Baouendi-Treves approximation theorem

Suppose $M$ is a real smooth manifold. Let $𝒱$ be a subbundle of the complexified tangent space $ℂTM$(that is $ℂ\otimes TM$). Let $n={dim}_{ℂ}𝒱$ and $d={dim}_{ℝ}M.$ We will say that $𝒱$ is integrable, if it is integrable in the following sense. Suppose that for anypoint $p\in M,$there exist $m=d-n$ smooth complex valued functions$z_1,\mathrm{\dots },z_m$ defined in a neighbourhood of $p$, such that the differentials $d{z}_1,\mathrm{\dots },d{z}_m$ are $ℂ$-linearly independent and for all sections $L\in \mathrm{\Gamma }(M,𝒱)$ we have $L{z}_k=0$ for$k=1,\mathrm{\dots },m.$ We say $z=(z_1,\mathrm{\dots },z_m)$ are near $p.$

We say $f$ is a if $Lf=0$ for every $L\in \mathrm{\Gamma }(M,𝒱)$ in the senseof 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 $p\mathrm{\in }M$ be fixed and let $z\mathrm{=}\mathrm{(}{z}_{\mathrm{1}}\mathrm{,}\mathrm{\dots }\mathrm{,}{z}_m\mathrm{)}$ be basic solutions near $p$. Then there exists a compactneighbourhood $K$ of $p$, such that for any continuous solution $f\mathrm{:}M\mathrm{\to }\mathrm{C}\mathrm{,}$there exists a sequence $p_j$ of polynomials in $m$ variables with complex coefficients such that

$$p_j(z_1,\mathrm{\dots },z_m)\to f\mathit{\text{ uniformly in }}K.$$

In particular we have the following corollary for CR submanifolds. A real smooth CR submanifoldthat is embedded in ${ℂ}^N$ has the CR vector fields as the integrable subbundle $𝒱$.Also the coordinate functions $z_1,\mathrm{\dots },z_N$ can be taken as the basic solutions. We will require that$M$ be a generic submanifoldrather than just any CR submanifold to make sure that ${ℂ}^N$ is of the minimal dimension.

Corollary.

Let $M\mathrm{\subset }{\mathrm{C}}^N$ be an embedded real smooth generic submanifold and $p\mathrm{\in }M$. Then there exists acompact neighbourhood $K\mathrm{\subset }M$ of $p$ such that any continuous CR function $f$ is uniformly approximated on $K$ by polynomialsin $N$ variables.

This result can be used to extend CR functions from CR submanifolds. For example, if we can fill a certain setwith analytic discs attached to $M$, we can approximate $f$ on $K\subset M$ and by the maximum principle we willbe 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 $M\mathrm{\subset }{\mathrm{C}}^{\mathrm{2}}$ is given in coordinates $\mathrm{(}z\mathrm{,}w\mathrm{)}$ by $\mathrm{Im}\mathit{}w\mathrm{=}{\mathrm{|}z\mathrm{|}}^{\mathrm{2}}\mathrm{.}$Note that forsome $t\mathrm{>}\mathrm{0}\mathrm{,}$the map $\xi \mathrm{\mapsto }\mathrm{(}t\mathit{}\xi \mathrm{,}t\mathrm{)}$ is an attached analytic disc. By taking different $t\mathrm{>}\mathrm{0}\mathrm{,}$we can fill the set $\mathrm{\{}\mathrm{(}z\mathrm{,}w\mathrm{)}\mathrm{\mid }\mathrm{Im}\mathit{}w\mathrm{\ge }{\mathrm{|}z\mathrm{|}}^{\mathrm{2}}\mathrm{\}}$ 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 sequenceof holomorphic polynomials converges uniformly on all the discs for for some $ϵ\mathrm{>}\mathrm{0}$ (such that the boundary of the disc lies in $K$).Hence $f$ extends to a holomorphic function on $ϵ\mathrm{>}\mathrm{Im}\mathit{}w\mathrm{>}{\mathrm{|}z\mathrm{|}}^{\mathrm{2}}$, and which iscontinuous on $ϵ\mathrm{>}\mathrm{Im}\mathit{}w\mathrm{\ge }{\mathrm{|}z\mathrm{|}}^{\mathrm{2}}$.

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

Corollary.

Suppose $M\mathrm{\subset }{\mathrm{C}}^N$ be a smooth strongly pseudoconvex hypersurface and $f$ a continuousCR function on $M\mathrm{.}$ Then $f$ extends to a small neighbourhood on the pseudoconvex side of $M$ as aholomorphic function.

Using the above corollary we can prove the Hartogs phenomenon for hypersurfaces by reducing to the standardHartogs phenomenon (although the theorem also holds without pseudoconvexity with a different proof).

Corollary.

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