IV Extensions of CR functions

Whilst studying Hartogs’ phenomenon in Chapter III we proved that if D is a simply connected bounded domain in ℂn,n ⩾ 2, then any holomorphic function defined on a neighbourhood of the boundary of D can be extended to a holomorphic function on D. It follows that the restriction to $$\partial$$ D of a holomorphic function defined in a neighbourhood of $$\partial$$ D is the boundary value of a holomorphic function on D which is continuous on $$\overline{D}$$ . We now try to characterise the boundary values of holomorphic functions on a bounded domain $$D \subset \mathbb{C}^n$$ which are continuous on $$\overline{D}$$ . The main result of this chapter is Bochner’s extension theorem for CR functions defined on the boundary of a domain. Its proof uses the Bochner–Martinelli transform which is studied in Section 1. We also prove our first generalisation of Bochner’s theorem to CR functions which are only defined on part of the boundary of the domain. This generalisation is also based on the properties of the Bochner–Martinelli transform but it requires two extra ingredients: Stokes’ formula for CR functions and the integrals of the Bochner–Martinelli kernel.