BaouendiTrevesApproximationTheorem 2007-12-04 20:04:40 2007-12-05 15:59:12 Theorem % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions \usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here \theoremstyle{theorem} \newtheorem*{thm}{Theorem} \newtheorem*{lemma}{Lemma} \newtheorem*{conj}{Conjecture} \newtheorem*{cor}{Corollary} \newtheorem*{example}{Example} \newtheorem*{prop}{Proposition} \theoremstyle{definition} \newtheorem*{defn}{Definition} \theoremstyle{remark} \newtheorem*{rmk}{Remark} 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 \emph{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 \emph{\PMlinkescapetext{basic solutions}} near $p.$ We say $f$ is a \emph{\PMlinkescapetext{continuous solution}} if $Lf = 0$ for every $L \in \Gamma(M,\mathcal{V})$ in the sense of distributions (or classically if $f$ is in fact smooth). \begin{thm}[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 \begin{equation*} p_j(z_1,\ldots,z_m) \to f \text{ ~~~~ uniformly in $K.$} \end{equation*} \end{thm} 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. \begin{cor} 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 $f$ is uniformly approximated on $K$ by polynomials in $N$ variables. \end{cor} 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. \begin{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 $\xi \mapsto (t z, t)$ 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 some small neighbourhood of $(0,0)$ intersected with $\operatorname{Im} w \geq \lvert z \rvert^2 .$ \end{example} Using methods of the example it is possible (among many other results) to prove the following. \begin{cor} 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. \end{cor} Using the above corollary we can prove the Hartogs phenomenon for hypersurfaces by reducing to the standard Hartogs phenomenon. \begin{cor} 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 $F|_{\partial U} = f .$ \end{cor} \begin{thebibliography}{9} \bibitem{ber:submanifold} M.\@ Salah Baouendi, Peter Ebenfelt, Linda Preiss Rothschild. {\em \PMlinkescapetext{Real Submanifolds in Complex Space and Their Mappings}}, Princeton University Press, Princeton, New Jersey, 1999. \bibitem{boggess} Albert Boggess. {\em \PMlinkescapetext{CR Manifolds and the Tangential Cauchy Riemann Complex}}, CRC, 1991. \end{thebibliography}