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.  Let  $$D\subset{\mathbb{C}}^n, G\subset{\mathbb{C}}^m$$ be open sets, let A (resp. B) be a subset of the boundary ∂D (resp. ∂G) and let W be the 2-fold boundary cross  $$((D\cup A)\times B) \cup (A \times (B \cup G))$$ . An open subset  $$X \subset {{\mathbb{C}}^{n+m}}$$ is said to be the “envelope of holomorphy” of W if it is, in some sense, the maximal open set with the following property: Any function locally bounded on W and separately holomorphic on  $$(A \times G) \cup (D \times B)$$ “extends” to a holomorphic function defined on X which admits the boundary values f a.e. on W. In this work we will determine the envelope of holomorphy of some boundary crosses.

Envelope of holomorphy for boundary cross sets