The famous 'edge of the wedge' theorem (N. N. Bogolubov 1956 [1]) asserts that if f(z) is a function holomorphic in a tubular domain Τ=Rn+iΓ, whose base Γ is the two-sided light cone y12>y22+ ... +yn2, and continuous in its closure, then f(z) admits an analytic continuation in Cn.
In [2] S.I.Pinchuk generalized Bogolubov`s result taking instead of the light cone an arbitrary wedge with the edge on a generating manifold bounded by smooth hypersurfaces in general position. In the paper [3] we studied the problem of holomorphic extension of functions into a neighborhood of the edge of a two-sided n-circled wedge in nongeneral position.
Analyzing the ideas of the proof in [3] we find out that in some cases the 'edge of the wedge' theorems could be interpreted as a question of an extension of sheafs.
Let Δ be a disjoint union of compact completely real manifolds Mrn that are subset of Cn, r>0. Denote
Δ* = r≠ 1 Mrn.
Assume that Δ is a fibering into holomorphic curves
lu={z:χj(z,u)=0, j=1,... ,n-1}, u is in Tn-1.
Theorem Let O(Δ*) be a sheaf of germs of holomorphic functions on Δ*. If its holomorphic section f extends continuously on M1n then f extends to a holomorphic section of the sheaf O(Δ).
References
1. Bogolubov N.N., Medvedev B.V. and Polivanov M.K., Problems of the Theory of Dispersion Relations, Fizmatgiz, 1958 (in Russian).
2. Pinchuk S.I., Bogolubov's theorem on the 'edge of the wedge' for generic manifolds, Math.Sb, 94(136), 1974, no. 3(7), 468-482 (in Russian).
3. Yurieva E.V., On the holomorphic extension into a neighborhood of the edge of a wedge in nongeneral position, Siberian Math. J., 52, 2011, no. 3, 563-568.