---
author:
- 'Nicolas Eisen  eisen@math.univ-poitiers.fr'
title: 'ON THE HOLOMORPHIC EXTENSION OF CR DISTRIBUTIONS FROM NON GENERIC CR SUBMANIFOLDS OF $\C^L$'
---

Introduction
============

Statement of Results
--------------------

Let $\n$ be a smooth generic submanifold of $\C^{k+m}$ of CR dimension $k$ and $h$ a smooth CR map from $\n$ into some $\C^{n}$ verifying $dh(0)=0$. Set $L=k+m+n$, we construct a CR submanifold $N$ of $\C^{p}$ near the origin as the graph of $h$ over $\n$, that is $N=\{(\n,h(\n))\}$. It turns out that any non generic CR submanifold of $\C^L$ can be obtained in that fashion, see for example [@[Bo]]. The main question we address in this paper is the possible holomorphic extension of CR functions of $N$ to some wedge $\W$ in a complex transverse direction. We say that a vector in $\C^L$ is [**complex transversal**]{} to $N$ at $p \in N$ if $v \not \in $ span$_{\C}T_pN$. For totally real submanifolds of $\C^L$, we have the following well known result (see the remarks following for different proofs of this result). If $N$ is a non generic totally real submanifold of $\C^L$ and $ v \in \C^L$ is complex transversal to $N$ at $p$. Then for any continuous function $f$ on a neighborhood of $p$ in $N$ there exists $\W_v$, a wedge of direction $v$ whose edge contains $N$ such that $f$ has a holomorphic extension to $\W_v$. In this paper, we shall study the case where $N$ is not totally real, that is we consider CR submanifolds of $\C^L$ given by $N=\{(\n,h(\n))\}$ where $\n$ is generic in $\C^{k+m}$ of CR dimension $k>0$. We say that a CR distribution extends holomorphically to a wedge $\W$ if there exists a holomorphic function $F$ in $W$ such that the boundary value of $F$ on $N$ is $f$, i.e $<f,\varphi>=
\lim_{\lambda \to 0^+} \int_{N}F(x+\lambda v)\varphi(x)dx$ for any $v \in  \W$.

\[th\] Let $N=\{(\n,h(\n))\} $ be a smooth ($\CC^{\infty})$ non generic CR submanifold of $\C^{k+1}\times \C^n$ such that $\n \subset \C^{k+1}$ is a hypersurface. If $f$ is a CR distribution on $N$ then for any $p \in
N$ and any $v$ complex transversal to $N$ at $p$, there exists a wedge $\W$ of direction $v$ whose edge contains a neighborhood of $p$ in $N$ and $F\in \O(\W)$ such that the boundary value of $F$ on $N$ is $f$.

We obtain some extension results for arbitrary $\n$ for decomposable CR distributions. Since the CR structure of $N$ is determined by $\n$, the CR distributions of $N$ are precisely the CR distributions of $\n$.

\[D1\] A CR distribution $u$ on $\n$ is decomposable at the point $Z\in \n$ if, near $Z$, $u=\sum_{j=1}^KU_j$ where the $U_j$ are CR distributions extending holomorphically to wedges $\W_j$ in $\C^{k+m}$ with edges $\n$. We shall say a CR distribution $u$ on $N$ is decomposable at a point $p=(p',h(p'))$ if $u$ is decomposable at $p'$ on $\n$.

\[t1\] Let $N=\{(\n,h(\n))\} $ be a smooth ($\CC^{\infty})$ non generic CR submanifold of $\C^{k+m+n}$ such that the function $h$ decomposable at some $p'_0 \in \n$. Let $v$ be a complex transversal vector to $N$ at $p_0=(p'_0,h(p'_0))$.

[**(A)**]{} If $f$ is a decomposable CR distribution at $p_0$, then, near $p_0$, there exists a wedge $\W$ of direction $v$ whose edge contains a neighborhood of $p_0$ in $N$ and $F\in \O(\W)$ such that the boundary value of $F$ on $N$ is $f$.

[**(B)**]{} There exists a wedge $\W$ of direction $v$ and $\{F_{\l}\}_{\l=1}^n$, $F_{\l}\in
\O(\W)$ such that $dF_1\wedge ...\wedge dF_n\not = 0$ on $\W$ and each $F_{\l}$ vanishes to order one on $N$.

We also get as a corollary:

\[t3\] Let $M$ be a smooth ($\CC^{\infty}$) generic submanifold of $\C^L$ containing through some $p_0 \in M$ a proper smooth ($\CC^{\infty}$) CR submanifold $N=(\n,h(\n))$ of same CR dimension, write $p_0=(p'_0,h(p'_0))$ with $p'_0\in \n$. Assume that the function $h$ decomposable at $p'_0$. Let $v\in T_{p_0}M \setminus 
\left [span_{\C}T_{p_0}N\right ]$. If $f$ is a decomposable CR distribution at $p_0$, then there exists a wedge $\W$ in $M$ of direction $v$ whose edge contains $N$ and $F$ smooth ($\CC^{\infty}$) CR functions on $\W$ such that the boundary value of $F$ on $N$ is $f$. Furthermore, there exists a collection of smooth CR functions $\{g_{\l}\}_{\l=1}^n$ vanishing to order one on $N$ and such that $dg_1\wedge....\wedge
dg_n \not=0 $ on $\W$.

Remarks
-------

The equivalent of theorem \[th\] for totally real manifolds (mentioned in the introduction) can be proved in several ways, one way is to follow the theory of analytic vectors of an elliptic operator due to Treves (see section 3.2). Another way is to use the following result: Let $N$ be a smooth submanifold of the boundary of $\Omega$, a strictly pseudoconvex domain in $\C^L$. If $N$ is complex tangential ($TN
\subset (T(\partial \Omega) \cap iT(\partial \Omega))$) then $N$ is a Pic interpolating set. See for example [@[Na]] or [@[Ru]]. Given $N$, a totally real non generic submanifold of $\C^L$, one can easily construct $\Omega$ as above and deduce the theorem.

The boundary value of a holomorphic function is well defined for functions of slow growth, that is there exists a constant $C>0$ and a positive integer $\ell$ such that $|F(z)|\leq {{C}\over{|dist(z,N)|^{\ell}}},$ where $dist(z,M)$ denotes the distance from a point $z$ to $N$. Conversely any CR distribution which extends holomorphically to a wedge $\W$ is given by the boundary value of a holomorphic function of slow growth. See for example [@[BER]].

As it is noted in [@[Tr-3]] on most CR submanifolds of $\C^L$ all CR functions are decomposable, hence the hypotheses of theorem \[t1\] hold in a generical sense. However, they are examples of CR submanifolds of $\C^L$ on which indecomposable CR functions exist, see [@[Tr-3]] for the general theory and [@[Rosay]] for an elementary explanation of such examples. It turns out that on Trépreau’s original example of a CR manifold where undecomposable CR functions exist we still do get holomorphic extension of CR functions to complex transversal wedges (see section 3.2).

Theorem \[t1\] implies that the extension obtained is not unique, which differs greatly with the holomorphic extension results obtained for generic submanifolds. We wish to point out the differences between our results and the previously known results on holomorphic extension of CR functions. The most general result on holomorphic extension is Tumanov’s theorem which states that if $M$ is a generic submanifold of $\C^L$ that is minimal at some $p_0$ then there exists a wedge $\W$ with edge $M$ (near $p_0$), such that one gets a “forced” unique local holomorphic extension of CR functions defined in a neighborhood of $p_0$ in $M$.

One should note that the question of CR extension can be viewed as a Cauchy problem with Cauchy data on a characteristic set $N$. In [@[Ei]] we constructed an example of an abstract CR structure where there is no such CR extension propriety. It is very easy to construct an abstract CR structure where there is no CR function vanishing on $N$ to finite order (see the example at the end of this paper).

Off course the situation is greatly different if we impose holomorphic extension to a full neighborhood of $N$, as the next example shows.

[**Example**]{} Let $I_n~~n>0$ be a sequence of disjointed intervals (separated by some open sets) in $\R$ accumulating to the origin and let $u$ be a smooth function such that $u|_{I_n}=1/n$. Let $\gamma=\{(s,u(s))\} \subset
\C^2$. Suppose $f$ is holomorphic on a neighborhood of the origin and that $f|_{\gamma}=0$, then $f(w_1,{{1} \over {n}})=0$ for all $n$ for $|w_1|$ small enough and thus $f \equiv 0$.

Background
----------

The first result on holomorphic extension of CR function is due to Lewy [@[Le]]. He proved that if a hypersurface was Levi nondegenerate at $p_0$ then CR functions extend holomorphically to one side of the hypersurface. This result was generalized by Boggess and Polking [@[Bo-Po]] for arbitrary dimensions. In the case of Levi flatness Trépreau [@[Tr-1]] proved that if a hypersurface in $\C^n$ is minimal at $p_0$ (it contains through $p_0$ no $n-1$ dimensional complex manifold), then CR functions extend holomorphically to one side of the hypersurface. The generalization of Trépreau’s result to arbitrary codimension is due to Tumanov [@[Tu-1]], in which he states that if the manifold $M$ is minimal at $p_0$ (it contains through $p_0$ no proper submanifolds of same CR dimension) then CR functions extend to a wedge in ${\C}^n$ with edge $M$. CR extension to manifolds of higher CR dimension has been dealt with in works by Taiani and Hill [@[Hi-Ta]], Tumanov [@[Tu-2]] as a well as in a recent paper by Eastwood and Graham [@[Ea-Gr]]. For general background on CR geometry, we recommend the books by Baouendi, Ebenfelt, Rothschild [@[BER]], Boggess [@[Bo]] and Jacobowitz [@[Ja]] and the survey paper on holomorphic extension by Trépreau [@[Tr-2]].

Outline of the Paper
--------------------

The main part of this paper is the proof of theorem\[t1\], we then obtain theorem \[th\] as a corollary. We show that it suffices to consider the case where $N$ is given as a trivial CR graph, that is $N=\{(\n,0)\}$. We then solve a Dirichlet problem on some open subset $\Omega$ of $\n \times \R^n$, by Treves’s analytic vector theory, this yields holomorphy in the $w$ variables. To obtain holomorphy in $z$ we need to proceed with a deformation on $\n$ so that the “boundary” of $\Omega$ (boundary with respect to the variables of the elliptic operator) $\partial
\Omega$ is a CR submanifold of $\n \times \R^n$ of same CR dimension.

Acknowledgements
----------------

The author would like to thank Jean-Pierre Rosay for some helpful remarks and fruitful discussions as well as Jean-Michel Bony for the references for the Dirichlet problem for elliptic operators and Joel Merker for the help with the pictures.

Complex Transversal Holomorphic Extension in Manifolds of type $N=(\n,0)$
=========================================================================

We first consider the case where the graph of $\n$ is trivial, i.e. $h
\equiv 0$.

\[Th2\] Let $N=\{(\n,0)\} $ be a smooth ($\CC^{\infty})$ non generic CR submanifold of $\C^{k+m+n}$. Let $\{v\}$ be complex transversal to $N$ at $p_0$. If $f$ is a CR distribution on $N$ that is decomposable at $p_0$, then there exists a wedge $\W$ of direction $v$ whose edge contains $N$ and $F\in \O(\W_{v})$ such that the boundary value of $F$ on $N$ is $f$.

The main tool to prove theorem \[Th2\] is the following proposition:

\[Prop1\] Let $N=\{(\n,0)\} $ be a smooth ($\CC^{\infty})$ non generic CR submanifold of $\C^{k+m+n}$ and let $f$ be a distribution on $N$ that admits a holomorphic extension to a wedge $\W$ in $\C^{k+m}$ with edge $\n$ near $p_0$, then for any $v$ complex transversal to $N$ at $p_0$, there exists a wedge $\W_v$ in $\C^{k+m+n}$ of direction $v$ whose edge contains $\n$ and $F \in \O(\W_v)$ such that the boundary value of $F$ on $N$ is $f$.

[**Proof of Theorem \[Th2\].**]{} We assume proposition \[Prop1\]. Let $v$ be a complex transversal vector and let $u$ be a CR distribution on $\n$. By hypothesis, $u=\sum_{j=1}^{K}
U_j$ where each $U_j$ is a CR distribution that extends holomorphically to a wedge $\W_j\subset
\C^{k+m}$ with edge $\n$. We thus apply proposition \[Prop1\] to each $U_j$ to obtain a holomorphic extension to wedges $\W'_j$ all in the direction $v$. Let $\W=\cap_{j=1}^K \W'_j$ we conclude that the function $\sum_{j=1}^{K}
U_j$ is holomorphic in $\W$ and $\sum_{j=1}^{K}U_j=u$ on $\n$. This concludes the proof of theorem \[Th2\]. $\blacksquare$

The remainder of this section will be devoted to the proof of proposition \[Prop1\].

Local Coordinates
-----------------

We begin with a choice of local coordinates on $\n$. $\n$ is a generic manifold in $\C^{k+m}$. We introduce local coordinates near $p_0$. We may choose a local embedding so that $p_0=0$ and $\n$ is parameterized in $\C^{k+m}=\C^k_z \times \C^m_{w'}$ by

$$\label{e1} 
\n=\{(z,w')\in \C^k \times \C^m:Im(w')=a(z,
Re(w')),~~ a(0)=da(0)=0\}.$$

We will denote by $s=Re(w') \in \R^m$, we thus have

$$\label{e2} 
\n=\{(z,s+ia(z,s)\}\subset \C^k \times \C^m,~~~ T_0\n=\C^k\times \R^m.$$

Define $\C T_p\n=T_p\n \otimes \C$ and $T^{0,1}_p\n=T^{0,1}_p\C^{k+m} \cap \C T_p\n$. We say that $\n$ is a CR manifold if dim$_{\C}T^{0,1}_p\n$ does not depend on $p$. The CR vector fields of $\n$ are vector fields $L$ on $\n$ such that for any $p\in \n$ we have $L_p \in T^{0,1}_p\n$. One can choose a basis $\L$ of $ T^{0,1}\n$ near the origin consisting of vector fields $L_j$ of the form

$$\label{e3} 
L_j={{\partial} \over {\partial \overline z_j}}+\sum_{\ell=1}^nF_{j
\ell} {{\partial} \over {\partial  s_{\ell}}}.$$

The wedge $\W$ in $\C^{k+m}$ with edge $\n$ on which we have holomorphic extension of the function $f$ in a neighborhood of the origin is of the form $$\W=\left (\U+i\Gamma\right ),$$ where $\U$ is a neighborhood of the origin in $\n$ and $\Gamma$ is a conic neighborhood of some vector $\mu$ in $\R^m\setminus
 \{0\}$. We make another linear change of variables, identifying the vector $\mu$ with $(1,1,...,1)$ thus not changing \[e2\]. We therefore have for some $\eta>0$ and $B_{\eta}(0)$, a ball centered at the origin in $\C^{k+m}$ of radius $\eta$

$$\label{e4}
 \begin{array}{l}
\W=\left (\U+i\Gamma\right ){\rm~~  ,where~} \U=B_{\eta}(0)\cap  \n~~
 {\rm and~} \Gamma~{\rm is~ a~conic~
 neighborhood~ of~}\\
 (1,1,....,1) ~{\rm in~} \R^m\setminus \{0\},\\
f~{\rm~extends~holomorphically~to~}\W=
 \left (\U+i\Gamma\right )\cap B_{\eta}(0).
\end{array}$$

Deformations
------------

Denote by $B_{r}$ the ball of radius $r$ centered at the origin in $\R^m$ and $\B_r$ the unit ball centered at the origin in $\C^k$. Let $d \in \R$ be such that for $\eta$ given by \[e4\] we have $$\label{e7}
0<d<{{\eta}\over {\sqrt m}}.$$ Let $\epsilon>0$ be given, let $\{b_j\}_{j=1}^m$ be $\CC^{\infty}$ functions so that

$$\label{e8}
\begin{cases}
b_j|_{\B_{\epsilon} \times B_{\epsilon}}=a_j|_{\B_{\epsilon} \times
B_{\epsilon}},\\
b_j=d~{\rm if}~s \in \R^{m} \setminus
B_{2\epsilon}
\end{cases}$$

Define the generic submanifold of $\C^k \times \C^m$ $\ti \n$ as follows

$$\label{e9}
\ti \n=\{(z,s+ib(z,s)):(z,s) \in \C^k \times \R^m\}.$$

[**Notation.**]{} We shall use the following convention, if $M$ is a manifold defined in a neighborhood of the origin, parametrized by $\C^k \times \R^{m}$, then $M|_{B_r}$ is defined by

$$M|_{B_r} =\{(Z(z,x)):x \in B_r\}.$$ With the above notation we then have $$\ti \n|_{ B_{\epsilon}}= \n|_{ B_{\epsilon}},$$

which we illustrate with the following picture.

p2.pstex\_t

\[p1\] For $\epsilon$ small enough, there exists $b_j$ as in \[e8\] so that $f$ (as in proposition \[Prop1\]) extends to a CR distribution on $\ti \n$ in a $\eta$ neighborhood of the origin.

\[coro1\] There is no loss of generality in assuming that the functions $a$ in \[e1\] verify \[e8\], i.e., one can replace $\n$ by $\ti \n$.

[**Proof of Proposition \[p1\].**]{} The main tool is the following lemma.

\[l1\] There exists $b_j $ so that for $\epsilon$ small enough we have

$$\left (\ti \n \cap V_{\eta}\right ) \subset \overline \W,$$ where $\W$ and $V_{\eta}$ are as in \[e4\].

[**Proof of Lemma \[l1\].**]{}

By \[e2\] we have $$\label{e10}
\|a_j\|_{L^{\infty}(\B_{3\epsilon} \times B_{3\epsilon})}<C\epsilon^2.$$

Choose $\vartheta$ and $\xi$ positive $\CC^{\infty}$ functions so that

$$\label{e11}
\begin{cases}
\vartheta=\vartheta(s),~~~\xi=\xi(s),\\
\vartheta=0~{\rm on}~B_{\epsilon}~{\rm and}~\vartheta=1
~{\rm on}~\R^m \setminus B_{{{4}\over {3}} \epsilon},\\
\xi=1 ~{\rm on}~B_{{{4}\over {3}} \epsilon},
\xi=0 ~{\rm on}~~\R^m \setminus B_{{{5}\over {3}} \epsilon}.
\end{cases}$$

Define the functions $b_j$ as follows, $$b_1=d \vartheta+(1-\vartheta)a_1$$ and $b_j$ for $1<j\leq m$ is given by $$b_j=a_j+\xi\vartheta (d-a_1)+(1-\xi)(d-a_j),$$ where $d$ is given by \[e7\]. We see that $b_j=a_j$ if $s\in B_{\epsilon}$ and $b_j=d$ if $s \in B_{3\epsilon}
\setminus B_{2\epsilon}$.

We need to show now that a point $(z,s+ib(z,s))$ is in the wedge $\W$. Write $ (z,s+ib(z,s))=(z,s+ia)+(0,iv)$ where $v$ is given by $$v=\vartheta(d-a_1) \left(1,...,1\right )+
(1-\xi)\left (0,a_1-a_2,...,a_1-a_j,...,a_1-a_m\right )$$

By \[e10\] $a_j=O(\epsilon^2)$, so if $\epsilon$ is small enough we see that $ (z,s+ib(z,s))\in \W$ To conclude the proof of lemma \[l1\], we need to make sure that $\epsilon$ is also small enough so that $(z,s+id) \in B_{\eta}(0)
$ when $(z,s)\in \B_{3\epsilon}
\times B_{3\epsilon}$, which we can do by \[e7\]. $\blacksquare$

The proof of proposition \[p1\] is now immediate, the CR extension of $f$ to $\ti \n$ is given by the restriction of the holomorphic extension of $f$ to $\ti
\n$, which is possible by the preceding lemma. $\blacksquare$

[**(Abuse of) Notation.**]{} From henceforth we shall be working on $\ti \n$. However, to keep the notation as simple as possible we will forget the tilde and keep the $\n$.

Resolution of a Dirichlet Problem
---------------------------------

Consider the generic manifold $\n \times
\R^n \subset \C^{k+m}\times \C^n$.

\[l2\] A basis $\L$ of the CR vector fields $\n \times
\R^n$ consists of vector field $ L_j$ of the form

$$\label{e17}
  L_j={{\partial} \over {\partial \overline z_j}}+\sum_{l=1}^mf_{j,l}
{{\partial} \over {\partial s_l}},$$

where the $  L_j$ can be chosen to verify the following property.

If $s \in \R^{m} \setminus B_{2\epsilon}$ we have $$L_j={{\partial} \over {\partial \overline z_j}}.$$

The proof of the first part of lemma \[l2\] can be found in any textbook on CR geometry. The second part is just a consequence of the construction of $  \n$.

### Construction of the operator $\Delta_{\n\times\R^n}$

\[l3\] There exists $m$ vector fields $R_j$ of the form $$R_j=\sum_{l=1}^m a_{jl}(z,s){{\partial} \over {\partial  s_l}}$$ where the $a_{jl}$ are smooth functions, such that

\(i) $R_{j}(  w'_l)=\delta_{jl}$, $j,l \in \{1,...,m\}$,

\(ii) $[R_j,R_l]=0$,

\(iii) $[  L_j,R_l]=0$,

\(iv) the set $\{  L_1,...,  L_k, \overline{   L_1},...,\overline
{  L_k},R_1,...,R_{m}\}$ span the complex tangent plane to $  \n $ near the origin.

The proof of the lemma is classic. (ii) (iii) and (iv) are a consequence of (i). We thus determine the $R_j$ by solving for their coefficients in (i) (see for example [@[BER]] Lemma 8.7.13 page 234).

\[r1\] We note that the $R_j$’s verifies the following property,

if $j \leq m$ then $$R_j|_{0}= R_j|_{\C^k\times \left [\R^{m}\setminus
B_{2\epsilon}\right ]}={{\partial}\over {\partial s_j}}.$$

The equivalent of lemma \[l3\] for $\n\times\R^n$, is given by

\[l’3\] The $m+n$ vector fields given by $\{R_1,....,R_m,{{\partial}\over{\partial
t_1}},...,{{\partial}\over{\partial t_n}}\}$ verify the same properties as in lemma \[l3\], where we replace $  \n$ by $\n\times\R^n$.

Define $\Delta_{\n\times\R^n}$ by

$$\label{e1.1}
\Delta_{\n\times\R^n}=\sum_{j=1}^{m}R_j^2 +\sum_{j=1}^{n}
{{\partial^2}\over{{\partial
t_j}}^2} .$$

From \[r1\] and lemma \[l3\] (iii) we deduce immediately the following:

\[p2\] The operator $-\Delta_{\n\times\R^n}$ is strongly elliptic of degree two on $\n\times\R^n$ with smooth ($\CC^{\infty}$) coefficients and no constant terms.

$\Delta_{\n\times\R^n}$ is a differential operator in the variables $s$ and $t$ whose coefficient functions depend smoothly on the $z$ variable. We are going to study a Dirichlet problem on $\n\times\R^n$. To do this, we need $\Omega$ some open set in $\n\times\R^n$ with boundary $\partial \Omega$ parametrized by some closed submanifold of $\R^{n+m}$ of codimension one (recall that $\Delta_{\n\times\R^n}$ is a differential operator in the variables $s$ and $t$).

The reason we went to the trouble of redefining $  \n$ was to be able to define an open set $\Omega$ whose boundary is a CR submanifold of $\n\times\R^n$ of same CR dimension containing near the origin $\n$.

### Construction of the open set $\Omega$

In $\R^{m+n}$ let $\omega$ be an open set contained in $\R^{m+n} \cap \{t_1>0\}$ and whose boundary (see figure) is the union of two submanifolds $\gamma_1$ and $\gamma_2$, verifying the following properties.

[**(1)**]{} $\gamma_1$ is given by $t_1=0$ for $(s,t)\in B_{2\epsilon}$.

[**(2)**]{} $\gamma_2 \subset B_{3\epsilon}\setminus B_{2\epsilon}$ is such that

\(i) $\gamma_1 \cup \gamma_2$ bounds an open set $\omega$,

\(ii) $\gamma_1 \cup \gamma_2$ is smooth.

Define $\Omega$ by $$\Omega=\{(z,  w(z,s,t): z\in \B_{3\epsilon}, (s,t) \in \omega\}.$$ The boundary of $\Omega$ on which we shall impose the Dirichlet data is defined to be $$\partial \Omega=\{(z ,  w(z,s,t):z\in \B_{3\epsilon} ,(s,t) \in  \gamma_1 \cup \gamma_2
\}.$$

\[l4\] $\partial \Omega$ is a smooth CR submanifold of $\n \times \R^n $ of same CR dimension.

[**Proof of Lemma \[l4\].**]{} The lemma is clear for the part of $\gamma_1$ which contains $\n
\times \R^{n-1}$. Out of $\n
\times \R^{n-1}$ by lemma\[l2\] the generators of the CR vector fields $L_j$ are equal to ${{\partial} \over {\partial \overline z_j}}$, hence the lemma. $\blacksquare$

We have the following picture.

p4.pstex\_t

Denote by $(D)$ the Dirichlet problem on $\Omega$, i.e. $${(D)}~~~~~~~~~~~~~~~~~
\begin{cases}
\Delta_{\n\times\R^n} (u)=0~~{\rm in}~\Omega,\\
u=g~~{\rm on}~\partial \Omega.
\end{cases}$$

We now have the following result, which seems well known but is not so easy to find in the literature.

\[proba\] For $g \in \D'(\partial \Omega)$, $(D)$ has a unique solution.

For a reference, we send the reader to [@[C-P]] theorem 5.2 and remark 5.3 page 263-265, as well as [@[Ho]] chapter 10.

Let $S$ be the solution operator for $(D)$, that is for $g \in \D'(\partial \Omega)$, we have

$$\label{ee2}
\begin{cases}
\begin{array}{l}
\Delta_{\n\times\R^n} (S(g))=0~~{\rm in}~~\Omega,\\
S(g)=g ~~{\rm on}~\partial \Omega.
\end{array}
\end{cases}$$

\[p3\] Let $  L_j \in  
\L$ and let $g$ be a CR distribution on $\partial {\Omega}$ then we have $$L_j(S(g))=0.$$

\[LL1\] If $f$ is a CR distribution on $\partial \Omega $ then for any $  L_j \in   \L$ we have $$L_j(S(f))|_{\partial \Omega}=0.$$

[**Proof of Lemma \[LL1\].**]{} The lemma holds since $\partial \Omega$ is a CR submanifold of $\n\times\R^n$ and $f$ is a CR distribution on $\partial \Omega$, so $ L_j(S(f))|_{\partial \Omega}=  L|_{\partial \Omega}(f|_{\partial \Omega})=0$.

[**Proof of Proposition \[p3\].**]{} Consider $  L_j(S(g))$. By lemma \[l3\] (iii) we have $$\Delta_{\n\times\R^n}   L_j(S(g))=  L_j\Delta_{\n\times\R^n}(
S(g)),$$ hence we have $$\label{e1.3}
\Delta_{\n\times\R^n} \left [  L_j(S(g))\right]=0.$$

By lemma \[LL1\] $  L_j(S(g))$ vanishes on $\partial \Omega$, hence by theorem \[proba\] we have $  L_j(S(g))=0$ in $\Omega$. $\blacksquare$

Analytic Vector Theory
----------------------

In this section we present the results developed by Baouendi and Treves [@[Ba-Tr]] and later by Treves in [@[Tre]]. We have included this section for the sake of completeness and claim no originality whatsoever. We will state the main results needed here, sending the reader who wishes to read the details of the proofs to [@[Tre]].

Using the notations $R_{m+j}={{\partial}\over{\partial t_j}}$ and $R^{\alpha}=R_1^{\alpha_1}...R_{m+n}^{\alpha_{m+n}}$ for $\alpha \in
\NN^{m+n}$, we shall say that a continuous function $f$ in $\omega$ is an analytic vector of the system of vector fields $\{R_1,...,R_{n+m}\}$ if $R^{\alpha}f \in \CC^0 $ for any $\alpha \in
\NN^{n+m}$, and if to every compact set $K$ of $\omega$ there is a constant $\rho>0$ such that, in $K$, $$\label{2.2.1}
\sup_{\alpha \in
\NN^{n+m}} \left ( \rho^{|\alpha|} {{ |R^{\alpha}f|} \over {\alpha!}}
\right )<\infty.$$

Let $p
=(z,  w(z,s_0,t_0)) \in \Omega$ so that with our previous notation $(s_0,t_0) \in \omega$. We are using the following convention

$$(z,  w(z,s_0,t_0)) \in \Omega \Leftrightarrow (z,s_0,t_0) \in
\B_{3\epsilon}\times \omega.$$

\[p5\] (Lemma 4.1 in [@[Ba-Tr]]). Let $p
=(z,  w(z,s_0,t_0)) \in \Omega$, let $\U$ be an open neighborhood of $(s_0,t_0)$ in $ \omega$, $K$ a compact subset of $\U$. To every pair of positive constants $C_1, C_2$, there is another pair of positive constants $C_1',C_2'$ with $C_2'$ independent of $C_1$ such that for any $\CC^{\infty}$ function $f$ in an open neighborhood of the closure of $\U$, if we have, for all $k\in \NN$ and all $z\in
\B_{3\epsilon}$, $$\label{e3.1}
\|\Delta_{\n\times\R^n} ^k f\|_{L^{\infty}(\U)} \leq C_1 C_2^k (2k)!,$$ then for every $z \in \B_{3\epsilon}$ and every $\alpha \in \NN^{n+m}$ $$\label{e3.2}
\|R^{\alpha} f\|_{L^{\infty}(K)}\leq C_1'{C_2'}^{|\alpha|}|\alpha|!$$

\[p4\] (Proposition II.4.1 in [@[Tre]]). Let $p \in \Omega$, in order that $f\in \CC^0$ be an analytic vector of the system of vector fields $\{R_1,...,R_{n+m}\}$ it is necessary and sufficient that there exists an open neighborhood $\V$ of $p$ in $\C^k \times \C^{n+m}$ and a continuous function $
F(z,w)$ in $\V$ holomorphic with respect to $w$ and such that $f(z,s,t)=F(z,  w(z,s,t))$.

The main difficulty in the proof of proposition \[p4\] is to show that the function defined by $$F(z,  w,s,t)=\sum_{\alpha \in \NN^{m+n}}{{R^{\alpha}f(z,s,t)} \over
{\alpha!}}
\left (  w-  w(z,s,t) \right )^{\alpha}$$ is equal to $f$ for $  w$ near $  w(z,s,t)$ in $\Omega$ if $f$ is an analytic vector of the vector fields $\{R_1,...,R_{n+m}\}$.

We shall use proposition \[p5\] to construct analytic vectors of the vector fields $\{R_1,...,R_{n+m}\}$ and then apply proposition \[p4\] to these vectors to obtain a holomorphic extension in the variables $  w$.

Proof of Proposition \[Prop1\]
------------------------------

We have $$\label{e'5} 
T_0N=\C^k\times \R^m \times \{0\}\subset \C^k\times \C^m\times\C^n.$$ If $v$ is complex transversal to $N$ at the origin, then $v\in\{0\}
\times \{0\}  \times\C^n$. Therefore after a linear change of variables in $\C^{k+m+n}$ (viewed as a linear change of variables in $\C^n$ being the identity on $\C^{k+m}$ so that \[e4\] is unaffected) we may assume that $v$ has the following form $$\label{cpx} 
v=(0,0,v'),~~v'=(1,0,...,0) \in \R^n \times\{0\}\subset \C^n.$$

We first show that $f$ extends holomorphically to a neighborhood of $\Omega$.

[**Notation**]{} Given $P=(z,  w)$ a point in $\Omega$, $\U$ will denote a neighborhood of $  w$ in $\Omega$ and $U$ will denote a neighborhood of $  w$ in $\C^{m+n}$.

[**(A) Holomorphy in $  w$**]{}

By construction, we have $\Delta_{\n\times\R^n} S(f)=0$ on $\Omega$. Let $Q=(z_0,  w_0) \in \Omega$. By proposition \[p5\], $S(f)$ is an analytic vector for the vector fields $R_j$ on some open neighborhood $\U_1$ of $  w_0$. By proposition \[p4\], $S(f)$ extends holomorphically as a function of $  w$ to a function denoted by $  F_1$ when $(z,  w)$ is in some open neighborhood $\B_{3\epsilon}\times
U_1$ of $(z_0,  w_0)$. Let $  w_1 \in \U_1$, $  w_1 \not =  w_0$. Consider then $P=(z_0,  w_1)$. By the above reasoning, we find the existence of neighborhoods $\U_2$ and $U_2$ of $  w_1 $ and a function $  F_2$ defined on $\B_{3\epsilon} \times
U_2$, holomorphic as a function of $  w$ such that $$S(f)=F_2|_{\B_{3\epsilon} \times
\U_2}.$$

[**Claim:**]{} $ 
F_1=  F_2$ on $\B_{3\epsilon} \times \left [U_1 \cap U_2\right ]$.

[**Proof of the Claim.**]{} For $z$ fixed, $(\U_1 \cap \U_2) $ can be viewed as a totally real submanifold of $\C^{m+n}$. The function $G=
  F_1-  F_2$ extends holomorphically to $(U_1 \cap U_2)$ as a function of $  w$ and is null on $(\U_1 \cap \U_2)$, and is thus null where it is defined.

Using the claim, we conclude that the function $S(f)$ extends holomorphically as a function of $  w$ to a neighborhood $\UU$ of $\Omega$ in $\C^{m+n}$.

[**(B) Holomorphy in $z$**]{}

Let $  L_j \in   \L$. Since $S(f)$ extends holomorphically as a function of $  w$, we have $$\label{e4.2}
  L_j(S(f))=\left [{{\partial} \over {\partial \overline {z_j}}} 
S(f) \right ]_{\Omega}.$$

By proposition \[p4\], $  L_j(S(f))=S(  L_j(f))$ and since $f$ is CR and $S(0)=0$, we have $$\label{e4.3}
\left [{{\partial} \over {\partial \overline {z_j}}}  
S(f) \right ]_{\Omega}=0.$$

Note then that ${{\partial} \over {\partial \overline {z_j}}} 
S(f)$ is a holomorphic function of $  w$, for $z$ fixed. It vanishes on a totally real generic submanifold of $\C^{n+m}$ and thus it is null where it is defined.

We note that $\ S(f)$ is holomorphic on a set of the form $\B_{3\epsilon} \times \left [\R_+^{n+m} \times i
\Gamma \right ]$ where $R_+^{n+m}=\R^{n+m} \cap \{t_1>0\}$ and $\Gamma \subset
\R^{n+m}$, thus by theorem 2.5.10 in [@[Ho]], we conclude that if $ch\Gamma$ stands for the convex hull of $\Gamma$ then $S(f)$ is holomorphic on $\B_{3\epsilon} \times \left [\R_+^{n+m} \times i ch\Gamma \right
] $.

To conclude the proof of proposition \[Prop1\] we need to show that the holomorphic extension obtained has slow growth and thus its boundary value agrees with our original CR distribution $f$. $S(f)$ is holomorphic in some wedge $\W$ and admits a boundary value when approaching $\n \times \{0\}$ along $\n \times \R^{n} \cap \{t_1>0\}$, hence $S(f)$ has at most slow growth when approaching $\n \times \{0\}$ along $\n \times \R^{n} \cap \{t_1>0\}$. By continuity, it has at most slow growth on any wedge $\W^0 \subset \W$, hence $S(f)$ admits a boundary value on $\n \times \{0\}$, which by uniqueness of boundary value is $f$. This concludes the proof of proposition \[Prop1\]. $\blacksquare$

Proof of the Main Results and Remarks
=====================================

Proof of the Results on Decomposable CR Distributions
-----------------------------------------------------

[**Proof of Theorem \[t1\].**]{} Recall that $  N=\{(  \n,h(  \n))\}$. Consider the CR map $h: 
\n \to \C^n$. By theorem \[Th2\], each $h_j$ extends holomorphically to some wedge $\W_j$. Set $\W =\cap_{j=1}^n \W_j$. Define $F:(  \n,0) \to (\n,\kappa h(  \n))$ where $\kappa \in \R^*$ by $$F(z,w',w'')=(z,w',w''+\kappa h(z,w')).$$ Clearly, there exists $\kappa\not = 0$ so that on $\overline \W$, the Jacobian of $F$ is non zero, without loss of generality, we can assume that $\kappa=1$. Hence $F$ is a biholomorphism from $\W$ to $F(\W)$ extending smoothly to a diffeomorphism from $\overline \W$ to $F(\overline \W)$. Since $dh(0)=0$, $F$ is tangent to the identity at the origin, hence there exists $\W'$ a wedge of direction $v$ such that $\W'\subset F(\W)$. We then conclude that any decomposable CR distribution on $N$ extends holomorphically to a complex transversal wedge of direction $v$.

To prove (B), we note that $f_j=w''_j-h_j$ are holomorphic in $\W$ and, since $dh_j(0)=0$, we conclude that $$d(w''_1-h_1)\wedge...\wedge d(w''_n-h_n)\not =0~~{\rm on} \W.$$ Each $f_j$ vanishes to order one on $N$ since $F^{-1}(z,w',w'')=(z,w',w''-h)=(z,w',t)$. $\blacksquare$

[**Proof of Corollary \[t3\].**]{} Let $M$ and $N$ be as in the hypothesis of theorem \[t3\]. After a linear of variables, we may assume that $p_0=0$ and that near the origin, $M$ is parametrized by $$M=\{z,u+iv(z,u):(z,u) \in \C^k \times \R^{p-k}\}.$$ By the implicit function theorem, we may assume that $N$ is given as a subset of $M$ by

$$\begin{cases}
u_{p-k-n}=\mu_1(z,u_1,...,u_{p-k-n-1}),...,
u_{p-k}=\mu_n(z,u_1,...,u_{p-k-n-1}),\\
\mu(0)=d\mu(0)=0.
\end{cases}$$ Denote by $s=(u_1,...,u_{p-k-n-1}) \in \R^m$ and $t=(u_{p-k-n},...,u_{p-k}) \in \R^n$. Setting $t'=t-\mu$, in the $(z,s,t')$ coordinates, we have $N$ given as a subset of $M$ by $t'=0$ and $$N=\{(z,w'(z,s),h(z,w'):(z,s)\in \C^k  \times \R^m\},$$ where $h$ is a CR map from $\n:=\{z,w'(z,s)\}$. We can now apply theorem \[t1\] to obtain the CR extension as the restriction of the holomorphic extension of $f$ to $\W \cap M$. The second part follows in the same manner. $\blacksquare$

The non Decomposable Case
-------------------------

[**Trépreau’s Example**]{}

We wish to note that on the particular class of manifolds of CR dimension $k$ containing through the origin $\C^k$, we do get a complex transversal extension result for any CR functions. Trépreau’s original example of a CR submanifold of $\C^3$ which admits non decomposable CR functions, is the following $$\label{e111}
\m=\{(z,s_1+is_2|z|^2,s_2-is_1|z|^2):(z,s_1,s_2)\in \C \times \R
\times \R\}$$ On this particular example, we do get holomorphic extension to [**complex transversal**]{} wedges of [**any**]{} CR distribution.

\[lader\] Let $M=\{(\m,h(\m)\}$ be a non generic CR submanifold of $\C^L$ where $\m$ is given by \[e111\], then the conclusions of theorem \[t1\] hold for CR distributions.

[**Proof of Proposition \[lader\].**]{} As seen previously, there is no loss of generality in assuming that $M=\{(\m,0)\} \subset \C^{k+m}\times \C^n$ and that if $v$ is a complex transversal vector to $M$ at the origin, then $v=(1,0,....,0)
\subset \R^n \times \{0\} \subset \C^n$. Let $f$ be a $\CC^0$ CR function on $M$, consider $\m\times
\R^n$ and the associated elliptic operator $\Delta_{\m\times
\R^n}$. As previously, we solve a Dirichlet problem with boundary data $f$. Denote the solution $S(f)$, by our previous arguments $S(f)$ is holomorphic in the variables $w$ on a neighborhood of $\Omega$ denoted $\V(\Omega)$. Note that since $\m$ contains through the origin $\C\times \{0\}$ then $\V(\Omega)$ contains $\left (\C\times \{0\}\times \R^n\right )\cap \{t_1>0\}$. Let $\U$ be a neighborhood of the origin in $\C_z$ and let $\xi$ be a smooth cut off function which is one on $\U$ and zero on a neighborhood of $\overline \U$. Set $F=\xi \overline \partial S(f)$ a $(0,1)$ form on a $\V(\Omega)$ with continuous coefficient. Since $S(f)$ is holomorphic in $w$, we have $\overline \partial S(f)={{\partial}\over
{\partial {\overline z}}}S(f)$. Since $f$ is CR, we get $F|_{(\m \cap \U) \times\R^n}=0$. Let $u={{\partial}\over
{\partial {\overline z}}}F\!*\!{{1}\over{\pi z}}
$ then $u$ is holomorphic in $w$ and $u|_{\m\times
\R^n}=0$. The desired extension for $f$ is thus obtained by considering $S(f)-u$. $\blacksquare$

We can generalize this result to any CR submanifold of CR dimension $k$ which contains through a point $p_0$ $\C^k\times \{0\}$. We state without proving the next result, since the proof of this result is identical to the proof of proposition \[lader\] one has to replace ${{1}\over{\pi z}}$ by an appropriate integral kernel solving $\overline \partial_z$ on a ball in $\C^k$.

\[der\] Let $M$ be a non generic smooth ($\CC^{\infty}$) CR submanifold of $\C^L$ of CR dimension $k$. If the reunion of the CR orbits through $ p_0\in M$ is a complex manifold then the conclusions of theorem \[t1\] hold for CR distributions.

We thus get as a corollary

(Theorem \[th\]) Let $N=\{(\n,h(\n))\} $ be a smooth ($\CC^{\infty})$ non generic CR submanifold of $\C^{k+m+n}$ such that $\n \subset \C^{k+m}$ is a hypersurface. If $f$ is a CR distribution on $N$ then for any $p \in N$ and any $v$ complex transversal to $N$ at $p$, there exists a wedge $\W$ of direction $v$ whose edge contains a neighborhood of $p$ in $N$ and $F\in \O(\W)$ such that $F|_N=f$.

[**Proof of the corollary**]{} To prove the corollary, we need the following result.

\[th1\] Let $\n$ be a generic submanifold of $\C^n$ minimal as some $p\in \n$. Then any CR distribution near $p$ admits a unique holomorphic extension to some wedge $\W$ with edge $\n$.

[**Proof of theorem \[th1\].**]{} This result is well known so we will only sketch the proof. Set $k=CRdim (\n)$, choose coordinates in $\C^n$ so that $\C^n=\C^k_z \times \C^{n-k}_w$. Construct the vector fields $R_j$ as in lemma \[l3\]. Denote by $\{L_j\}_{j=1}^{k}$ a basis for the CR vector fields. By proposition \[p4\] (using the terminology of Baouendi and Treves) the system of vector fields $R_j$ is normal at $p$ (definition 4.1 in [@[Ba-Tr]]). Let $f$ be a CR distribution on $\n$, by theorem 4.1 in [@[Ba-Tr]] there exists $u\in \CC^0$ and $\ell \in \NN$ such that $$\left [\Delta_{\n \times \R^n}\right ]^{\ell} u=f,$$ $$L_j(u)=0.$$ By Tumanov’s theorem $u$ extends holomorphically to some wedge $\W$, denote by $U$ the holomorphic extension of $u$ to $\W$. The holomorphic extension of $f$ is then given by $(\sum_{j=1}^{n-k}{{\partial^2} \over {\partial w_j^2}})^{\ell}U$. $\blacksquare$

We now proceed with the proof of the corollary. Let $p\in N$, $p=(p',h(p'))$, if $\n$ is minimal at $p'$, then any CR distribution is decomposable (it extends holomorphically to a wedge by the above theorem), hence by theorem \[t1\] we are done. If $\n$ is not minimal at $p'$, then it contains a proper submanifold $\n'$ of same CR dimension, since $\n$ is a hypersurface, $\n'$ is a complex manifold, hence by theorem \[der\] we are done. $\blacksquare$

We can now construct functions holomorphic in a complex transversal wedge vanishing to infinite order on a non generic CR submanifold of $\C^L$.

\[coro\] Let $N=\{(\n,h(\n))\}$ be a non generic smooth CR submanifold of $\C^L$ such that either $\n$ is a hypersurface or $h$ is decomposable at $p'\in \n$. Then for any $v$ complex transversal to $N$ at $p=(p',h(p'))$ there exists a wedge $\W$ of direction $v$ and $g \in \O(\W)$ such that $g \not \equiv 0$ and $g$ vanishes to infinite order on $N$.

[**Proof of Corollary \[coro\].**]{} Choose coordinates $(z,w',w'') \in \C_z^k\times \C_{w'}^m\times \C_{w''}^n$ such that $\n \subset  \C_z^k\times \C_{w'}^m$. After a linear change of variables, we can assume that $v$, the complex transversal vector is given by $v=(0,0,v'') \in \C^{k}\times \C^m \times \C^n$ with $v''=(1,0,...,0)$. Consider the function $\mu$ defined by $$\mu=
\begin{cases}
e^{-{{1}\over{w''_1}}}~~~~w''_1\not=0,\\
0~~~~w''_1\not=0.
\end{cases}$$ Then $\mu$ is holomorphic on any wedge with direction $v$ off edge $\n\times\{0\}$ and vanishes to infinite order on $\n\times\{0\}$. Consider $F$ the biholomorphism constructed in the proof of theorem \[t1\], $F(z,w',w'')=(z,w',w''-h(z,w'))$. The desired function $g$ is then given by $g=\mu(F^{-1}(z,w',w''))$. $\blacksquare$

[**Note.**]{} The function $g$ does not extend holomorphically to any neighborhood of $N$. This situation differs greatly from the holomorphic extension situation in the generic case. Indeed, let $M$ be a generic manifold where any CR function in a neighborhood of some point $p\in
M$ extends holomorphically to a neighborhood of $p$. Let $\W$ be a wedge attached to $M$, then any function which is holomorphic in $\W$ and at least $\CC^1$ on $M$ extends holomorphically to a full neighborhood of $p$. However, if we now consider $\M=\{(M,h(M))\}$ then the function $g$ in corollary \[coro\] does not extend holomorphically to a full neighborhood of $\M$.

In [@[Ei]] we constructed an example of an abstract CR structure where there is no CR extension. It is even easier to construct an example where there is no non trivial CR function vanishing on $N$.

[**Example.**]{} Let $L$ be a real analytic vector field (the Lewy operator for example) of the form $$L={{\partial} \over {\partial \bar z}}+f(z,s){{\partial} \over
{\partial s}},$$ that is not solvable (Hörmander’s theorem [@[Ho]]). Let $g$ be a smooth function not in the image of $L$, then define the abstract CR structure $(M,\L)$ where $\L$ is given by $$\L=L+tg{{\partial} \over
{\partial t}},$$ then the equation $\L(tu)=0$ is not solvable. Indeed, suppose it was, then we would have $$t\L(u)+u\L(t)=t(\L(u)+ug)=0,$$ decomposing $u=u_0+tu_1$ where $u_0=u_0(z,s)$ we obtain $$L(Log(u_0))+g=0.$$

By real analycity of $L$, we see that there are plenty of non trivial functions in the kernel of $\L$.

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Nicolas Eisen\
Département de Mathématiques, UMR 6086 CNRS\
Université de Poitiers\
eisen@math.univ-poitiers.fr
