Boundary cross theorem in dimension 1 with singularities

Abstract

. Let D and G be copies of the open unit disc in C, let A (resp. B) be a measurable subset of ∂D (resp. ∂G), let W be the 2-fold cross ((D u A) x B) u (A X (B u G)), and let M be a relatively closed subset of W. Suppose in addition that A and B are of positive one-dimensional Lebesgue measure and that M is fiberwise polar (resp. fiberwise discrete) and that M∩ (A X B) = ∅. We determine the envelope of holomorphy W?M of W M in the sense that any function locally bounded on W M, measurable on A × B, and separately holomorphic on ((A × G) u (D x B)) M extends to a function holomorphic on W?M.