<span class="mathjax-formula">$C^k$</span>-estimates for the <span class="mathjax-formula">$\overline{\partial }$</span>-equation on concave domains of finite type

ANNALES

DE LA FACULTÉ DES SCIENCES

Mathématiques

WILLIAMALEXANDRE

C^k-estimates for the∂-equation on concave domains of finite type

Tome XV, n^o3 (2006), p. 399-426.

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Article mis en ligne dans le cadre du

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pp. 399–426

C^k

-estimates for the

-equation on concave domains of finite type

William Alexandre⁽¹⁾

ABSTRACT.—C^kestimates for convex domains of finite type inC^nare

known from [7] fork= 0 and from [2] fork >0. We want to show the same result for concave domains offinite type. As in the case ofstrictly pseudoconvex domain, we fit the method used in the convex case to the concave one by switchingzandζin the integral kernel ofthe operator used in the convex case. However the kernel will not have the same behavior on the boundary as in the Diederich-Fischer-Fornæss-Alexandre work. To overcome this problem we have to alter the Diederich-Fornæss support function. Also we have to take care of the so generated residual term in the homotopy formula.

R^´ESUM´E.— Les estim´eesC^kpour les domaines convexes de type fini ont

´et´e ´etablies dans [7] pour k = 0 et dans [2] pour k >0. Nous voulons ici ´etudier le cas des domaines concaves de type fini. Comme pour le cas strictement pseudoconvexe, nous adaptons les outils utilis´es par K.

Diederich, B. Fisher et J.E. Fornæss et W. Alexandre en ´echangeant le rˆole des variables dans les noyaux int´egraux de leurs op´erateurs. Cependant le comportement au bord des nouveaux noyaux n’est plus le mˆeme et il faut modifier la fonction de support de K. Diederich et J.E. Fornæss. Elle perdra son holomorphie et g´en´erera un terme r´esiduel dans la formule d’homotopie dont il faudra tenir compte.

1. Introduction

For any convex finite type domain was constructed in [7] a ∂-solving operator which satisfies the best H¨older estimates. These estimates were generalized in [2] toC^k-estimates for all k∈N. Each article used Cauchy- Fantappi`e type integral operators based on the Diederich-Fornæss support function constructed in [6]. In those two articles rose the problem of the

(∗) Re¸cu le 1 octobre 2004, accept´e le 4 avril 2005

(1) LMPA, Universit´e du Littoral - Cˆote d’Opale, maison de la recherche Blaise Pascal, B.P. 699, 62 228 Calais Cedex (France).

E-mail: [email protected]

bad behavior of the normal component of the kernel in the integration vari- able. In case of H¨older estimates this bad behavior did not matter because the main difficulty was the control of a boundary integral, so the normal component did not play any role. However in [2] was to be controlled an integral over a volume. The problem of the normal component was avoid in two times. First were shown new estimates of the derivatives of the defining function for the domain in the normal direction to the boundary and then the author integrated by parts many times.

In this article we are interested inC^k-estimates on concave finite type domain. We use definitions ofC^k-norms used in [11] and show

Theorem 1.1. — LetD ⊂Cⁿ be a smooth bounded convex domain of finite type mandq= 1, . . . , n−2. There exist a neighborhoodU ofbDand a linear operator T_q :C_0,q(U \D)→C_0,q−1(U \D) such that for allk∈N and all ∂-closedf ∈C_0,q^k (U \D)with support included inU \D, we have

i)∂Tqf =f,

ii)Tqf belongs toC^k+

1

0,q−1m(U \D)and there exists a constantck>0, not depending onf, such thatTqf_k+m¹_,U\Dckf_k,U\D.

In order to prove theorem 1.1 we try to fit the operator used in [7]

for convex domains by exchanging in the integral kernel the integration variables ζ and z. After this manipulation the normal component of the kernel inz has a bad behavior and does not disappear by integrating over the boundary ofDand it seems to be impossible to show H¨older estimates for such a kernel.

To overcome the problem we have to alter the Diederich-Fornæss support function. We add some terms to the support function in order to improve the behavior of the normal component of the Cauchy-Fantappi`e kernel gen- erated with it. Because the new support function will not be holomorphic, a residual term will appear in the homotopy formula. We will show that the modification is made in such a way that the residual term is extremely reg- ular and∂-closed in a neighborhood ofCⁿ\D. Then we solve the∂-problem for this residual term and get the operatorTq from the theorem 1.1.

2. Integral operators

We recall the definition of the support function F of [6]. Let D be a bounded convex domain inCⁿof finite typemandraC^∞defining function for D. We set Dα := {z ∈ Cⁿ, r(z) < α}, α ∈ R, and we assume that r is convex onCⁿ and thatgrad ris non zero in a bounded neighborhood V of bD. We fix some ζ in V and denote by T_ζ^CbD_r(ζ) the complex tangent space tobDr(ζ) at ζ and byηζ the outer unit normal atζ to bDr(ζ). Then we choose an orthonormal basis w₁, . . . , w_n such that w₁ =η_ζ and we set r_ζ(ω) =r(ζ+ω₁w₁+. . .+ω_nw_n) and

F_ζ(ω) = ∂r_ζ

∂ω₁(0)ω₁+K ∂r_ζ

∂ω₁(0)ω₁ 2

−K m j=2

κ_jM^2j

|β|=j β1 =0

1 β!

∂^jr_ζ

∂ω^β(0)ω^β whereK, K, M are positive constantsκ_j = 1 whenj ≡ 0mod4,−1 when j≡2mod4 and 0 otherwise. We also set

T_ζ(ω) =−K m j=2

κ_jM^2jj+ 1 j

|β|=j β1 =0

1 β!

∂^j+1r_ζ

∂ω1∂ω^β(0)ω₁ω^β+2 n i=1

∂²r_ζ

∂ω1∂ωi

(0)ω_iω₁

−K m j=2

κjM^2jj+ 2 2j

|β|=j β1 =0

1 β!

∂^j+2rζ

∂ω²₁∂ω^β(0)ω12ω^β+3 2

n i=1

∂³rζ

∂ω²₁∂ωi

(0)ω²₁ωi,

F_ζ(ω) =−K m j=2

κ_jM^2j

|β|=j β1=0

1 β!

∂^jr_ζ

∂ω^β(0)ω^β, F˜_ζ(ω) =F_ζ(ω) +F_ζ(ω) +T_ζ(ω).

We should notice that Tζ and F_ζ do not depend on the basis w₁, . . . , w_n providedw₁ =ηζ. Forz=ζ+ωz,1w₁+. . .+ωz,nw_n we set

F(ζ, z) := Fζ(ωz,1, . . . , ωz,n), T(ζ, z) := T_ζ(ω_z,1, . . . , ω_z,n), F˜(ζ, z) := F˜_ζ(ω_z,1, . . . , ω_z,n).

The following theorem was shown in [7].

Theorem 2.1. — There exist a neighborhoodV ofbDand positive con- stants M,K,K, k,c+,c₋ andR such that for all ζ∈ V, all unit vector v∈T_ζ^CbD_r(ζ) and allw= (w1, w2)∈C², with|w|< R, we have

F(ζ, ζ+w₁η_ζ+w₂v)

−

w1

2 −K

2(w₁)²−Kk 4

m j=2

α+β=j

∂^jr(ζ+λv)

∂λ^α∂λ^β

λ=0

|w₂|^j

−c_±(r(ζ)−r(ζ+w1ηζ+w2v))

wherec_±=c+ whenr(ζ)−r(ζ+w1ηζ+w2v)>0 andc_± =c₋ otherwise.

With no restriction we assume thatV =Dρ\D₋ρ, ρ > 0 sufficiently small.

For ε > 0 sufficiently small and z ∈ V we denote by w^∗₁, . . . , w_n^∗ an ε-extremal basis atz as defined in [3]. We denote by ζ^∗= (ζ₁^∗, . . . , ζ_n^∗) the ε-extremal coordinates atzof a pointζand Φ_∗the unitary transformation such thatζ^∗= Φ_∗(ζ−z). We also define the following complex directional level distancesτ(ζ, v, ε) := sup{τ, r(ζ+λv)−r(ζ)< ε for allλ∈C,|λ|< τ}, we writeτ_i(z, ε) =τ(z, w^∗_i, ε),i= 1, . . . , n, and set

Pε(z) :={ζ ∈Cⁿ,|ζ_i^∗|< τ_i(z, ε), i= 1, . . . n} the polydisc of McNeal (see [12, 3, 2, 7]). We haveτi(z, ε)ε¹² for alli= 1 andτ1(z, ε)εuniformly with respect to z andε (see proposition 3.1 from [7]). As in [7, 2] we use some kind of poly-annuli defined by

P_ε⁰(z) :=Pε(z)\c1Pε(z)

wherec1>0 given by the proposition 3.1 of [7] is such thatc1Pε(z)⊂ P^ε₂(z) for allε >0.

Proposition 2.2. — For all ε >0 sufficiently small, the following in- equality holds uniformly for allz∈ V allζ∈ P_ε⁰(z)

|F(z, ζ)|ε+c_±(r(z)−r(ζ))

with c_±=c+ whenr(z)−r(ζ)>0 andc_± =c₋ whenr(z)−r(ζ)0.

Proof. — We write ζ ∈ P_ε⁰(z) as ζ = z+ληz+µv, v unit vector in T_z^CbD_r(z). We have|λ|c1εor|µ|c1τ(z, v, ε). Indeed by proposition 3.1 of [7], we have _τ(z,v,ε)^|µ| _n

i=2

|ζi^∗|

τi(z,ε). Therefore when |µ| < c₁˜cτ(z, v, ε),

˜

c > 0 sufficiently small and not depending on ζ and z, we have

|ζ_i^∗|< c1τi(z, ε) fori= 2, . . . , n. So we have|λ|=|ζ₁^∗|c1τ1(z, ε)c1ε.

We first assume|µ|< c₁˜cτ(z, v, ε) so that|λ|c₁ε. We have with the proposition 3.1 (vi) of [7] and the theorem 2.1

|F(z, ζ)| |λ| −K|λ|²−K m j=2

M^2j

k+l=j

|µ|^j k!l!

∂^jr(z+µv)

∂µ^k∂µ^l

µ=0

+ +c_±(r(z)−r(ζ))

ε(1−ε−c) +˜ c_±(r(z)−r(ζ)) ε+c_±(r(z)−r(ζ))

provided ˜candεare sufficiently small. We now fix such constants ˜cand ε.

When|µ|c₁˜cτ(z, v, ε) we have again with the proposition 3.1 (vi) of [7] and the theorem 2.1 :

|F(z, ζ)| K m j=2

M^2j

k+l=j

|µ|^j k!l!

∂^jr(z+µv)

∂µ^k∂µ^l

µ=0

+c_±(r(z)−r(ζ)) ε+c_±(r(z)−r(ζ)).

Corollary 2.3. — For all ε > 0 sufficiently small, the following in- equality holds

|F˜(z, ζ)|ε+c_±(r(z)−r(ζ))

uniformly for all z ∈ V and all ζ∈ P_ε⁰(z), c_± =c+ when r(z)−r(ζ)>0 andc_±=c₋ whenr(z)−r(ζ)0.

Proof. — We use w^∗₁, . . . , w_n^∗ as a basis for the definition ofTz and F_z. Using|ζ₁^∗|< τ₁(z, ε)εand|ζ_i^∗|< τ_i(z₀, ε)ε^m1 for allζ∈ Pε(z) we get uniformly with respect to z and ζ |F_z(ωζ)|+|Tz(ωζ)| ε^1+m1. Therefore whenεis sufficiently small the proposition 2.2 gives the estimate.

Corollary 2.4. — There exists a constant R > 0 such that for all (z, ζ)∈ V ×Cⁿ with|ζ−z|< R the following inequality holds

|F(z, ζ)˜ ||ζ−z|^m+c_±(r(z)−r(ζ))

uniformly with respect toζandz,c_± =c+whenr(z)−r(ζ)>0andc_±=c₋ whenr(z)−r(ζ)0.

Proof. — We denote by ε0>0 a constant such that for allε∈]0, ε0] the corollary 2.3 holds and by R > 0 a constant such that B(z, R) ⊂ Pε0(z)

for all z ∈ V. We fix ζ ∈B(z, R). Ifζ =z, the corollary is obvious. Oth- erwise we have B(z, R)⊂ Pε0(z)⊂ {z} ∪_+∞

i=0P₂⁰−iε0(z) thus there exists i₀ ∈ N such that ζ belongs to P₂⁰−i0ε0(z). The corollary 2.3 then implies that|F(z, ζ)˜ |2⁻ⁱ⁰ε₀+c_±(r(z)−r(ζ)).

Sinceζ/ c∈ 1P2⁻ⁱ⁰ε0(z), we have |ζ−z|^mc12⁻ⁱ⁰ε0 and so|F˜(z, ζ)|

|ζ−z|^m+c_±(r(z)−r(ζ)).

AsF, ˜F is a local support function. However we need a global support function to use integral formulas. Therefore we will give a global support function which has locally the same behavior.

We set ˜R := _4c¹

−

_R

4

m and assumeV so small that r(z) −R˜ for all z∈ V. Now letχbe aC^∞function such that for allζ∈Cⁿ, 0χ(ζ)1, χ(ζ) = 1 when|ζ|^R₂ andχ(ζ) = 0 when|ζ|R,Rgiven by the corollary 2.3. We set for all (ζ, z)∈Cⁿ× V

S(ζ, z)˜ = F˜

z, χ(ζ−z)ζ+ (1−χ(ζ−z))

z+ R

2|ζ−z|(ζ−z)

.

Proposition 2.5. — For all(ζ, z)∈Cⁿ× V we have

i) |S(ζ, z)˜ | 1 uniformly with respect to ζ ∈ D_R˜ and z ∈ V with

|ζ−z| ^R₄.

ii) ˜S(ζ, z) = ˜F(z, ζ)when|ζ−z| ^R₂.

Proof. — (ii) follows immediately from the definition of χ. To show (i) We set ˜ζ=χ(ζ−z)ζ+ (1−χ(ζ−z))

z+_2|ζ^R_−z|(ζ−z)

.

We have |ζ˜−z| = ^R₂ when |ζ−z| R and ^R₄ |ζ˜−z| R when

R

4 |ζ−z|R. So the corollary 2.4 gives

|S(ζ, z)˜ | R

4 m

+c_±(r(z)−r(˜ζ)). (2.1) Moreover when |ζ −z| ^R₄ we have r(˜ζ) max(r(z), r(ζ)). Indeed ζˆ:= _2|ζ−z|^R ζ+

1−_2|ζ−z|^R

z is a point of [ζ, z] and by convexity we have r(ˆζ)max(r(ζ), r(z)). Again by convexity we haver(˜ζ)max(r(ζ), r(ˆζ)) max(r(ζ), r(z)). Therefore forζ∈DR˜ andz∈ V we havec_±(r(z)−r(˜ζ))

−¹_2R

4

m. Now (2.1) implies (i).

We now define the Hefer section and the kernel we use. Let U be an arbitrary unitary matrix and set forζ, ω∈Cⁿ andz∈ V

Σ(z, ω)˜ = S(z˜ +U ω, z), (2.2)

Θ(z, ω) = T(z+U ω, z), (2.3)

˜

σi(z, ω) = 1

0

∂Σ˜

∂ω_i(z, tω)dt, (2.4)

θi(z, ω) = 1

0

∂Θ

∂ω_i(z, tω)dt, (2.5)

Q(ζ, z)˜ = U(˜σ₁(z, U^t(ζ−z)), . . . ,σ˜_n(z, U^t(ζ−z))), (2.6) Q(ζ, z) = Q(ζ, z)˜ −U(θ1(z, U^t(ζ−z)), . . . , θn(z, U^t(ζ−z))).(2.7) One can easily check that ˜Q and Q do not depend on U and satisfies S(ζ, z) =˜ n

i=1(ζi−zi) ˜Qi(ζ, z). MoreoverQis holomorphic with respect to ζ ∈ B(z,^R₂). Later on we will choose U := U(z) such that U^tη_z= (1,0, . . . ,0). Now we set

˜

η1(ζ, z) = n i=1

Q˜i(ζ, z)dζi,

η₀(ζ, z) = n i=1

ζ_i−z_idζ_i,

˜

η(ζ, λ, z) = (1−λ)η0(ζ, z)

|ζ−z|² +λη˜1(ζ, z)

S(ζ, z)˜ , for(ζ, z)∈D× Vwith ˜S(ζ, z)= 0,

=η0(ζ, z)

|ζ−z|² for (ζ, z)∈(Cⁿ− V)× V, Ω˜_n,q=(−1)^q(q−1)2

(2iπ)ⁿ

n−1 q

˜

η∧(∂_ζ,λη)˜ ^n−q−1∧(∂_zη)˜ ^q forn= 0, . . . , n−1 and ˜Ωn,−1= ˜Ωn,n= 0.

We also define for i = 0,1 ι_i : Cⁿ× {i} ×Cⁿ → Cⁿ×[0,1]×Cⁿ the canonical injection and we set B_n,q = ι^∗₀( ˜Ω_n,q) and K_n,q = ι^∗₁( ˜Ω_n,q). We finally define ˜Tq andRq by setting forf ∈C_0,q⁰ (Cⁿ\D) andz∈ V \D

T˜qf(z) = −

bD×[0,1]f(ζ)∧Ω˜n,q−1(ζ, λ, z)−

V\Df(ζ)∧Bn,q−1(ζ, z), Rqf(z) = −

bD

f(ζ)∧Kn,q(ζ, z);

Whensuppf, the support off, is included inV \Dthe Stokes theorem gives (see [15])

f =Rqf+∂T˜qf + ˜Tq+1∂f. (2.8) At the end of section 3 we show thatR_qand ˜T_qsatisfy the following theorem.

Theorem 2.6. —i) For all k ∈N and all ∂-closed f ∈ C_0,q⁰ (Cⁿ\D), 1 q n−2, Rqf belongs to C_0,q^k (V) and Rqfk,V f0,bD, ck not depending onf. Moreover R_qf is∂-closed onV.

ii)For allf ∈C_0,q⁰ (Cⁿ\D),1qn−2,T˜_qf belongs toC_0,q−1^m1 (V \D) and there exists some constant c > 0 not depending on f such that T˜_qf_m¹,V\Dcf0,V\D.

We may not have T˜_qfk+_m¹,V\D cfk,V\D for all ∂-closed f ∈ C_0,q^k (V \D), k ∈ N^∗. Thus we modify ˜Tq as in [11] or [2]. Even if the method is almost the same, there are some differences because ˜S is not holomorphic.

We setG=V ∩D andE the Seeley type extension operator given by the following lemma (see [11] or [16]).

Lemma 2.7. — There exists a linear extension operator E:C⁰(V \D)→C⁰(V)such that

i)for allu∈C⁰(V \D) Eu|_V\D=uandsupp Eu⊂Cⁿ−(D\ V), ii)for all k∈Nand all u∈C^k(V \D) Eu belongs toC^k(V)and there exists a constant ck >0 not depending onusuch that

Euk,Vcku_k,V\D.

We set forz∈ V \D andf ∈C_0,q⁰ (V \D) R˜^∗_qf(z) =

G

Ef(ζ)∧Kn,q−1(ζ, z), q1, M˜_qf(z) = ∂_z

G×[0,1]

Ef(ζ)∧Ω˜_n,q−2(ζ, λ, z) forq >1,

= 0 forq= 1, and

T˜_q^∗f = ˜Tqf −M˜qf−R˜^∗_qf.

Since ˜M_qf is trivially ∂-closed, for all ∂-closed f ∈ C_0,q⁰ (Cⁿ \D) with suppf ⊂ V, (2.8) becomes

f = ∂T˜_q^∗f +∂R˜^∗_qf+R_qf. (2.9) Moreover whenf ∈C_0,q¹ (V \D) is∂-closed the Stokes theorem gives

G×[0,1]∂_ζ,λ(Ef∧Ω˜_n,q−1) =

=

bD×[0,1]Ef∧Ω˜n,q−1−

G

Ef∧Bn,q−1+

G

Ef∧Kn,q−1.

Since∂_ζ,λΩ˜_n,q−1= (−1)^q−1∂_zΩ˜_n,q−2, we get T˜_q^∗f(z) =−

G×[0,1]

∂_ζEf(ζ)∧Ω˜_n,q−1(ζ, λ, z)−

V

Ef(ζ)∧B_n,q−1(ζ, z).(2.10) Using this expression of ˜T_q^∗, we will show the following theorem.

Theorem 2.8. —i) For q = 1, . . . n−2, k ∈ N and f ∈ C_0,q^k (V \D)

∂-closed withsuppf ⊂ V \D, T˜_q^∗f belongs to C_0,q−1^k+m1 (V \D) and there ex- ists a constant ck > 0, not depending on f, such that T˜_q^∗f_k+m¹_,V\D ckf_k,V\D.

ii)Forq= 1, . . . , n−2andf ∈C_0,q⁰ (V \D),R˜^∗_qf belongs to C_0,q^k (V \D) for all k and there exists a constant ck >0 not depending on f such that R˜^∗_qf_k,V\Dckf_0,V\D.

Proof of theorem 1.1. — Let U and W be neighborhoods of bD such that U ⊂ W ⊂ W ⊂ V = Dρ \D₋ρ, ρ > 0. There exists an operator T_q^∗ : C_0,q⁰ (W) → C_0,q−1⁰ (U \D) such that for all k ∈ N and all ∂-closed f ∈ C_0,q^k (W) ∂T_q^∗f = f and T_q^∗f belongs to C_0,q^k ₋₁(U) and satisfies T_q^∗fk,Ufk,W uniformly with respect tof (see [15]).

We setT_q = ˜T_q^∗+ ˜R^∗_q+T_q^∗R_q. For allk∈Nand all∂-closedf∈C_0,q^k (U\D) withsupp f ⊂ U \D, we have∂Tqf =f. Moreover the theorem 2.6 (i) and 2.8 imply thatT_qf belongs toC_0,q^k+₋^m1₁(U \D) and satisfiesT_qfk+_m¹,U\D fk,U\D.

3. C⁰-estimates

As in [7, 9, 2] we use ε-extremal basis to estimate the kernel. We fix ε > 0, z0 ∈ V \D and an ε-extremal basis w₁^∗, . . . , w^∗_n at z0. We assume that εis so small that_∂w^∂r∗

1(ζ)cfor all ζ∈ Pε(z0),c not depending on z₀,ζ and ε. We denote by (ζ₁^∗, . . . , ζ_n^∗) the extremal coordinates of a point ζ∈Cⁿ and by Φ_∗ a unitary matrix such thatζ^∗= Φ_∗(ζ−z₀). In order to write ˜Ω_n,q in theε-extremal basis we setQ^∗ = Φ_∗Qand ˜Q^∗ = Φ_∗Q. T hus˜ we have

˜

η1(ζ, z) = n j=1

Q˜^∗_j(ζ, z)dζ_j^∗,

∂_zη˜₁(ζ, z) = n i,j=1

∂Q˜^∗_j

∂z^∗_i (ζ, z)dz^∗_i ∧dζ_j^∗,

∂ζη˜1(ζ, z) = n i,j=1

∂Q˜^∗_j

∂ζ^∗_i (ζ, z)dζ^∗_i ∧dζ_j^∗.

We also need a unitary matrix Ψ(z) such that Ψ(z)η_z= (1,0, . . . ,0) for all z∈ Pε(z0). We use the matrix defined and studied in [2]. In the definition of ˜Q (see (2.2), (2.4) and (2.6)) we use U = Ψ(z)Φ_∗^t and set ω(z, ζ) = Ψ(z)(ζ^∗−z^∗). We notice thatω(z₀, ζ) =ζ^∗ and that for|ζ−z|< ^R₂

Q^∗₁(ζ, z) = ∂r

∂z₁^∗(z) +K ∂r

∂z^∗₁(z) n k=1

∂r

∂z_k^∗(z)(ζ_k^∗−z_k^∗), Q^∗_j(ζ, z) = ∂r

∂z_j^∗(z) +K ∂r

∂z^∗_j(z) n k=1

∂r

∂z_k^∗(z)(ζ_k^∗−z_k^∗)

−K m k=2

κkM^2k

|β|=k

βj

kβ!

∂^kr

∂z^∗β(z)(ζ^∗−z^∗)^β

ζ_j^∗−z_j^∗ , j= 2, . . . , n.

For later use (see section 4) we introduceδ^∗_j = _∂z^∂_∗

j

+_∂ζ^∂_∗

j

, j= 1, . . . , n

Lemma 3.1. — For i = 1, . . . , n, j = 2, . . . , n and ζ ∈ Pε(z₀) we have uniformaly with respect toζ,z₀ andε

|ω_i(z₀, ζ)| τ_i(z₀, ε) ∂ω1

∂z_i^∗(z0, ζ)

ε τ_i(z0, ε)

|δ^∗_iωj(z0, ζ)|+ ∂ωj

∂z^∗_i (z0, ζ)

τj(z0, ε) ∂ω1

∂z^∗_i (z0, ζ)

ε τ_i(z₀, ε) ∂ω1

∂z^∗₁(z0, ζ) + 1

ε¹² whereτ_i(z₀, ε) =τ_i(z₀, ε)if i= 1 andτ₁(z₀, ε) =ε¹².

Proof. — We have for allk ω_k(z, ζ) =n

l=1Ψ_kl(z)(ζ_l^∗−z_l^∗). The propo- sition 4.2 of [2] implies all the inequalities.

Lemma 3.2. — For all multiindices β with |β| 1 and β₁ = 0 and j= 1, . . . , nwe have uniformly with respect to z₀ andε

∂^|β|+1rz0

∂ω1∂ω^β (0)

_n ε¹²

k=1τ_k(z₀, ε)^βk,

∂

∂z_j^∗

∂^|β|+1rz0

∂ω1∂ω^β (0)− ∂^|β|+2r

∂z_j^∗∂z₁^∗∂z^∗β(z0)

ε τ_j(z₀, ε)_n

k=1τ_k(z₀, ε)^βk,

∂

∂z^∗_j

∂^|β|+1r_z0

∂ω1∂ω^β (0)− ∂^|β|+2r

∂z^∗_j∂z₁^∗∂z^∗β(z0)

ε τ_j(z0, ε)_n

k=1τk(z0, ε)^βk. Proof. — Sincer_z(ω) =r(z+Ψ(z)^tω) and Ψ(z₀) =Id_Cn, for allα₁, . . . , α_l, l1, we have

∂^l+1r_z0

∂ω1∂ωα1. . . ∂ωαl

(0) = ∂^l+1r

∂z^∗₁∂z_α^∗₁. . . ∂z_α^∗_l(z₀).

Therefore the first inequality is a straightforward consequence of the corol- lary 3.4 of [2]. In order to prove the second inequality we compute

∂

∂z^∗_j

∂^l+1r_z0

∂ω1∂ωα1. . . ∂ωαl

(0) =

1^pl 1^sn

∂^l+1r

∂z^∗₁∂z_α^∗₁. . .[∂z^∗_αp]. . . ∂z^∗_αl(z0)∂Ψ_αps

∂z_j^∗ (z0) +

+

1^sn

∂^l+1r

∂z^∗_s∂z_α^∗₁. . . ∂z^∗_αl(z₀)∂Ψ_1s

∂z^∗_j (z₀) +

+ ∂^l+2r

∂z_j^∗∂z^∗₁∂z_α^∗₁. . . ∂z_α^∗_l(z0). (3.1) where the term between [·] is omitted.

The proposition 4.2 of [2] gives

^{∂Ψ∂zαks∗}j (z)

_τ ^ε

αk(z0,ε)τ_j(z0,ε) for alls,αk

andj.

The corollary 3.4 of [2] gives _∂z∗ ^∂p+1r s∂z_α^∗

1...∂z_αp^∗

p ^ε12

k=1τ_αk(z0,ε) and by proposition 4.2 of [2] we have ^∂Ψ_∂z^1s∗

j (z) _τ^ε12

j(z0,ε) from which follows the estimates of the second sum in (3.1). This proves the second inequality. The third can be shown in the same way.

Proposition 3.3. —i) For all ζ ∈ B(z0,^R₂), i = 2, . . . , n and j = 1, . . . , nwe have

∂Q˜^∗_j

∂ζ^∗_i (ζ, z₀) = 0.

ii) For allζ∈ Pε(z0)∩B(z0,^R₂)andj = 1, . . . , nholds uniformly

∂Q˜^∗_j

∂ζ^∗₁(ζ, z₀)

ε¹² τj(z0, ε).

Proof. — We have ˜Q^∗_j(ζ, z) =Q^∗_j(ζ, z) +_n

k=1Ψkj(z)θk(z, ω(z, ζ)) and θ1(z, ω) = 0

θ_k(z, ω) = −ω₁K m l=2

κ_lM^2l

|β|=l β1 =0

β_k lβ!

ω^β ω_k

∂^l+1r_z

∂ω₁∂ω^β(0) + ∂²r_z

∂ω₁∂ω_k(0)ω₁

−ω²₁K m l=2

κ_lM^2l 2

|β|=l β1 =0

β_k lβ!

ω^β ωk

∂^l+2r_z

∂ω²₁∂ω^β(0) +1 2

∂³r_z

∂ω²₁∂ω_k(0)ω²₁.

Q^∗(·, z₀) is holomorphic forζ∈B(z₀,^R₂) and Ψ(z₀) =Id_Cn thus ^∂Q˜∗j

∂ζ^∗_i (ζ, z₀)

= ^{∂θj(z0,ω(z0,ζ))}

∂ζ^∗_i . On the other hand ^∂ω1

∂ζ^∗_i(z0, ζ) = Ψ1i(z0). Therefore (i) is obvious and (ii) follows from the lemma 3.1 and 3.2.

Proposition 3.4. — For all ζ ∈ Pε(z0) ∩B(z0,^R₂), j = 1, . . . , n, i= 2, . . . , nwe have uniformly with respect to ζ,z0 andε

|Q˜^∗_j(ζ, z0)| ε τj(z0, ε),

∂Q˜^∗_j

∂z^∗_i (ζ, z₀)

ε

τj(z0, ε)τi(z0, ε),

∂Q˜^∗_j

∂z^∗₁(ζ, z₀)

ε τj(z0, ε).

Proof. — Sinceτ1(z0, ε)ε(see proposition 3.1 (v) of [7]), the proposi- tion is obvious whenj = 1. Therefore we only consider the casesj = 2, . . . , n.

We use ˜Q^∗_j(ζ, z) =Q^∗_j(ζ, z) +n

k=1Ψkj(z)θk(z, ω(z, ζ)). |θk(z0, ω(z0, ζ)|

|ω1(z0, ζ)|εfor allkand the corollary 3.4 of [2] implies thatQ^∗_j(ζ, z0)

ε

τj(z0,ε). Therefore the first inequality holds. Because of the proposition 4.2 of [2], to show the others inequalities it remains to estimate ^∂Q_∂z∗^∗j

i (ζ, z₀) +

∂θj(z,ω(z,ζ))

∂z^∗_i

z=z0

. The corollary 3.4 of [2] gives^∂Q_∂z∗^∗j

i (ζ, z0)_τj(z0,ε)τ^ε i(z0,ε). The lemma 3.1 then implies

^{∂∂zQ˜∗}i^∗j(ζ, z0)

_τj(z0,ε)τ^ε_i(z0,ε) for i, j = 2, . . . , n.

Wheni= 1 we notice that ^∂_∂ω^|β|+1rz0

1∂ω^β (0) = _∂z^∂|β|+1∗ ^r

1∂z^∗β(z0). Therefore lemma 3.1 and 3.2 imply for all multiindicesβ withβ₁= 0 and|β|1 that

βj

ω(z₀, ζ)^β ωj(z0, ζ)

∂ω₁

∂z^∗₁(z0, ζ)∂^|β|+1r_z0

∂ω1∂ω^β (0) + ∂^|β|r

∂z^∗₁∂z^∗β(z0)

ε τj(z0, ε). The inequality|ω1(z0, ζ)|εthen implies that for allβ with β1= 0

∂

∂z^∗₁

β_jω(z0, ζ)^β

ω_i(z₀, ζ)ω₁(z₀, ζ)∂^|β|+1r_z0

∂ω₁∂ω^β (0)+β_jζ^∗β ζ_j^∗

∂^|β|r

∂z^∗β(z₀)

ε

τ_j(z₀, ε).(3.2) Since|ζ₁^∗|εwe have for all multiindicesβ withβ₁= 0

∂

∂z^∗₁

βj

ζ^∗β ζ_j^∗

∂^|β|r

∂z^∗β(z0)

ε. (3.3)

The corollary 3.4 of [2] gives

∂²r

∂z^∗₁∂z_j^∗(z₀) n k=1

∂r

∂z^∗_k(z₀)ζ_k^∗+ ∂r

∂z^∗_j(z₀) n k=1

∂²r

∂z^∗₁∂z^∗_k(z₀)ζ_k^∗

ε³²

τ_j(z₀, ε).(3.4) All together (3.2), (3.3) and (3.4) hield to

^∂Q∂z∗1^∗j(ζ, z0)+^{∂θj(z,ω(z,ζ))}_∂z∗ 1

z=z0

ε

τj(z0,ε) and so

^{∂∂zQ˜∗}1^∗j(ζ, z0)

_τj(z^ε_0,ε).