Local Peak Sets in Weakly Pseudoconvex Boundaries in <span class="mathjax-formula">$\mathbb{C}^n$</span>

ANNALES

DE LA FACULTÉ DES SCIENCES

Mathématiques

BORHENHALOUANI

Local Peak Sets in Weakly Pseudoconvex Boundaries inCⁿ

Tome XVIII, n^o3 (2009), p. 577-598.

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pp. 577–598

Local Peak Sets in Weakly Pseudoconvex Boundaries in

Borhen Halouani⁽¹⁾

ABSTRACT.—We give a sufficient condition for aC^ω(resp.C^∞)-totally real, complex-tangential, (n−1)-dimensional submanifold in a weakly pseudoconvex boundary of classC^ω(resp.C^∞) to be a local peak set for the classO(resp.A^∞). Moreover, we give a consequence of it for Catlin’s multitype.

R^´ESUM´E.—On donne une condition suffisante pour qu’une sous vari´et´e C^ω (resp. C^∞), totalement r´eelle, complexe-tangentielle, de dimension (n−1) dans le bord d’un domaine faiblement pseudoconvexe deC^{n, soit}

un ensemble localement pic pour la classe O (resp. A^∞). De plus, on donne une cons´equence de cette condition en terme de multitype de D.

Catlin.

1. Introduction and basic definitions

This article is a part of the Ph.D thesis of the author. TheO part was motivated by the paper of Boutet de Monvel and Iordan [B-I] andA^∞ part by the methods of Hakim and Sibony [H-S]. LetDbe a domain inCⁿ with C^ω(resp.C^∞)-boundary. We denote for an open setU byO(resp.A^∞) the class of holomorphic functions onU(resp. the class of holomorphic functions in U which have aC^∞-extension toU ).

We say thatM⊂bD is a local peak set at a pointp∈Mfor the class O (resp.A^∞), if there exist a neighborhood U of p in Cⁿ and a function

(∗) Re¸cu le 1 novembre 2007, accept´e le 10 janvier 2008

(2) LMPA, Centre Universitaire de la Mi-Voix. Bˆat H. Poincar´e, 50 rue F. Buisson, B.P. 699, F-62228 Calais C´edex, France.

halouani@lmpa.univ-littoral.fr

f ∈ O(U) (resp.A^∞(D ∩ U)) such that|f|<1 on (D ∩ U)\Mand f = 1 on M ∩ U. Or equivalently, if there exists a function g ∈ O(U) (resp.

A^∞(D ∩ U)) such thatg= 0 onM ∩ U andg <0 on (D ∩ U)\M.

We say thatM ⊂bD is a local interpolation set at a pointp∈M for the classA^∞, if there exists a neighborhoodU ofpsuch that each function f ∈C^∞(M∩ U) is the restriction toM∩ Uof a functionF ∈A^∞(D∩ U).

A submanifold Mof bD is complex-tangential if for everyp∈M we have Tp(M)⊆T_p^C(bD), whereT_p^C(bD) is the complex tangent space ofTp(bD).

If for every p ∈ M, Tp(M) ∩ iTp(M) = {0} , we say that M is totally real. Letρ:U −→Rbe a local C^∞ defining function of D, D ∩ U ={z∈ U/ρ(z)<0},dρ(p)= 0, whereU is a neighborhood ofp∈bD. We sayD is (Levi) pseudoconvex atpif

Levρ(p)[t] =

1i,jn

∂²ρ

∂zi∂zj

(p)titj 0,

for every t ∈ T_p^C(bD).Levρ(p)[t] is called the Levi form or the complex hessian ofρ.

Let D be Levi pseudoconvex at p. The point p is said to be strongly pseudoconvex if the Levi form is positive definite whenever t = 0, t ∈ T_p^C(bD). Otherwise it is said to be weakly pseudoconvex. A domain is called pseudoconvex if its boundary points are pseudoconvex.

We need the following terminology due to L. H¨ormander. A function φ∈C^∞(U) is almost-holomorphic with respect to a setE⊂ Uif∂φvanishes to infinite order at points ofE.

The paper is organized as follows: In §2, we introduce the hypotheses (H1) and (H2). In§3 and§4, we give the equivalent more handy sufficient condition (H) for the existence of local peak set for the classOand for the classA^∞. In the final section, we give some consequences for the multitype onMof the sufficient hypotheses.

2. Preliminaries

Let D be a pseudoconvex domain with C^ω (resp. C^∞)-boundary. Let M be an (n−1) dimensional submanifold of bD which is totally real and complex- tangential in a neighborhood of a pointp∈M. Let (V, γ) be aC^ω (resp. C^∞)-parametrization ofM at p, where V is a neighborhood of the origin in Rⁿ⁻¹ such thatγ(0) =p. Let Xbe a C^ω (resp.C^∞)-vector field onMsuch that X(p) = 0. Denote byζ= (ζ1, . . . , ζn−1) the coordinates of

a point inV. ThenXcan be written asX=

i

di(ζ) ∂

∂ζ_i wheredi areC^ω (resp. C^∞)-functions on V. We set D0 the Jacobian matrix at the origin:

∂di

∂ζi

(0)

ii,jn−1

. Now, we introduce our first hypothesis:

(H1) The matrixD0 is diagonalizable and hasm1. . .m_n−1 eigenval- ues withmi∈N^∗ for all i.

We say that M admits a peak-admissible C^ω (resp. C^∞)-vector field X of weights (m1, . . . ,mn−1) at p ∈ M for the class O (resp.A^∞). (H1) is independent of the choice of the parametrization and the mi and their multiplicities are uniquely determined. Using hypothesis (H1), one can easily prove that there exists a C^ω (resp. C^∞)-change of coordinates on V such that X=

i

m_iζ_i ∂

∂ζi

. This representation of Xis invariant if we apply a

“weight-homogeneous” polynomial transformation of coordinates as below:

Lemma 2.1. — Let Λ = (Λ1, . . . ,Λ_n−1) be a C^ω (resp. C^∞)-change of coordinates on V such that Λ(0) = 0 and dΛ(X) = X. Then Λ is a poly- nomial map. More precisely, if ζ= (ζ₁, . . . , ζ_n−1)∈V,I= (i₁, . . . , i_n−1)∈ Nⁿ⁻¹ and we set |I|∗ =

ν

i_νm_ν then for every 1 j n−1, Λ_j(ζ) =

|I|∗=^mj

a^j_Iζ₁ⁱ¹. . . ζ_nⁱⁿ⁻¹₋₁ with a^j_I ∈ R. Conversely, any Λ of this form pre- serves X.

Proof. — The integral curves ofXareκζ(λ) = (λ^m1ζ1, . . . , λ^mn−1ζn−1), λ∈R. SincedΛ(X) =X, Λ transforms an integral curve passing throughζ to an integral curve passing throughη = Λ(ζ). So we obtain

(λ^m1Λ1(ζ), . . . , λ^mn−1Λ_n−1(ζ)) = (Λ1(κζ(Λ)), . . . ,Λ_n−1(κζ(λ))). (2.1) Let 1j n−1 be fixed. We write Λ_j as: Λ_j(ζ) = Λ^∗(ζ) +R(ζ) where Λ^∗(ζ) :=

|I|∗=^m

a^∗_{i1,...,in−1}ζ₁ⁱ¹. . . ζ_n−1ⁱⁿ⁻¹ is non identically zero for a smallest integer m that satisfies this condition: there exists a constant C >0 such that |R(κζ(λ))|C|λ|^m+1 . From (2.1), we have

λ^mjΛ_j(ζ) = Λ_j(κ_ζ(λ)) =λ^mΛ^∗(ζ) +R(κ_ζ(λ)). (2.2) Now we divide (2.2) by λ^m. When λ tends to 0 we obtain m = mj and Λj(ζ) = Λ^∗(ζ) for allζ∈Rⁿ⁻¹.

So let the coordinates be chosen such that X =

im_iζ_∂ζ^∂

i. For ζ = (ζ₁, . . . , ζ_n−1),η= (η₁, . . . , η_n−1)∈Rⁿ⁻¹andλ,µ∈R, we setσ:=ζ+i.η∈ Cⁿ⁻¹,κζ(λ) := (λ^m1ζ1, . . . , λ^mn−1ζn−1) andκσ(µ, λ) :=κζ(µ)+i.κη(λ). Let ρbe a local defining function ofD atp∈bDandγ:V −→θ(V) :=M be a holomorphic-extension (resp. almost-holomorphic extension) of the para- metrization γ of M. In the C^ω-case M is a complexification of M and V is an open neighborhood of the origin in Cⁿ⁻¹. Let M, K ∈ N^∗ be such that M K and mj := M/mj ∈ N^∗, kj := K/mj ∈ N^∗. We set E={ζ∈Rⁿ⁻¹/

j

ζ_j^2mj = 1}. Now, we introduce our second hypothesis:

(H2) There exist constants ε > 0, 0 < c C such that for every σ = ζ+i.η∈E+i.E,|λ|< ε,|µ|< ε, we have:c|λ|^2M(|µ|+|λ|)^2(K−M) ρ(γ(κσ(µ, λ)))C|λ|^2M(|µ|+|λ|)^2(K−M).

Definition 2.2. — If a C^∞ (resp. C^∞)-vector field X on M verifies (H1)and(H2)we say thatXis peak-admissible of peak-type(K, M;m1, . . . ,

mn−1)atp∈Mfor the class O(resp. A^∞).

Remark 2.3. —

1) The hypothesis (H2) does not depend neither on the choice of the defining function of the boundarybD nor the choice of the almost- holomorphic extension (see Lemma 4.3 in section 4).

2) The geometric meaning of (H2) will become clear in inequality (H).

3. A sufficient condition for the existence of local peak set for the class O

Theorem 3.1. — Let D be a pseudoconvex domain in Cⁿ with C^ω- boundary. Let M be an (n−1)-dimensional C^ω-submanifold in bD that is totally real and complex-tangential atp∈M. We suppose thatMadmits a peak-admissibleC^ω-vector fieldXof peak-type(K, M;m1, . . . ,mn−1)atp forO. ThenM is a local peak set at pfor the class O.

Proof. — The proof is based on Propositions 3.2 and 3.4 below after several holomorphic coordinates changes. Also we allow shrinkings of M.

Proposition 3.2. — Let D be a domain in Cⁿ with C^ω (resp. C^∞)- boundarybD. LetMbe an(n−1)-dimensionalC^ω-submanifold inbDwhich

is totally real and complex-tangential near p. Then there exists a holomor- phic change (resp. an almost-holomorphic change) of coordinates(Z, w)with Z =X+i.Y ∈Cⁿ⁻¹ and w=u+iv ∈C, such that pcorresponds to the origin and in an open neighborhood U of the origin, we have:

i) M = {(Z, w) ∈ U/Y =w = 0} . Moreover, M is contained in an n-dimensional totally real submanifoldN={(Z, w)∈ U/Y =u= 0}

ofbD.

ii) For everyc∈R,Mc ={(Z, w)∈N/v=c} is complex-tangential or empty.

iii) D∩ U={(Z, w)∈ U/ρ(Z, w)<0}with

ρ(Z, w) =u+A(Z) +vB(Z) +v²R(Z, v).

iv) AandB vanish of order2 whenY = 0.

Proof. — We give the proof in theC^ω-case. Letγbe aC^ω-parametrization of Mdefined on a neighborhood of the origin in Rⁿ⁻¹. After a translation and a rotation of the coordinates inCⁿ we may assume that pis the ori- gin and the real tangent space at 0 to bD is T0(bD) = Cⁿ⁻¹×iR. We set L(Z, w) = in(Z, w) where n is the vector field of the outer exterior normal to bD. Then, for every (Z, w) ∈ bD, there exists a C^ω-integral curve l_(Z,w)(λ) ∈ bD of L satisfying l_(Z,w)(0) = (Z, w) and dl_(Z,w)

dλ (λ) = L(l_(Z,w)(λ)). Now, we consider the mapθ: (t, λ)−→l_γ(t)(λ). It is clear that θis aC^ω-diffeomorphism from a neighborhoodU of the origin inRⁿinto an n-dimensional submanifoldN:=θ(U) ofbDwhich is totally real. By com- plexification ofθin a neighborhoodWof the origin inCⁿ, we obtain in the new holomorphic coordinates (Z, w),M ={(Z, w)∈ W/Y =w = 0}

and N = {(Z, w) ∈ W/Y = v = 0}. We remark that the system {Σq =Tq(N)∩T_q^C(bD), q∈ W}isC^ωand involutive. By Frobenius theorem [Bo] the leavesM_c ={(Z, w)∈ W ∩N/v=c}c∈Rare complex-tangential tobD. Now, we change coordinates again by defining:Z=Z andw=iw. We obtain in a neighborhoodU of the origin i) and ii). RepresentingbDas a graph overCⁿ⁻¹×iR, we obtain iii). SinceM⊂bDis complex-tangential A vanishes of order2 ifY = 0. As ∂

∂v is tangent to Nand the complex gradient∇ρ= (0_Cn−1,−1) is constant alongN, we obtain thatB vanishes of order2 ifY = 0. This achieves iv) and the proposition.

Let the change of coordinates of Proposition 3.2 for the vector field X which verifies hypothesis (H2) be achieved. Now we show the impact of (H2). We set κ := K/M = kj/mj 1. Since κ is independent of j,

we define in a sufficiently small neighborhoodV of the origin inCⁿ⁻¹ the following pseudo-norms of theZ= (z₁, . . . , z_n−1) coordinates of Proposition 3.2: ||Y||=





j

y^2m_j ^j





1/2M

and||Z||∗=

j|z_j|^2kj1/2K

. We note that A(Z) =ρ(γ(κ_σ(µ, λ))) whereZ =X+i.Y =κ_σ(µ, λ). Therefore, from now on we may assume thatAverifies:

(H) There exist two constants 0< cC such that, for every Z =X+ iY ∈Cⁿ⁻¹near the origin, we have:

c||Y||^2M_∗ .||Z||^2K_∗ ^−2M A(Z)C||Y||^2M_∗ .||Z||^2K_∗ ^−2M

Remark 3.3. —

1) The proof of Proposition 3.2 remains true in theC^∞-case. We indicate the modification in Lemma 4.2 (section 4).

2) If Z = (z1, . . . , zn−1) ∈ V where V is a small open neighborhood of the origin in Cⁿ⁻¹ , then

j

|zj|^2(kj−mj) ≈





j

|zj|^2mj





κ−1

. Moreover, we may replacekj bymj andKbyM in the definition of the pseudo-norm||Z||_∗ .

3) IfK=M =m1=. . .=m_n−1 = 1, we find the property onA for a strongly pseudoconvex boundary.

Proposition 3.4. — 1) If the real hyperplaneH =Cⁿ⁻¹×R={(Z, iv)/

Z ∈Cⁿ⁻¹, v∈R} lies outside of D in a neighborhoodU of the origin, then there exists a constant T >0such that B²T Anear the origin.

2) If there exists a constant T > 0 such that B² T A near the ori- gin, then there exist a sufficiently small neighborhood U of the origin and a holomorphic functionψonU (resp. an almost-holomorphic function with respect toN∩ U) which satisfies: ψ <0 onD∩ U if w= 0and ψ= 0if w= 0. Hereψ= w

1−2K1w with a suitable constantK₁>0.

Proof. — The proof is elementary. See also [B-I].

In order to apply Proposition 3.4 2), we should determine the order of vanishing for certain functions on Mat p= 0∈M. We begin by defining theZ-weights and theY-weights for polynomial functions.

Definition 3.5. — Letχ=a_I,Jz₁ⁱ¹z^j₁¹. . . z_n−1ⁱⁿ⁻¹z^j_n−1ⁿ⁻¹, witha_I,J = 0, be a monomial. We define theZ-weightPZ(χ)ofχas :PZ(χ) =

ν

mν(iν+jν).

If g ≡ 0 is a polynomial function in Z and Z we define the Z-weight of g as the smallest Z-weight in the decomposition of g by monomials. If g is a sum of monomials which have the same Z-weight L, we say that g is homogeneous with respect to the Z-weight. Let X ∈ Rⁿ⁻¹ be fixed and Ξ =αI,J(X)y₁ⁱ¹. . . y_nⁱⁿ⁻¹₋₁, withαI,J(X)≡0, be a monomial atY. We define theY-weightP_Y(Ξ)ofχas

ν

m_νi_ν. Ifh≡0is a polynomial function inY we define theY-weight ofhto be the smallestY-weight in the decomposition of h. If his a sum of monomials which have the sameY-weight L, we say that h is homogeneous with respect to theY-weight of order L.

Lemma 3.6. — Let R,S ∈N,RS andF(X, Y) =

I,J

F_I,JY^IX^J be aC^ω-function on an open neighborhood of the origin ofCⁿ⁻¹such that, for all multi-indices I= (i₁, . . . , i_n−1),J = (j₁, . . . , j_n1) in Nⁿ⁻¹,F_I,J = 0or PY(F_I,JY^IX^J) S and PZ(F_I,JY^IX^J) R S. Then, there exists a constant C >0such that, |F(Z)|C||Y||^S_∗.||Z||^R−S_∗ ,∀Z=X+i.Y near the origin.

Proof. — This can be seen by Taylor expansion and standard arguments.

Lemma 3.7. — With the notations of Lemma 3.6, if S M andR K=κM, then |F|²

A is uniformly bounded on a sufficiently small neighbor- hood of the origin.

Proof. — This follows immediately from Lemma 3.6 and inequality (H).

In order to know the weights of A and B we analyze the restrictions which are imposed on the functionsAandBby the pseudoconvexity ofbD.

We assume that B ≡ 0 and we set (PY(B),PZ(B)) = (S, R). From (H) we have (PY(A),PZ(A)) = (2M,2K). Next, a simple computation of the Levi form at a point near the origin to bDfor t=

νmνyνχν ∈T^C(bD), with χν = i

∂

∂zν − i η

∂ρ

∂zν

∂

∂w

and η = 1 2

i+B+ 2vR+v²∂R

∂v

, gives Levρ[t] =A(Z) +vB(Z) +v²R(v, Z),Z varying onM, the complexifica- tion ofM. By pseudoconvexity ofbDand Proposition 3.4 1) there exists a positive constantT^∗>0 such that

BT^∗A. (3.1)

It remains to study the Z-weight and Y-weight of A and B and their re- lationship with the weights of A and B and finally to show S M and RK. Some necessary auxiliaries results are given in Lemmas 3.8 and 3.9 below. We denote by ∂_νµ² the partial derivative _∂z^∂2

ν∂zµ and OY(L) (resp.

OZ(L)) is the set of functions that admit an Y-weight (resp. a Z-weight)

L(L∈N).

• Suppose thatS < M.

The expressions ofAandBare:

A =

ν,µ

∂_νµ² Amνmµyνyµ+OY(2M+ 1)

B =

ν,µ

∂_νµ² Am_νm_µy_νy_µ+O_Y(2S).

By Lemma 3.8 A = A_2M +A with PY(A_2M) = 2M and every term of A has anY-weight > 2M. We put A2M :=

ν,µ∂_νµ² A2Mmνmµyνyµ. By Lemma 3.9 we obtainA2M ≡0 andPY(A2M) = 2M. Similary, we have B

= BS+BS where every term ofBS has anY-weight>2M. We putBS :=

ν,µ∂_νµ² BSmνmµyνyµ. We obtain BS ≡ 0 and PY(BS) = S. Inequality (3.1) becomes:

(BS+O_Y(S+ 1))²T^∗(A2M +O_Y(2M+ 1)). (3.2) Since BS ≡0 there existsZ₀ =X₀+i.Y₀ withY₀ = (y_0,1, . . . , y_0,n−1)≡0 such thatBS(Z₀)= 0. Since every term in the decomposition ofBS has an Y-weight S, we consider for λ > 0, φY0(λ) = (λ^m1y0,1, . . . , λ^mn−1y_0,n−1).

Then BS(X0+i.φY0(λ)) becomes an homogeneous polynomial in λof de- gree S (i.e. BS(X0+i.φY0(λ)) = λ^SBS(X0+i.Y0)). Therefore, we obtain

λlim→0⁺

1

λ^SBS(X0+i.φY0(λ))= 0. Now we replace Z byX0+i.φY0(λ) in in- equality (3.2) and divide by λ^2S. We obtain B_S²(X0+i.φY0) 0 when λ tends to 0⁺. SoBS(X₀+i.Y₀) = 0 which is a contradiction. Thus,S M.

•The caseR < K can be falsified in an analogous way by using Lemma 3.9.

Now Lemma 3.7 shows that |B|²

A is uniformly bounded. Then Proposition 3.4 implies the theorem.

Lemma 3.8. — Let X = (x1, . . . , x_n−1) ∈ Rⁿ⁻¹ be fixed and P_X ∈ R[y1, . . . , yn−1] be homogeneous with respect to the Y-weight L. Then we have the following equations:

1)

n−1

ν=1

∂P_X

∂yν

(y₁, . . . , y_n−1)m_νy_ν=LP_X(y₁, . . . , y_n1).

2)

ν,µ

∂²PX

∂yν∂yµ

(y1, . . . , yn−1)mνmµyνyµ+

n−1 ν=1

∂PX

∂yν

(y1, . . . , yn−1)m²_νyν = L²PX(y1, . . . , yn−1).

Proof. — For 1 ν n−1, we set yν = y^mν

ν . Now, we consider the polynomial QX defined by :QX(y1, . . . ,yn−1) = PX(y^m1

1 , . . . ,y^mn−1

n−1 ). QX

is an homogeneous polynomial atY = (y₁, . . . ,y_n−1) in the classic sense, of degreeL. Then the result follows from Euler’s equation.

Lemma 3.9. — IfPX ≡0is a polynomial inR[y1, . . . , yn−1]not contain- ing neither constant nor linear terms which is homogeneous with respect to the Y-weight L2then

ν,µ

∂²P_X

∂yν∂yµ

(y₁, . . . , y_n−1)m_νm_µy_νy_µ≡0.

Proof. — LetPX be a polynomial which depends exactly on (n−r−1)- variables, where 0 r n−2. By a permutation of variables we may assume that PX(yr+1, . . . , y_n−1) =

I=(ir+1,...,i_n−1)

aI(X)yⁱ_r+1^r+1. . . yⁱ_n−1ⁿ⁻¹. We suppose that the assertion of lemma is false. From Lemma 3.8, we have

n−1

ν=r+1

∂P_X

∂yν m²_νyν = L²PX. Since

n−1

ν=r+1

∂P_X

∂yν mνyν = LPX we get, for all (y_r+1, . . . , y_n−1):

n−1

ν=r+1

mν(L−mν)∂PX

∂y_ν (yr+1, . . . , yn−1)yν = 0 (3.3)

Now, for everyr+ 1νn−1, we setτ_ν=m_ν(L−m_ν). We haveτ_ν >0.

In fact, let us suppose thatτ_µ = 0 for aµwithr+ 1µn−1.

For every term ofPX we have:L=

n−1 ν=r+1

mνiν. Then, two cases are possible for this term:

• i_µ= 1 andi_ν = 0 for allν=µ.

• iµ= 0.

Since there are no linear terms, the first case is impossible. So, i_µ = 0 for this term. But, this is also impossible from the choice of variables.

Now we show that PX vanishes identically. In fact, letY = 0 be fixed. We considerf(λ) =PX(λ^τr+1yr+1, . . . , λ^τn−1y_n−1),λ >0. So, we have:

f(λ) =

n−1

j=r+1

∂PX

∂yj

(λ^τr+1yr+1, . . . , λ^τn−1yn−1)τjλ^τj−1yj. Forr+ 1jn−1, we setwj=λ^τjyj. We get by (3.3):

f(λ) = 1 λ

n−1

j=r+1

τ_jw_j∂P_X

∂yj

(w_r+1, . . . , w_n−1) = 0.

So,f is constant. Asf(1) =P_X(y_r+1, . . . , y_n−1) = lim

λ→0f(λ) =P_X(0) = 0, P_X vanishes identically. Therefore, we obtain a contradiction.

4. A sufficient condition for the existence of a local peak sets for the class A^∞

This part was inspired by the article of Hakim and Sibony [H-S]. The fol- lowing lemma can be shown by standard methods [Na].

Lemma 4.1. — Let UX be a neighborhood of the origin in Rⁿ and h : (X, Y)−→h(X, Y)aC^ω-function onUX×Rⁿ. We suppose thathism-flat where Y = 0. Then there exist a neighborhood VY of the origin in Rⁿ , a neighborhood U_X ⊂⊂U_X of the origin and a function g ∈C^∞(U_X×Rⁿ) which vanishes onU_X×V_Y and verifies for ε >0 :||g−h||^Um^X ×R< ε.

Lemma 4.2. — Let θ: U −→ Cⁿ be a C^∞-parametrization of the sub- manifoldNin a neighborhood of the origin inRⁿ. Thenθhas an extension θ defined on a neighborhood U of the origin in Cⁿ and which is almost- holomorphic with respect toN∩U.

Proof. — Let T_m(X, Y) =

|α|m

1

α!D^α_Xθ(X)(iY)^α and U_X ⊂⊂ U_X be a neighborhood of the origin inRⁿ. Fork∈Nit is clear thatTk+1−Tk isk- flat atY whenY = 0. Now we apply the preceding Lemma 4.1 toTk+1−Tk.

Then there exist a neighborhoodV_Y^k of the origin inRⁿ and aC^∞-function g_k(X, Y) which vanishes onU_X×V_Y^k such that

||Tk+1−Tk−gk||^U_k^X×Rn<2^−k. (4.1)

Form∈N^∗, we set Tm:=T0+ m k=0

(Tk+1−Tk−gk)∈C^∞(UX×Rⁿ). By (4.1)

k

(T_k+1−T_k−g_k) is a normal series for all normsC^l onU_X×Rⁿ, l∈N. So, the sequence (Tm)mconverges uniformly toθ∈C^∞(UX×Rⁿ). It is clear that formandk,Tm(X,0) =θ(X),gk(X,0) = 0. Hence,θ(X, 0) =

m→+∞lim Tm(X,0) =θ(X). So θis an C^∞-extension ofθ on UX×Rⁿ. That θis almost-holomorphic with respect to UX×Rⁿ can be seen by similar arguments as in [H-S].

The following lemma shows that (H2) does not depend of the choice of the almost-holomorphic extension.

Lemma 4.3. — Letγ:V −→Cⁿ⁻¹ be an almost-holomorphic extension of γwith respect to V∩Rⁿ⁻¹ which satisfies the hypothesis(H2)(hereγis the C^∞-parametrization of M defined in section 2). Let φ: W −→ Cⁿ⁻¹ be an another almost-holomorphic extension ofγ with respect toW∩Rⁿ⁻¹. Then, the hypothesis (H2)is satisfied for φ.

Proof. — The passage fromγtoφis given by the transformationψ: W −→

V which is almost-holomorphic with respect to W∩Rⁿ⁻¹. So, we have ψ

W∩Rⁿ⁻¹ =Idandφ=γ◦ψ. It is sufficient to prove for every σ∈W and for alll∈N:|ψ(σ) −σ||Iσ|^l.

Letσ=ζ+i.ηwithζ∈W∩Rⁿ⁻¹ andl∈Nbe fixed. Then, we have ψ(σ) =

|I|l

1 I!

∂^|I|ψ

∂η^I (ζ)η^I+O(|η|^l+1).

ψ(σ) = ζ+

1|I|l

1 I!

∂^|I|ψ

∂η^I (ζ)η^I +O(|η|^l+1). So we can writeψas ψ(σ) = ζ+

l j=1

ψ^(j)(σ) +O(|η|^l+1) withψ^(j)(σ) =

|I|=j

1 I!

∂^jψ

∂η^I(ζ)η^I. In particular,

we have

ψ(σ) = ζ+ψ⁽¹⁾(σ) +O(|η|²) =

n−1

i=1

∂ψ

∂η_i(ζ)ηi+O(|η|²).

Since∂ψ=O(|η|), we haveδ_kj+i∂ψ_j

∂ηk

(ζ) =O(|η|),∀1k, jn−1. This impliesψ⁽¹⁾(σ) =iη. Consequently,ψ(σ) = σ+

l j=2

ψ^(j)(σ) +O(|η|^l+1). Let 2j0l be the smallest integer such thatψ^(j0) is non zero. Then we get:

ψ(σ) = σ+ψ^(j0)(σ) +O(|η|^j0+1). Now, ∂ψ=∂ψ^(j0)+O(|η|^j0) =O(|η|^j0).

Thus, for all 1kn−1, we have

∂ψ^(j0)

∂σk

=−1 2i

∂ψ^(j0)

∂ηk

+O(|η|^j0) =O(|η|^j0).

This implies ∂ψ^(j0)

∂ηk

= O(|η|^j0) for all 1 k n−1. As ∂ψ^(j0)

∂ηk

is a polynomial with respect toη of degree (j₀−1) we get, for all 1kn−1,

∂^{ψ(j0 )}

∂η_k ≡0. Soψ^(j0) is independent ofη. This contradicts our choice ofj0. Therefore, we obtainψ(σ) = σ+O(|η|^l+1).

Before stating our theorem for the A^∞-case, we need a condition to guar- antee the pseudoconvexity of the boundary under an almost-holomorphic change of coordinates. It is the aim of the following lemma.

Lemma 4.4. — Suppose that the hypotheses of Proposition 3.2 are ful- filled. We denote byψ:(Z, w)−→(Z, w)the almost-holomorphic change of coordinates. We suppose that there exist two constants C >0 andL∈N such that, in an open neighborhood U of p∈M, we have

(H3)

Lev ρ(q)[t]C|t|²dist(q,N)^L, ∀q∈U ∩ bD.

Then, D =θ(D∩U)is a locally pseudoconvex at the origin.

Proof. — We set N = θ(N) and M = θ(M). Since θ is a local C^∞- diffeomorphism on an open neighborhoodUofp,θpreserves the distances.

In particular, we have: dist(q, N)≈dist(q,N) with q =θ(q) and q∈U.

Set Ψ =θ⁻¹,w=z_nandw=z_n. Sinceθis an almost-holomorphic change of coordinates the matrix

∂Ψi

∂z_j

1ⁱⁿ 1^jn

is nonsingular (4.2)

on a sufficiently small neighborhood of the origin.

For 1in, we have

∂

∂z_j = n j=1

∂Ψ_j

∂z_i

∂

∂zj

+ n j=1

∂Ψ_j

∂z_i

∂

∂zj

= n j=1

∂Ψ_j

∂z_i

∂

∂z_j + n j=1

O

dist(q,N)^L+1 ∂

∂z_j

The domain D is defined by ρ =ρ◦Ψ. Let t = (t₁, . . . , t_n)∈ T_q^C(bD).

Thus n j=1

∂ρ(q)

∂z_j t_j= 0. This implies n

i,j=1

∂ρ

∂zi

∂Ψ_i

∂z_it_j+O

dist(q,N)^L+1

= 0.

For 1inwe set t_i= n i,j=1

∂Ψ_i

∂z_it_j . From (4.2) we get:

n i=1

∂ρ

∂zi

t_i=O

|t|dist(q,N)^L+1

=O

|t|dist(q,N)^L+1 . Now we decomposetinto tangential componentt^Hand a normal component t^N. So,t=t^H+t^N witht^H∈T_q^C(bD),t^N⊥Tq^C(bD) and|t^H|+|t^N|2|t|.

Moreover, t^N = κ(q)n(q) with κ(q) ∈ C and, for all 1 i n, we have t^N_i =κ(q)^∂ρ(q)_∂z

i . This implies κ(q)

n i=1

∂ρ(q)

∂z_i ² =

n i=1

∂ρ(q)

∂z_i κ(q)∂ρ(q)

∂z_i

= n i=1

∂ρ(q)

∂z_i t^N_i = n i=1

∂ρ

∂z_iti

= O

|t|dist(q,N)^L+1 . Consequently,

|t^N|=|κ(q)|=O

|t|dist(q,N)^L+1

. (4.3)

Now, we compute the Levi form ofρ. As

∂ρ(q)

∂z_i = n i=1

∂ρ(q)

∂z_κ

∂Ψκ(q)

∂z_i +O

dist(q,N)^L+1 and by replacingLbyL+ 1, we get

∂²ρ(q)

∂z_i∂z_j = n k,l=1

∂²ρ(q)

∂zk∂zl

∂Ψk(q)

∂z_i

∂Ψl(q)

∂z_j +O

dist(q,N)^L+1 By (4.3) it follows that

n i,j=1

∂²ρ(q)

∂z_i∂z_j t_it_j = n k,l=1

∂²ρ(q)

∂zk∂zl

_n

i=1

∂Ψ_k(q)

∂z_i t_i 

ⁿ

j=1

∂Ψ_l(q)

∂z_j t_j





+ O

dist(q,N)^L+1

= n k,l=1

∂²ρ(q)

∂zk∂zl

t^H_i t^H_l +O

|t|²dist(q,N)^L+1 .

From (H3) and (4.3) we get:

n k,l=1

∂²ρ(q)

∂zk∂zl

t^H_i t^H_l C|t^H|²dist(q,N)^L C|t|²dist(q,N)^L+O

|t|²dist(q,N)^L+1 . Thus there exists a constant C > 0 such that Lev ρ(q)[t] C|t|² dist(q, N)^L. This means that D is a locally pseudoconvex at the origin.

Definition 4.5. — LetF be a C^∞-function on a neighborhoodV of the origin in Cⁿ⁻¹. We say that F has Y-weight PY(F)S (S ∈N) if there exists a constant C >0 such that|F(X, Y)|C||Y||^S_∗,∀Z=X+i.Y ∈ V. Also, we say thatF hasZ-weightPZ(F)RS (R∈N) if there exists a constant c >0 such that|F(X, Y)|c||Z||^R_∗,∀Z =X+i.Y ∈ V.

In the sequel we have to take into account the following obvious assertions.

Remark 4.6. —

1) LetF be a polynomial function with respect to Y. Then PY(F) S⇐⇒F(X, Y) =

I=(i1,...,i_n−1)

FI(X)Y^I with

n−1

ν=1

mνiν S.