A boundary cross theorem for separately holomorphic functions

Abstract

Abstract. Let D ⊂ C n and 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 crossA (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 to N -fold crosses is also given.

Boundary cross theorem in dimension 1

“…Although our results have been stated only for the case of a 2-fold cross, they can be formulated for an N -fold cross with any N ≥ 2 (see also [9,10]). Now we present the main ideas of the proof of Theorems A and B.…”

“…In fact, we keep the main notation from [10]. In particular E denotes the open unit disc in C and mes the linear measure (i.e.…”

“…Step 3: Proof of (2) Therefore, the two-constant theorem (see Theorem 2.2 in [10]) implies that (6.12) |f…”

“…The main ingredient for the second step is a mixed cross type theorem (see also [10]) valid for measurable boundary sets on complex manifolds of dimension 1. We prove it using a recent work of Zeriahi (see [15]) and the classical method of doubly orthogonal bases of Bergman type.…”

Envelope of holomorphy for boundary cross sets

“…In a series of articles [6,7,8] the authors establish various "boundary cross theorems". These results deal with the continuation of holomorphic functions of several complex variables which are defined on some boundary crosses.…”