Extension of bounded holomorphic functions¶ in convex domains

manuscripta math. 105, 1 – 12 (2001)

© Springer-Verlag 2001

Klas Diederich · Emmanuel Mazzilli

Extension of bounded holomorphic functions in convex domains Received: 7 July 2000 Abstract. The E. Amar and G. Henkin theorem on the bounded extendability of bounded holomorphic functions from certain closed complex submanifolds of strictly pseudoconvex domains to the whole domain is generalized to the case of finite type convex domains and their intersections with affine linear hyperplanes. Suitable integral operators of Berndtsson– Andersson type are constructed and estimated for this purpose.

1. Introduction and main results In this article we deal with a special case of the following general problem: Let D ⊂⊂ Cn be a pseudoconvex domain and D  ⊂ D a closed complex hypersurface. Is any bounded holomorphic function f on D  the restriction of a bounded holomorphic function fˆ on D to D  ? It is known, that the answer to this question is, in general, negative even in rather harmless situations like D being convex of finite type with a smooth real-analytic boundary and D  = D ∩ A where A is a closed complex submanifold of an open neighborhood of D intersecting ∂D everywhere transversally. On the other hand, as proved by E. Amar [1] and by G. Henkin [8] , the answer is positive, if D is strictly pseudoconvex with C 2 -smooth boundary and, again, D  = D ∩ A where A is a closed complex submanifold of an open neighborhood of D (which, in the case of the work of Henkin has to intersect ∂D transversally, a hypothesis which was eliminated by Amar). However, there are also cases of weakly pseudoconvex domains D with C ∞ smooth boundaries, for which the answer is positive. In [9] the case of real pseudoellipsoids D with intersections D  with affine complex hyperplanes is treated positively by the second author. The article [11] contains necessary conditions for a positive answer to the above question, if D ⊂⊂ C3 is a convex domain of finite type and D  = D ∩ A with a closed smooth complex hypersurface A of an open neighborhood of D. More precisely, it is proved: K. Diederich: Mathematik, Universität Wuppertal, Gausstr. 20, 42095 Wuppertal, Germany e-mail: [email protected] E. Mazzilli: UFR de Mathématiques Pures et Appliqués, Université de Lille 1, 59655 Villeneuve d’Ascq Cedex, France. e-mail: [email protected] Mathematics Subject Classification (2000): 32A35, 32A26, 32F32, 32T25, 32T27

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K. Diederich, E. Mazzilli

Theorem 1.1. Let D ⊂⊂ C3 be a smooth convex domain of finite type and A a closed smooth complex submanifold of a neighborhood of D intersecting ∂D transversally at every point a ∈ ∂D  , D  := D ∩ A. If every bounded holomorphic function f on D  has a bounded holomorphic extension to D, then one has for each point a ∈ ∂D  the equality N1 (a, A) = N2 (a, A)

(1.1)

where N1 (a, A) is the maximal order of contact between smooth analytic curves through a tangent at a to A with ∂D and N2 (a, A) is the maximal order of contact between smooth analytic curves through a in A with ∂D. It is also shown in [11] , that for affine linear hypersurfaces A the equality 1.1 always holds. Hence, affine-linear hypersurfaces seem to be natural candidates for positive results. However, this is only the case for convex domains D, as the results from [6] show. In this article we will now show, that one has indeed Theorem 1.2. Let D ⊂⊂ Cn be a smooth convex domain of finite type m and A a complex affine linear hypersurface such that D  := D ∩ A = ∅. Then there is a bounded linear extension operator E : H ∞ (D  ) → H ∞ (D), where H ∞ (·) denotes, of course, the Banach space of bounded holomorphic functions on the corresponding domain. Of course, by induction one obtains from this Theorem the Corollary 1.3. Let D be as in the Theorem and let A be an affine linear subspace (of arbitrary dimension) of Cn with D  := D ∩ A = ∅. Then there is a bounded linear extension operator E : H ∞ (D  ) → H ∞ (D). Our extension operator will be constructed as an integral operator ofAndersson– Berndtsson type (see [3]) using the support functions for convex domains of finite type from [5] in a similar way as in [7].

2. Construction of the extension operator Let now always D ⊂⊂ Cn be a smoothly bounded convex domain of finite type. We write 
D = z ∈ Cn : ρ(z) < 0 with a C ∞ convex function ρ on Cn . Let A be a smooth complex hypersurface in an open neighborhood V of D of the form A = { z ∈ V : f (z) = 0} with a holomorphic function f on V with ∂f = 0 everywhere on A and ∂f ∧∂ρ = 0 at all points of A ∩ ∂D. For a positive number η0 > 0 which has been chosen small

Extension of bounded holomorphic functions in convex domains

3

enough we let φ be a C ∞ cut-off function with φ(ζ ) = 0 for ρ(ζ ) ≥ − η20 and φ(ζ ) = 1 if ρ(ζ ) ≤ −η0 . We let n 

Qi (z, ζ )dζi

i=1

be the form introduced in [4] and S(z, ζ ) the support function from [5]. We use the formalism of Berndtsson–Andersson from [3] for the construction of the integral kernel used for the extension of bounded holomorphic functions from D  := D ∩ A to D and, therefore, put for z, ζ ∈ D   n n   ∂ρ 1 1 Q (ζ, z) := Qi (z, ζ ) dζi + φ(ζ ) (ζ ) dζi . (2.1) (1 − φ(ζ )) ρ(ζ ) ∂ζi i=1

i=1

Furthermore, we decompose f as f (z) − f (ζ ) =

n 

hi (ζ, z)(zi − ζi )

i=1

with holomorphic functions hi and put Q2 (ζ, z) :=

f (ζ )

n 

|f (ζ )|2 + ε i=1

hi (ζ, z) dζi

(2.2)

with a small ε > 0. According to the theorem of Berndtsson–Andersson the following Proposition holds: Proposition 2.1. For any N > 1 the integral operator E N defined on any function g ∈ H ∞ (D  ) for any z ∈ D by  g (ζ ) E N g(z) := D

ρ N+n−1 (ζ ) N+n−1  ∂ρ ρ (ζ ) + (1−φ (ζ )) S (z, ζ ) + φ(ζ ) ni=1 ∂ζ −ζ (ζ )(z ) i i i n−1 ∂f (ζ ) ∧ n hi (ζ, z) dζi   i=1 × ∂Q1 (ζ, z) ∧ dλ ∂f 2 (2.3) ×

where dλ denotes the surface measure on D  , is a linear extension operator. Furthermore, the function E N g(·) is continuous on D \ (∂D ∩ A). Proof. The proof of the fact, that E N is, indeed, an extension operator is completely contained in [2] . The fact, that we obtain a function continuous on D \ (∂D ∩ A) is clear, since the only singularity of the intergal kernel occurs for ζ = z.   Remark 2.2. Precisely speaking, the integrand in the definition of E N is an (n, n)differential form with measures as coefficients. What we mean in (2.3) is: we take the unique coefficient of this differential form, multiply it by the Lebesgue-measure dλ on D  and integrate over D  . This is also the original notation of [2] .

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3. Rewriting E N and some geometry of convex domains of finite type For the proof of Theorem 1.2 we have to use Proposition 2.1 for convex domains of finite type with C ∞ -smooth boundaries and for f (z) = zn (after a linear change of coordinates). We rewrite at first the kernel E N for z ∈ ∂D \ A in the following way: Proposition 3.1. For each z ∈ ∂D \ A and each N > 2 one has E N g(z) =

 D

zn g(ζ )

n−1  ∧ dζ n ρ(ζ )N+n−1 ni=1 (Qi (ζ, z) dζi ) ∧ ∂ Q1 (ζ, z)  N−n−1 dλ.  ∂ρ S(ζ, z) ρ(ζ )+(1−φ(ζ ))S(z, ζ )+φ(ζ ) ni=1 ∂ζ −ζ (z ) i i i (3.1) Proof. We choose a family (πγ )δ≥0 of C ∞ cut-off functions on R, such that πγ ≡ 1 on {x ≤ γ2 } and πγ ≡ 0 on {x ≥ γ }. Furthermore, we observe, that, because of the convexity of D, for a > 0 sufficiently small the “lense” 
La := z : |zn | ≤ a|ρ(z)|, z := (z1 , . . . , zn−1 ) ∈ D  is contained in D. We extend the given function g from D  to La by putting g(z1 , . . . , zn ) := g(z1 , . . . , zn−1 , 0) for z ∈ La . Then the function hγ (g) :=

| πγ ( ρ|z2n(z) )g(z) is well-defined if γ is small enough, hγ (g) ∈ L∞ (D) ∩ C ∞ (D) and 2

M

∂(hγ (g)) ≤ |ρ|γ , where Mγ is a constant depending on γ and going to +∞ for γ → 0. We apply the formula of Berndtsson–Andersson (see [3] and [2] ) to the function hγ (g) and use their formalism with the data  n n   

   s(ζ, z) = −ρ(z) ζ i − zi dζi + (1 − φ(z))S(ζ, z) Qi (ζ, z) dζi ,     i=1 i=1       n n     1 ∂ρ  1  Qi (z, ζ ) dζi + φ(ζ ) (ζ ) dζi , (1 − φ(ζ ))  Q (ζ, z) = ρ(ζ ) ∂ζi i=1 i=1    G1 (z) = z−N (N ≥ 2),      dζn   Q2 (ζ, z) = ,   ζn     G2 (z) = z. We obtain the following representation formula holding for all z ∈ D  hγ (g)(z) =

D

 hγ (g)(ζ )P (ζ, z) +

D



∂ hγ (g) (ζ ) ∧ K(ζ, z)

(3.2)

Extension of bounded holomorphic functions in convex domains

5

where the kernels P and K are defined by   n −N−n z  n  1 1  < Q , z − ζ > +1 ∂Q P = C n   ζn     n−1 −N −n+1    1 1   < Q + D , z − ζ > +1 ∂Q ∧ dζn ∧ dζ n dλ n       =: P0 (ζ, z) + P1 (ζ, z),    

n−1−k  n−1  −N−k z  k   s ∧ ∂s  n  1 1  K = An ∂Q ∧ < Q , z − ζ > +1   ζn < s, ζ − z >n−k                            

k=0 n−2  

−N−k 

< Q1 , z − ζ > +1

+ Bn

∂Q1

k

k=0

n−2−k s ∧ ∂s ∧ ∧ dζn ∧ dζ n dλ < s, ζ − z >n−k−1   =: K0k (ζ, z) + K1k (ζ, z). k

k

If we now use that hγ (g)(ζ ) and ∂hγ (g)(ζ ) vanish for ζ far away from A, it follows, that the representation formula (3.2) remains true for z ∈ ∂D\A (notice, that the only singularities of the kernels occur for ζ = z). Hence we have  0= hγ (g)(ζ )(P0 (ζ, z) + P1 (ζ, z)) D   (3.3)



  + ∂ hγ (g) (ζ ) ∧ K0n−1 + ∂ hγ (g) (ζ )K1n−2 . D

D

Furthermore, the form K1n−2 has its support on D  and ∂(hγ (g)) is zero there since  g is holomorphic. This implies D ∂(hγ (g))(ζ )K1n−2 = 0. Altogether, we obtain  0=

D

 hγ (g)(ζ )(P0 (ζ, z) + P1 (ζ, z)) +

D

 ∂ hγ (g) (ζ ) ∧ K0n−1 .

(3.4)

Notice, that, because of the Lebesgue-Theorem and Proposition (2.1), we have   lim hγ (g)(ζ )(P0 (ζ, z) + P1 (ζ, z)) = lim hγ (g)(ζ )P1 (ζ, z) = E N g(z). γ →0 D

γ →0 D

(3.5) For the proof of Proposition 3.1 it, therefore, suffices to show, that the second term of (3.4) goes to minus of the right side of 3.1 if γ goes to 0. For this, we observe, that K0n−1 is of the form K0n−1 =

zn ζ n zn T. T = lim ε→0 |ζn |2 + ε ζn

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K. Diederich, E. Mazzilli

After this regularization we apply Stokes to the expression  ∂D

zn ζ n hγ (g)(ζ )T (ζ, z). |ζn |2 + ε

Since T is zero for ζ ∈ ∂D, we obtain 

 zn ζ n 0= ∂ hγ (g) (ζ ) T (ζ, z) + 2 D |ζn | + ε  zn ζ n hγ (g)(ζ )∂ (T (ζ, z)) . + 2 D |ζn | + ε

 zn ζ n hγ (g) (ζ ) T (ζ, z) ∂ |ζn |2 + ε D





z ζ

n ) converges weakly to the residual current ∂( ζ1n ) = Notice next, that ∂( |ζ n|2 +ε n

dζ n dλ (see for instance [2] ). Furthermore, we have  zn hγ (g)(ζ )∂(T (ζ, z)) = 0 lim γ →0 D ζn

since hγ (g)(ζ ) = 0 if ζ is far from A and T is regular outside {ζ = z}. Altogether the claim follows from this immediately.   Remark 3.2. Using the formalism of currents we also can write Proposition 2.1 in the form    n   1 N E g(z) = ∂ ∧ hi (ζ, z) dζi , T1 (ζ, z) , f i=1

where T1 is defined by Proposition 2.1 . If f is linear, we can take hi (ζ, z) =  such that ni=1 hi (ζ, z) dζi = ∂f (ζ ). Hence       1 N ∧ ∂f, T1 (ζ, z) = [f = 0], T1 (ζ, z) E g(z) = ∂ f

∂f ∂ζi (ζ ),

where [f = 0] denotes the current  of integration on {f = 0}. In the case of nonlinear f , the current ∂( f1 ) ∧ ni=1 hi (ζ, z) dζi is a perturbation of the current of integration on {f = 0}, namely, ∂

     n 1 1 hi (ζ, z) dζi = [f = 0] + O(|ζ − z|) ∧ ∂ ∧ . f f i=1

This perturbation is the reason why the extension problem for non-linear varieties in convex domains of finite type has such a different behavior (see [10] ). Finally, for the convenience of the reader, we formulate here the following results concerning the geometry of convex domains of finite type, which will be needed later. They are all contained in [4] and have first been proved in [12] . We use the notations of [4] :

Extension of bounded holomorphic functions in convex domains

7

Proposition 3.3. Let D ⊂⊂ Cn be a C ∞ -smooth convex domain of finite type and ρ a C ∞ convex defining function of D. Let, furthermore, for a point z ∈ D and a sufficiently small number ε > 0, (v1 , . . . , vn ) be an ε-extremal basis at z and  v = nj=1 aj vj ∈ Cn a vector. Then one has   n aj   1 ≈ (3.6) τ (z, v, ε) τj (z, ε) j =1

and, for ε1 ≥ ε2 sufficiently small and v ∈ Cn a unit vector one has  1/m   ε1 ε1 τ (z, v, ε2 ). τ (z, v, ε2 )
τ (z, v, ε1 )
ε2 ε2

(3.7)

Finally, if ζ = z + λv with a unit vector v and d(ζ, z) denotes the pseudodistance between ζ and z, one has |λ|
τ (z, v, d(ζ, z)).

(3.8)

In the inequalities (3.6)–(3.8) all constants are independent of the choices of z, v, ε, ε1 , ε2 . 4. Proof of Theorem 1.2 Since we will have to make several linear coordinate changes in the proof of Theorem 1.2, we prefer to write from now on f (z) instead of zn and ∂f instead of dzn . For the proof of the Theorem we have to show that for all z ∈ ∂D \ A the following inequality holds:    N  E g(z)
gL∞ (D  ) . We associate to every point z ∈ ∂D \ A a point ζ0 ∈ A ∩ D satisfying
 d(ζ0 , z) = inf d(ζ, z) : ζ ∈ A ∩ D and we write d := d(ζ0 , z). We denote by Pr (z) the pseudoball of radius r around z and decompose      N |f (z)g(ζ )K(ζ, z)| E g(z)
D  ∩P1 (z)  |f (z)g(ζ )K(ζ, z)|
D  ∩(P1 (z)\Pd/2 (z))  |f (z)g(ζ )K(ζ, z)| + D  ∩Pd/2 (z)

where K(ζ, z) := n

ρ n+N −1 (ζ )

i=1 Qi (ζ, z) dζi

n−1   1 n ∧ ∂ ρ(ζ ∧ ∂f i=1 Qi (z, ζ ) dζi )

S(ζ, z)(ρ(ζ ) + S(z, ζ ))n+N−1

dλ.

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K. Diederich, E. Mazzilli

Since for ζ ∈ D  always d(ζ, z) ≥ d and, hence, ζ ∈ Pd/2 (z), we get     N  |f (z)g(ζ )K(ζ, z)|. E g(z)
D  ∩(P1 (z)\Pd/2 (z))

(4.1)

We now use the covering family Pεi (z) := C1 P2−i ε (z) \ 21 P2−i ε (z) from (11) in [4]. As stated in (12) of that article we obtain i0 

P1i (z) ⊃ P1 (z) \ Pd/2 (z)

(4.2)

i=0

with i0 < − log2 (c1 d8 ). Furthermore, it has been shown in [7] , that the following estimates hold: Lemma 4.1. One has (i) |S(ζ, z)|  2−i ε (ii) |ρ(ζ ) + S(z, ζ )|  2−i ε for ζ ∈ Pεi (z) ∩ D. Putting (4.2) and Lemma 4.1 into (4.1) we get i0      N  |f (z)K(ζ, z)| E g(z)
gL∞ i=0


gL∞

D  ∩P1i (z)

 i0 i=0



1 n+1

2−i

 D  ∩P1i (z)

|f (z)|

 

n−1   ∧ ∂f  dλ ρ (ζ ) Q (ζ, z) ∧ ∂Q (z, ζ ) +

i0  i=0



1 n+1

2−i

(4.3)

 D  ∩P1i (z)

|f (z)|

  

n−2   ∧ ∂f  dλ Q (ζ, z) ∧ Q (z, ζ ) ∧ ∂ρ (ζ ) ∧ ∂Q(z, ζ ) The final proof of Theorem 1.2 will be based on the estimates of the following Lemma, which will be proved later in this article: Lemma 4.2. For ε > 0 small enough and z ∈ ∂D \ A we have    n−1

  |f (z)|ρ(ζ )Q(ζ, z) ∧ ∂Q(z, ζ ) (i) ∧ ∂f  dλ D  ∩Pε (z)  τ z, ζζ00 −z ,d −z ;
εn+1  , ε τ z, ζζ00 −z −z   

n−2   |f (z)|Q(ζ, z) ∧ Q(z, ζ ) ∧ ∂ρ(ζ ) ∧ ∂Q(z, ζ ) (ii) ∧ ∂f  dλ D  ∩Pε (z)  τ z, ζζ00 −z , d −z .
εn+1  ζ0 −z τ z, ζ0 −z , ε

Extension of bounded holomorphic functions in convex domains

9

In order to finish at first the proof of Theorem 1.2 we apply the estimates (i) and (ii) of Lemma 4.2 with ε = c2−i to the two expression on the very right-hand side of (4.3). We obtain:   i0   ζ0 − z   N  g g(z)
τ z, , d  E L∞ ζ0 − z τ z, i=o

1 ζ0 −z −i ζ0 −z , c2

.

The inequality (3.7) of Proposition 3.3 then leads to  1 −i0  1 i   i0   2m 2m ζ0 − z  N   ,d E g(z)
gL∞ τ z, ζ0 −z ζ0 − z −i i=0 τ z, ζ0 −z , c2 0  τ z, ζζ00 −z , d −z .
gL∞  ζ0 −z τ z, ζ0 −z , c2−i0 As stated after (4.2), we have 2−i0 > c1 d8 , hence, 

2−i0 c1 d/8

  m1    ζ0 − z ζ0 − z , cc1 d/8
τ z, , c2−i0 , τ z, ζ0 − z ζ0 − z

in other words, we have 1 1 1
 
 . ζ0 −z ζ −z ζ τ z, ζ0 −z , c2−i0 τ z, ζ00 −z , cc1 d/8 τ z, ζ00 −z −z , d Altogether we get

    N E g(z)
gL∞ .

Theorem 1.2 is proved.   5. Proof of Lemma 4.2 The inequality (i) of the Lemma follows exactly as (ii) if one uses in addition, that for z ∈ ∂D and ζ ∈ Pε (z) always |ρ(ζ )|
ε holds. We will, therefore, just prove (ii). We let (v1 , . . . , vn ) be an ε-extremal basis at z and denote by (v1∗ , . . . , vn∗ ) the corresponding coordinates. Then we have

−1 v ∗ = 5−1 (ζ − z) = 5−1 ◦ t(ζ ) = 5∗ (ζ ), where 5 is a unitary transformation and t the translation by −z. It follows from Lemma 5.1 of [4] , that the integral in (ii) is dominated by a sum of terms of the form  A dλ, ε n+1 D  ∩Pε (z)

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K. Diederich, E. Mazzilli

where

      

 n ∗ ∗ ∧ ∂ (f ◦ 5∗ ) f ◦ 5∗ v ∗ − f ◦ 5∗ (0)  n dv dv  j =1 j =1 0 j j n

A=

j =1 τj (z, ε)

j =k

n

j =1 τj (z, ε)

,

j =k

v0∗

where denotes the coordinates of ζ0 in the ε-extremal base and k ∈ {1, . . . , n}. Since f is linear we have   n  ∂(f ◦ 5∗ )   

   f ◦ 5∗ v ∗ − f ◦ 5∗ (0) =   0 ∗   ∂vi i=1

We will show that for all i ∈ {1, . . . , n}          ∂ (f ◦5∗ ) n n ∗ ∗   ∂v ∗  j =1 dvj∗ j =1 dvj ∧ ∂(f ◦ 5 )  i   j =k n n dλ D  ∩Pε (z) j =1 τj (z, ε) j =1 τj (z, ε) 

ζ0 −z ζ0 −z , d

j =k



τ z, .
 τ z, ζζo0 −z −z , ε This will, in fact, finish the proof of Lemma 4.2. – For this, we call the integral on ∗) ∗ = 0, there is nothing to = 0 or if vi,0 the left side of this expression I . If ∂(f∂v◦5 ∗ ∗

i

) ∗ = 0. We get prove. So we may assume that ∂(f∂v◦5

= 0 and vi,0 ∗ i          ∗  n n ∗ ∗ ∗ ∗ ) dv dv ∧ ∂(f ◦ 5 ∧ ∂(f ◦ 5 ) v   j =1  i,0  j =1 j j  j =i j =k n n I
dλ. D  ∩Pε (z) j =1 τj (z, ε) j =1 τj (z, ε) j =k

(5.1) Let us put vj∗ = xj + iyj and denote by δi the Kroneckersymbol at i. The right side of (5.1) is then dominated by  n  ∗ 

 v | dxj ∧ dyj ∧ ( dxi − i dyi ) ∧ ( dxk + i dyk ) I ≤ (1 − δi (k)) i,0 D  ∩Pε (z)

j =1 j =i j =k

d Re (f ◦ 5∗ ) ∧ d Im (f ◦ 5∗ ) n ∧ n | dλ j =1 τ (z, ε) j =1j =k τj (z, ε)      ∗  n

∗ ∗   vi,0  j =1 dxj ∧ dyj ∧ d Re (f ◦ 5 ) ∧ d Im (f ◦ 5 ) j =i n n dλ. + δi (k) j =1 τj (z, ε) j =1 τj (z, ε) D  ∩Pε (z)

j =i

(5.2)

Extension of bounded holomorphic functions in convex domains

11

The two terms of this sum can be treated in a similar way. Therefore, we consider here only the second term, meaning the last line of the right side. We observe, that the canonical volume form of the normal bundle of {f ◦ 5∗ = 0} is given by d Re(f ◦5∗ )∧ d Im(f ◦5∗ ). Hence we can estimate dλ by (see, for example, [13], p. 247–248)  dx1 dy1 · · · dx i dyi · · · dxn dyn  .     n

  j =1 dxj ∧ dyj ∧ d Re (f ◦ 5∗ ) ∧ d Im (f ◦ 5∗ )  j =i  Based on this, we obtain after integration for the second term I2 of the sum in (5.2) an upper estimate which can be written as     ∗  ∗  vi,0  v0 I2
 ∗  . v τi (z, ε) 0 Finally, using (3.6) and (3.8) of Proposition 3.3 together with the fact, that ∗  vi,0 i v ∗  vi , we obtain 0

ζ0 −z ζ0 −z

=

 ζ0 −z , d τ z, ζ0 −z ζ0 − z o = 
 . I2
 ζ0 −z ζ0 −z τ z, ζ0 −z , ε τ z, ζ0 −z , ε τ z, ζζ00 −z , ε −z  ∗ v 

This finishes the proof of Lemma 4.2 .   References [1] Amar, E.: Extension de fonctions holomorphes et courants. Bull. Sci. math. 107, 25–48 (1983) [2] Berndtsson, B.: A formula for interpolation and division in Cn . Math. Ann. 263, 399– 418 (1983) [3] Berndtsson, B., Andersson, M.: Henkin–Ramirez formulas with weight factors, Ann. Inst. Fourier 32, 91–110 (1982) [4] Diederich, K., Fischer, B., Fornæss, J.E.: Hölder estimates on convex domains of finite type. Math. Z. 232, 43–61 (1999) [5] Diederich, K., Fornæss, J.E.: Support functions for convex domains of finite type. Math. Z. 230, 145–164 (1999) [6] Diederich, K., Mazzilli, E.: Extension and restriction of holomorphic functions. Ann. Inst. Fourier 47, 1079–1099 (1997) [7] Diederich, K., Mazzilli, E.: Zero varieties for the Nevanlinna class on all convex domains of finite type. Preprint, 1998 [8] Henkin, G. M.: Continuation of bounded holomorphic functions from submanifolds in general position to strictly pseudoconvex domains. Math.USSR Izvestija 6, 536–563 (1972) [9] Mazzilli, E.: Extension des fonctions holomorphes. C. R. Acad. Sci. Paris 321, 837–841 (1995)

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