Baouendi-Treves approximation theorem
Suppose is a real smooth manifold. Let be a subbundle of the complexified tangent space (that is ). Let and We will say that is integrable, if it is integrable in the following sense. Suppose that for any point there exist smooth complex valued functions defined in a neighbourhood of , such that the differentials are -linearly independent and for all sections we have for We say are near
We say is a if for every in the sense of distributions (or classically if is in fact smooth).
Suppose is a smooth manifold of real dimension and an integrable subbundle as above. Let be fixed and let be basic solutions near . Then there exists a compact neighbourhood of , such that for any continuous solution there exists a sequence of polynomials in variables with complex coefficients such that
In particular we have the following corollary for CR submanifolds. A real smooth CR submanifold that is embedded in has the CR vector fields as the integrable subbundle . Also the coordinate functions can be taken as the basic solutions. We will require that be a generic submanifold rather than just any CR submanifold to make sure that is of the minimal dimension.
Let be an embedded real smooth generic submanifold and . Then there exists a compact neighbourhood of such that any continuous CR function is uniformly approximated on by polynomials in variables.
This result can be used to extend CR functions from CR submanifolds. For example, if we can fill a certain set with analytic discs attached to , we can approximate on and by the maximum principle we will be able to use the fact that uniform limits of holomorphic functions (in this case polynomials) are holomorphic. A key point is that while is not arbitrary, it does not depend on , it only depends on and .
Suppose is given in coordinates by Note that for some the map is an attached analytic disc. By taking different we can fill the set by analytic discs attached to If is a continuous CR function on , then there exists some compact neighbourhood of such that is uniformly approximated on by holomorphic polynomials. By maximum principle we get that this sequence of holomorphic polynomials converges uniformly on all the discs for for some (such that the boundary of the disc lies in ). Hence extends to a holomorphic function on , and which is continuous on .
Using methods of the example it is possible (among many other results) to prove the following.
Using the above corollary we can prove the Hartogs phenomenon for hypersurfaces by reducing to the standard Hartogs phenomenon (although the theorem also holds without pseudoconvexity with a different proof).
Let be a domain with smooth strongly pseudoconvex boundary. Suppose is a continuous CR function on . Then there exists a function holomorphic in and continuous on such that
- 1 M. Salah Baouendi, Peter Ebenfelt, Linda Preiss Rothschild. , Princeton University Press, Princeton, New Jersey, 1999.
- 2 Albert Boggess. , CRC, 1991.