Baouendi-Treves approximation theorem

Suppose M is a real smooth manifold. Let 𝒱 be a subbundle of the complexified tangent space ℂ⁒T⁒M (that is β„‚βŠ—T⁒M). Let n=dimℂ⁑𝒱 and d=dimℝ⁑M. We will say that 𝒱 is integrable, if it is integrable in the following sense. Suppose that for any point p∈M, there exist m=d-n smooth complex valued functionsMathworldPlanetmath z1,…,zm defined in a neighbourhood of p, such that the differentialsMathworldPlanetmath d⁒z1,…,d⁒zm are β„‚-linearly independentMathworldPlanetmath and for all sections LβˆˆΞ“β’(M,𝒱) we have L⁒zk=0 for k=1,…,m. We say z=(z1,…,zm) are near p.

We say f is a if L⁒f=0 for every LβˆˆΞ“β’(M,𝒱) in the sense of distributions (or classically if f is in fact smooth).

Theorem (Baouendi-Treves).

Suppose M is a smooth manifold of real dimension d and V an integrable subbundle as above. Let p∈M be fixed and let z=(z1,…,zm) be basic solutions near p. Then there exists a compact neighbourhood K of p, such that for any continuousMathworldPlanetmath solution f:Mβ†’C, there exists a sequence pj of polynomials in m variables with complex coefficients such that

pj⁒(z1,…,zm)β†’f⁒ uniformly inΒ K.

In particular we have the following corollary for CR submanifolds. A real smooth CR submanifold that is embedded in β„‚N has the CR vector fields as the integrable subbundle 𝒱. Also the coordinatePlanetmathPlanetmath functions z1,…,zN can be taken as the basic solutions. We will require that M be a generic submanifold rather than just any CR submanifold to make sure that β„‚N is of the minimal dimensionPlanetmathPlanetmath.


Let MβŠ‚CN be an embedded real smooth generic submanifold and p∈M. Then there exists a compact neighbourhood KβŠ‚M of p such that any continuous CR function f is uniformly approximated on K by polynomials in N 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 M, we can approximate f on KβŠ‚M and by the maximum principle we will be able to use the fact that uniform limits of holomorphic functionsMathworldPlanetmath (in this case polynomials) are holomorphic. A key point is that while K is not arbitrary, it does not depend on f, it only depends on M and p.


Suppose MβŠ‚C2 is given in coordinates (z,w) by Im⁑w=|z|2. Note that for some t>0, the map ξ↦(t⁒ξ,t) is an attached analytic disc. By taking different t>0, we can fill the set {(z,w)∣Im⁑wβ‰₯|z|2} by analytic discs attached to M. If f is a continuous CR function on M, then there exists some compact neighbourhood K of (0,0) such that f is uniformly approximated on K by holomorphic polynomials. By maximum principle we get that this sequence of holomorphic polynomials converges uniformly on all the discs for t<Ο΅ for some Ο΅>0 (such that the boundary of the disc lies in K). Hence f extends to a holomorphic function on Ο΅>Im⁑w>|z|2, and which is continuous on Ο΅>Im⁑wβ‰₯|z|2.

Using methods of the example it is possible (among many other results) to prove the following.


Suppose MβŠ‚CN be a smooth strongly pseudoconvex hypersurface and f a continuous CR function on M. Then f extends to a small neighbourhood on the pseudoconvex side of M as a holomorphic function.

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 UβŠ‚CN be a domain with smooth strongly pseudoconvex boundary. Suppose f is a continuous CR function on βˆ‚β‘U. Then there exists a function f holomorphic in U and continuous on UΒ―, such that F|βˆ‚β‘U=f.


  • 1 M. Salah Baouendi, Peter Ebenfelt, Linda Preiss Rothschild. , Princeton University Press, Princeton, New Jersey, 1999.
  • 2 Albert Boggess. , CRC, 1991.
Title Baouendi-Treves approximation theorem
Canonical name BaouendiTrevesApproximationTheorem
Date of creation 2015-05-07 16:14:43
Last modified on 2015-05-07 16:14:43
Owner jirka (4157)
Last modified by jirka (4157)
Numerical id 7
Author jirka (4157)
Entry type Theorem
Classification msc 32V25