Hartogs companions and holomorphic extensions in arbitrary dimension

2020/09/07 by Vlad Timofte, Timofte, Vlad
Mathematics · #46E50 #Advanced Banach Space Theory #Advanced Operator Algebra Research #Complex Variables (math.CV) #FOS: Mathematics #Holomorphic and Operator Theory #Primary 32D15 #Secondary 46G20

paper · pdf · doi:10.48550/arxiv.2009.03086

openalex publication_date 2020/09/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We show that every holomorphic map f\inH(Ω∖ K) (K⊂Ω⊂ℂⁿ, with K compact, Ω open, and n≥2), has a unique "Hartogs companion" f\inH(Ω) matching f on an open subset C_K,Ω⊂Ω∖ K. Furthermore, f extends f, if and only if ℂⁿ∖ K is a connected set; this equivalence proves the converse implication from the Hartogs Kugelsatz. The existence of vector-valued Hartogs companions in any dimension yields a Hartogs-type extension theorem for Gâteaux holomorphic maps f\inH_G(Ω∖ K,Y) on finitely open sets in arbitrary complex vector spaces. The equivalence is very similar to that for K⊂Ω⊂ℂⁿ and leads to a corresponding Hartogs Kugelsatz in arbitrary dimension and to extension theorems for five types of holomorphy (Gâteaux, Mackey/Silva, hypoanalytic, Fréchet, locally bounded). We also show that the range f(Ω) of a vector-valued Hartogs companion cannot leave a domain of holomorphy containing f(Ω∖ K). We establish a boundary principle for maps f\inH_G(Ω,Y)\capC(Ω,Y) on finitely bounded open sets. For Y=ℂ, the principle states that f(Ω)=f(∂Ω) (hence sup_x∈Ω|f(x)|=sup_x∈∂Ω|f(x)|). Several results require a new identity theorem, which yields a maximum norm principle and a "max-min" seminorm principle.

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