domains in several complex variables

2021/01/23 by Sylwester Zając, Zając, Sylwester
Mathematics · #32A05 #32D15 #Analytic and geometric function theory #Complex Variables (math.CV) #FOS: Mathematics #Holomorphic and Operator Theory #Meromorphic and Entire Functions #math.CV #msc:32A05 #msc:32D15

paper · pdf · doi:10.48550/arxiv.2101.09586

openalex publication_date 2021/01/23 · arxiv created 2021/03/01 · arxiv updated 2021/03/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this article we continue the research, carried out in \citezajac, on computing the *-product of domains in \CC^N. Assuming that 0∈ G⊂\CC^N is an arbitrary Runge domain and 0∈ D⊂\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.

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