# Baouendi-Treves approximation theorem

Suppose $M$ is a real smooth manifold. Let $\mathcal{V}$ be a subbundle of the complexified tangent space $\mathbb{C}TM$ (that is $\mathbb{C}\otimes TM$). Let $n=\dim_{\mathbb{C}}\mathcal{V}$ and $d=\dim_{\mathbb{R}}M.$ We will say that $\mathcal{V}$ is integrable, if it is integrable in the following sense. Suppose that for any point $p\in M,$ there exist $m=d-n$ smooth complex valued functions  $z_{1},\ldots,z_{m}$ defined in a neighbourhood of $p$, such that the differentials  $dz_{1},\ldots,dz_{m}$ are $\mathbb{C}$-linearly independent  and for all sections $L\in\Gamma(M,\mathcal{V})$ we have $Lz_{k}=0$ for $k=1,\ldots,m.$ We say $z=(z_{1},\ldots,z_{m})$ are near $p.$

We say $f$ is a if $Lf=0$ for every $L\in\Gamma(M,\mathcal{V})$ 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 $\mathcal{V}$ an integrable subbundle as above. Let $p\in M$ be fixed and let $z=(z_{1},\ldots,z_{m})$ be basic solutions near $p$. Then there exists a compact neighbourhood $K$ of $p$, such that for any continuous  solution $f\colon M\to\mathbb{C},$ there exists a sequence $p_{j}$ of polynomials in $m$ variables with complex coefficients such that

 $p_{j}(z_{1},\ldots,z_{m})\to f\text{ ~{}~{}~{}~{} uniformly in K.}$

In particular we have the following corollary for CR submanifolds. A real smooth CR submanifold that is embedded in ${\mathbb{C}}^{N}$ has the CR vector fields as the integrable subbundle $\mathcal{V}$. Also the coordinate  functions $z_{1},\ldots,z_{N}$ 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 ${\mathbb{C}}^{N}$ is of the minimal dimension  .

###### Corollary.

Let $M\subset{\mathbb{C}}^{N}$ be an embedded real smooth generic submanifold and $p\in M$. Then there exists a compact neighbourhood $K\subset M$ 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 $K\subset M$ 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 $K$ is not arbitrary, it does not depend on $f$, it only depends on $M$ and $p$.

###### Example.

Suppose $M\subset{\mathbb{C}}^{2}$ is given in coordinates $(z,w)$ by $\operatorname{Im}w=\lvert z\rvert^{2}.$ Note that for some $t>0,$ the map $\xi\mapsto(t\xi,t)$ is an attached analytic disc. By taking different $t>0,$ we can fill the set $\{(z,w)\mid\operatorname{Im}w\geq\lvert z\rvert^{2}\}$ 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 $t<\epsilon$ for some $\epsilon>0$ (such that the boundary of the disc lies in $K$). Hence $f$ extends to a holomorphic function on $\epsilon>\operatorname{Im}w>\lvert z\rvert^{2}$, and which is continuous on $\epsilon>\operatorname{Im}w\geq\lvert z\rvert^{2}$.

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

###### Corollary.

Suppose $M\subset{\mathbb{C}}^{N}$ 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).

###### Corollary.

Let $U\subset{\mathbb{C}}^{N}$ be a domain with smooth strongly pseudoconvex boundary. Suppose $f$ is a continuous CR function on $\partial U$. Then there exists a function $f$ holomorphic in $U$ and continuous on $\bar{U},$ such that $F|_{\partial U}=f.$

## References

• 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 BaouendiTrevesApproximationTheorem 2015-05-07 16:14:43 2015-05-07 16:14:43 jirka (4157) jirka (4157) 7 jirka (4157) Theorem msc 32V25