PlanetMath: Baouendi-Treves approximation theorem

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Baouendi-Treves approximation theorem

(Theorem)

Suppose 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 smooth complex valued functions $z_1,\ldots,z_m$ defined in a neighbourhood of , 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 for $k = 1,\ldots,m.$ We say $z=(z_1,\ldots,z_m)$ are basic solutions near

We say is a continuous solution if for every $L \in \Gamma(M,\mathcal{V})$ in the sense of distributions (or classically if is in fact smooth).

Theorem 1 (Baouendi-Treves) Suppose is a smooth manifold of real dimension 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 . Then there exists a compact neighbourhood of , such that for any continuous solution $f \colon M \to \mathbb{C},$ there exists a sequence of polynomials in variables with complex coefficients such that

$\displaystyle p_j(z_1,\ldots,z_m) \to f$ uniformly in

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 be a generic submanifold rather than just any CR submanifold to make sure that ${\mathbb{C}}^N$ is of the minimal dimension.

Corollary 1 Let $M \subset {\mathbb{C}}^N$ be an embedded real smooth generic submanifold and $p \in M$ . Then there exists a compact set $K \subset M$ 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 $K \subset M$ 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.

Example 1 Suppose $M \subset {\mathbb{C}}^2$ is given in coordinates by $\operatorname{Im} w = \lvert z \rvert^2 .$ Note that for some the map $\xi \mapsto (t z, t)$ is an attached analytic disc. By taking different we can fill the set $\{ (z,w) \mid \operatorname{Im} w \geq \lvert z \rvert^2 \}$ 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 $t < \epsilon$ for some $\epsilon > 0$ (such that the boundary of the disc lies in ). Hence extends to a holomorphic function on some small neighbourhood of intersected with $\operatorname{Im} w \geq \lvert z \rvert^2 .$

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

Corollary 2 Suppose $M \subset {\mathbb{C}}^N$ be a smooth strongly pseudoconvex hypersurface and a continuous CR function on Then extends to a small neighbourhood on the pseudoconvex side of as a holomorphic function.

Using the above corollary we can prove the Hartogs phenomenon for hypersurfaces by reducing to the standard Hartogs phenomenon.

Corollary 3 Let $U \subset {\mathbb{C}}^N$ be a domain with smooth strongly pseudoconvex boundary. Suppose is a continuous CR function on $\partial U$ . Then there exists a function holomorphic in and continuous on $\bar{U},$ such that $F\vert _{\partial U} = f .$

Bibliography

1: M. Salah Baouendi, Peter Ebenfelt, Linda Preiss Rothschild. Real Submanifolds in Complex Space and Their Mappings, Princeton University Press, Princeton, New Jersey, 1999.
2: Albert Boggess. CR Manifolds and the Tangential Cauchy Riemann Complex, CRC, 1991.

"Baouendi-Treves approximation theorem" is owned by jirka.

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Cross-references: domain, Hartogs Phenomenon, side, pseudoconvex, hypersurface, strongly pseudoconvex, boundary, discs, converges uniformly, attached analytic disc, holomorphic functions, limits, maximum principle, analytic discs, CR function, compact set, minimal, generic submanifold, coordinate, CR vector fields, CR submanifolds, coefficients, variables, polynomials, sequence, continuous, compact, solutions, fixed, dimension, distributions, near, sections, independent, neighbourhood, functions, complex, smooth, point, tangent space, subbundle, smooth manifold, real

This is version 3 of Baouendi-Treves approximation theorem, born on 2007-12-04, modified 2007-12-05.
Object id is 10093, canonical name is BaouendiTrevesApproximationTheorem.
Accessed 259 times total.

Classification:

AMS MSC:

32V25 (Several complex variables and analytic spaces :: CR manifolds :: Extension of functions and other analytic objects from CR manifolds)

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