A boundary cross theorem for separately holomorphic functions

2004/11/30 by Peter Pflug Viet-Anh Nguyen, Nguyen, Peter Pflug Viet-Anh
Mathematics · #32D15 32D10 #Analytic and geometric function theory #Complex Variables (math.CV) #FOS: Mathematics #Geometry and complex manifolds #Holomorphic and Operator Theory #math.CV #msc:32D10 #msc:32D15

paper · pdf · doi:10.48550/arxiv.math/0411657

arxiv created 2004/11/30 · openalex publication_date 2004/11/30 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let D⊂ \Cⁿ, G⊂ \C^m be pseudoconvex domains, let A (resp. B) be an open subset of the boundary ∂ D (resp. ∂ G) and let X be the 2-fold cross ((D∪ A)× B)∪ (A×(B∪ G)). Suppose in addition that the domain D (resp. G) is \it locally C² smooth on A (resp. B). We shall determine the "envelope of holomorphy" X of X in the sense that any function continuous on X and separately holomorphic on (A× G) ∪ (D× B) extends to a function continuous on X and holomorphic on the interior of X. A generalization of this result for an N-fold cross is also given.

Related

• Envelope of holomorphy for boundary cross sets
• Boundary cross theorem in dimension 1
• Cross theorem
• Boundary cross theorem in dimension 1 with singularities
• An extension theorem for separately holomorphic functions with singularities