Baouendi-Treves approximation theorem

Baouendi-Treves approximation theorem

Suppose

M

is a real smooth manifold. Let

𝒱

be a subbundle of the complexified

tangent space

ℂ

⁢

T

⁢

M

(that is

ℂ

⊗

T

⁢

M

). 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 any
point

p

∈

M

,

there exist

m

=

d

-

n

smooth

complex

valued

functions

z

1

,

…

,

z

m

defined in a neighbourhood of

p

, such that the

differentials

d

⁢

z

1

,

…

,

d

⁢

z

m

are

ℂ

-

linearly independent

and for all sections

L

∈

Γ

⁢

(

M

,

𝒱

)

we have

L

⁢

z

k

=

0

for

k

=

1

,

…

,

m

.

We say

z

=

(

z

1

,

…

,

z

m

)

are

near

p

.

We say

f

is a

if

L

⁢

f

=

0

for every

L

∈

Γ

⁢

(

M

,

𝒱

)

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

V

an integrable subbundle as above.
Let

p

∈

M

be fixed and let

z

=

(

z

1

,

…

,

z

m

)

be basic solutions near

p

. Then there exists a

compact

neighbourhood

K

of

p

, such that for any

continuous

solution

f

:

M

→

C

,

there exists a sequence

p

j

of

polynomials

in

m

variables

with complex coefficients such that

p

j

⁢

(

z

1

,

…

,

z

m

)

→

f

⁢

uniformly in

K

.

In particular we have the following

corollary

for

CR submanifolds

. A real smooth CR submanifold
that is embedded in

ℂ

N

has the CR vector fields as the integrable subbundle

𝒱

.
Also the

coordinate

functions

z

1

,

…

,

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

ℂ

N

is of the minimal

dimension

.

Corollary

.

Let

M

⊂

C

N

be an embedded real smooth generic submanifold and

p

∈

M

. Then there exists a
compact neighbourhood

K

⊂

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

⊂

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.
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

⊂

C

2

is given in coordinates

(

z

,

w

)

by

Im

⁡

w

=

|

z

|

2

.

Note that for
some

t

>

0

,

the map

ξ

↦

(

t

⁢

ξ

,

t

)

is an attached analytic disc. By taking different

t

>

0

,

we can fill the set

{

(

z

,

w

)

∣

Im

⁡

w

≥

|

z

|

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

<

ϵ

for some

ϵ

>

0

(such that the boundary of the disc lies in

K

).
Hence

f

extends to a holomorphic function on

ϵ

>

Im

⁡

w

>

|

z

|

2

, and which is
continuous on

ϵ

>

Im

⁡

w

≥

|

z

|

2

.

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

Corollary

.

Suppose

M

⊂

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.

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

⊂

C

N

be a domain with smooth strongly pseudoconvex boundary.
Suppose

f

is a continuous CR function on

∂

⁡

U

. Then there exists a function

f

holomorphic in

U

and continuous on

U

¯

,

such that

F

|

∂

⁡

U

=

f

.

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

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