An Edge-of-the-wedge Theorem for Hypersurface Cr Functions

The Lewy extension theorem asserts the holomorphic extendability of CR functions defined in a neighborhood of a point on a hypersurface in C. The edge-of-the-wedge theorem asserts the extendability of holomorphic functions defined in wedges in C with edge a maximally real submanifold. In this article we prove under suitable hypotheses the holomorphic extendability to an open set in C of CR functions defined in the intersection of a hypersurface with a wedge whose edge is contained in the hypersurface. Unlike the situation for the classical edge-of-the-wedge theorem, for this hypersurface version extendability generally depends on the direction of the wedge. Equip R with its standard inner product and let σ ∈ R be a unit vector. By the round cone in R of aperture δ > 0, extent l > 0, and axis σ, we shall mean