# Baouendi-Treves approximation theorem

Suppose $M$ is a real smooth manifold. Let $\mathrm{\pi \x9d\x92\pm}$ be a subbundle of the complexified tangent space $\mathrm{\beta \x84\x82}\beta \x81\u2019T\beta \x81\u2019M$
(that is $\mathrm{\beta \x84\x82}\beta \x8a\x97T\beta \x81\u2019M$). Let $n={dim}_{\mathrm{\beta \x84\x82}}\beta \x81\u2018\mathrm{\pi \x9d\x92\pm}$ and $d={dim}_{\mathrm{\beta \x84\x9d}}\beta \x81\u2018M.$ We will say that $\mathrm{\pi \x9d\x92\pm}$ is *integrable*, if it is integrable in the following sense. Suppose that for any
point $p\beta \x88\x88M,$
there exist $m=d-n$ smooth complex valued functions^{}
${z}_{1},\mathrm{\beta \x80\xa6},{z}_{m}$ defined in a neighbourhood of $p$, such that the differentials^{} $d\beta \x81\u2019{z}_{1},\mathrm{\beta \x80\xa6},d\beta \x81\u2019{z}_{m}$ are $\mathrm{\beta \x84\x82}$-linearly independent^{} and for all sections $L\beta \x88\x88\mathrm{\Xi \x93}\beta \x81\u2019(M,\mathrm{\pi \x9d\x92\pm})$ we have $L\beta \x81\u2019{z}_{k}=0$ for
$k=1,\mathrm{\beta \x80\xa6},m.$ We say $z=({z}_{1},\mathrm{\beta \x80\xa6},{z}_{m})$ are near $p.$

We say $f$ is a if $L\beta \x81\u2019f=0$ for every $L\beta \x88\x88\mathrm{\Xi \x93}\beta \x81\u2019(M,\mathrm{\pi \x9d\x92\pm})$ 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 $\mathrm{V}$ an integrable subbundle as above.
Let $p\mathrm{\beta \x88\x88}M$ be fixed and let $z\mathrm{=}\mathrm{(}{z}_{\mathrm{1}}\mathrm{,}\mathrm{\beta \x80\xa6}\mathrm{,}{z}_{m}\mathrm{)}$ be basic solutions near $p$. Then there exists a compact
neighbourhood $K$ of $p$, such that for any continuous^{} solution $f\mathrm{:}M\mathrm{\beta \x86\x92}\mathrm{C}\mathrm{,}$
there exists a sequence ${p}_{j}$ of polynomials in $m$ variables with complex coefficients such that

$${p}_{j}\beta \x81\u2019({z}_{1},\mathrm{\beta \x80\xa6},{z}_{m})\beta \x86\x92f\beta \x81\u2019\mathit{\text{\Beta uniformly in\Beta}}K.$$ |

In particular we have the following corollary for CR submanifolds. A real smooth CR submanifold
that is embedded in ${\mathrm{\beta \x84\x82}}^{N}$ has the CR vector fields as the integrable subbundle $\mathrm{\pi \x9d\x92\pm}$.
Also the coordinate^{} functions ${z}_{1},\mathrm{\beta \x80\xa6},{z}_{N}$ 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 ${\mathrm{\beta \x84\x82}}^{N}$ is of the minimal dimension^{}.

###### Corollary.

Let $M\mathrm{\beta \x8a\x82}{\mathrm{C}}^{N}$ be an embedded real smooth generic submanifold and $p\mathrm{\beta \x88\x88}M$. Then there exists a compact neighbourhood $K\mathrm{\beta \x8a\x82}M$ 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 $K\beta \x8a\x82M$ 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 $M\mathrm{\beta \x8a\x82}{\mathrm{C}}^{\mathrm{2}}$ is given in coordinates $\mathrm{(}z\mathrm{,}w\mathrm{)}$ by $\mathrm{Im}\mathit{\beta \x81\u2018}w\mathrm{=}{\mathrm{|}z\mathrm{|}}^{\mathrm{2}}\mathrm{.}$ Note that for some $t\mathrm{>}\mathrm{0}\mathrm{,}$ the map $\mathrm{\Xi \u038e}\mathrm{\beta \x86\xa6}\mathrm{(}t\mathit{\beta \x81\u2019}\mathrm{\Xi \u038e}\mathrm{,}t\mathrm{)}$ is an attached analytic disc. By taking different $t\mathrm{>}\mathrm{0}\mathrm{,}$ we can fill the set $\mathrm{\{}\mathrm{(}z\mathrm{,}w\mathrm{)}\mathrm{\beta \x88\pounds}\mathrm{Im}\mathit{\beta \x81\u2018}w\mathrm{\beta \x89\u20af}{\mathrm{|}z\mathrm{|}}^{\mathrm{2}}\mathrm{\}}$ by analytic discs attached to $M\mathrm{.}$ If $f$ is a continuous CR function on $M$, then there exists some compact neighbourhood $K$ of $\mathrm{(}\mathrm{0}\mathrm{,}\mathrm{0}\mathrm{)}$ 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 $$ for some $\mathrm{{\rm O}\u0385}\mathrm{>}\mathrm{0}$ (such that the boundary of the disc lies in $K$). Hence $f$ extends to a holomorphic function on $\mathrm{{\rm O}\u0385}\mathrm{>}\mathrm{Im}\mathit{\beta \x81\u2018}w\mathrm{>}{\mathrm{|}z\mathrm{|}}^{\mathrm{2}}$, and which is continuous on $\mathrm{{\rm O}\u0385}\mathrm{>}\mathrm{Im}\mathit{\beta \x81\u2018}w\mathrm{\beta \x89\u20af}{\mathrm{|}z\mathrm{|}}^{\mathrm{2}}$.

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

###### Corollary.

Suppose $M\mathrm{\beta \x8a\x82}{\mathrm{C}}^{N}$ be a smooth strongly pseudoconvex hypersurface and $f$ a continuous CR function on $M\mathrm{.}$ 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 $U\mathrm{\beta \x8a\x82}{\mathrm{C}}^{N}$ be a domain with smooth strongly pseudoconvex boundary. Suppose $f$ is a continuous CR function on $\mathrm{\beta \x88\x82}\mathit{\beta \x81\u2018}U$. Then there exists a function $f$ holomorphic in $U$ and continuous on $\stackrel{\mathrm{{\rm B}\u2015}}{U}\mathrm{,}$ such that ${F\mathrm{|}}_{\mathrm{\beta \x88\x82}\mathit{\beta \x81\u2018}U}\mathrm{=}f\mathrm{.}$

## 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 |
---|---|

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 |