William Alexandre
email: william.alexandre@math.univ-lille1.fr

Emmanuel Mazzilli
email: emmanuel.mazzilli@math.univ-lille1.fr

Laboratoire Paul

DIVISION OF HOLOMORPHIC FUNCTIONS AND GROWTH CONDITIONS

Let D be a strictly convex domain of C n , f1 and f2 be two holomorphic functions defined on a neighborhood of D and set X l = {z, f l (z) = 0}, l = 1, 2. Suppose that X l ∩bD is transverse for l = 1 and l = 2, and that X1 ∩X2 is a complete intersection. We give necessary conditions when n ≥ 2 and sufficient conditions when n = 2 under which a function g can be written as g = g1f1+g2f2 with g1 and g2 in L q (D), q ∈ [1, +∞), or g1 and g2 in BM O(D). In order to prove the sufficient condition, we explicitly write down the functions g1 and g2 using integral representation formulas and new residue currents.

Introduction

In this article, we are interested in ideals of holomorphic functions and corona type problems. More precisely, if D is a domain of C n and f 1 , . . . , f k are k holomorphic functions defined in a neighborhood of D, we are looking for condition(s), as close as possible to being necessary and sufficient, under which a function g, holomorphic on D, can be written as

g = f 1 g 1 + . . . + f k g k , (1)
with g 1 , . . . , g k holomorphic on D and satisfying growth conditions at the boundary of D. This kind of problem has been widely studied by many authors under different assumptions.

When D is strictly pseudoconvex and when f 1 , . . . , f k are holomorphic and bounded functions on D, which satisfy |f | 2 = |f 1 | 2 + . . . + |f k | 2 ≥ δ 2 > 0, for a given holomorphic and bounded function g, finding functions g 1 , . . . , g k bounded on D is a question known as the Corona Problem. When D is the unit ball of C, the Corona Problem was solved in 1962 by Carleson in [8]. This question is still open for n > 1, even for two generators f 1 and f 2 , and even when D is the unit ball of C n .

For p ∈ [1, +∞), we denote by H p (D) the Hardy space of D. When n > 1, k = 2 and |f | ≥ δ > 0, Amar proved in [START_REF] Amar | On the corona problem[END_REF] that for any g ∈ H p (D), (1) can be solved with g 1 and g 2 in H p (D). Andersson and Carlsson in [START_REF] Andersson | Wolf type estimates and the H p corona problem in strictly pseudoconvex domains[END_REF] generalized this result to any strictly pseudoconvex domain in C n and to any k ≥ 2 and also obtained the BM O-result already announced by Varopoulos in [START_REF] Varopoulos | BMO functions and the ∂-equation[END_REF]. In [START_REF] Andersson | Estimates of solutions of the H p and BMOA corona problem[END_REF], they studied the dependence of the g i 's on the lower bound δ of |f | and they explicitly obtained a constant c δ such that for all i, g i H p (D) ≤ c δ g H p (D) . Of course c δ goes to infinity when δ goes to 0. In [START_REF] Amar | On H p -solutions of the Bezout equation in the ball[END_REF], when |f | does not have a positive 2000 Mathematics Subject Classification. 32A22, 32A26, 32A27, 32A37, 32A40, 32A55. The first author is partially supported by A.N.R. BL-INTER09-CRARTIN. lower bound, Amar and Bruna formulated a sufficient condition in term of the admissible maximum function of |f | 2 |log |f || 2+ε , ε > 0, under which the g i 's belong to H p (D).

The corona problem was also studied in the case of the Bergman space A p (D), the space of holomorphic functions which belong to L p (D), and in the case of the Zygmund space Λ γ (D) by Krantz and Li in [START_REF] Krantz | Some remarks on the corona problem on strongly pseudoconvex domains in C n[END_REF], and in the case of Hardy-Sobolev spaces by Fàbrega and Ortega in [START_REF] Fàbrega | Multipliers in Hardy-Sobolev spaces[END_REF].

In the above papers, the first step of the proof in the case of two generators f 1 and f 2 , is to find two smooth functions on D, ϕ 1 and ϕ 2 , such that

ϕ 1 f 1 + ϕ 2 f 2 = 1;
(2) and then to solve the equation

∂ϕ = g f 1 ∂ϕ 2 -f 2 ∂ϕ 1 |f 1 | 2 + |f 2 | 2 . ( 3
)
Then setting g 1 = gϕ 1 + ϕf 2 and g 2 = gϕ 2 -ϕf 1 , (1) holds and, provided ϕ belongs to the appropriate space, g 1 and g 2 will belong to H p (D), A p (D), . . . So the problem is reduced to solve the Bezout equation ( 2) and then to solve the ∂-equation (3) with an appropriate regularity.

In [START_REF] Andersson | H p -estimates of holomorphic division formulas[END_REF], Andersson and Carlsson used an alternative technique. They constructed a division formula g = f 1 T 1 (g) + . . . + f k T k (g) where for all i, T i is a well chosen Berndtsson-Andersson integral operator, and, still under the assumption |f | ≥ δ > 0, they proved that T i (g) belongs to H p (D) (resp. BM O(D)) when g belongs to H p (D) (resp. BM O(D)). The same kind of technics was also used in [START_REF] Bonneau | Division dans les espaces de Lipschitz de fonctions holomorphes, Séminaire d'[END_REF] by Bonneau, Cumenge and Zériahi who studied the equation [START_REF] Alexandre | Extension with growth estimates of holomorphic functions defined on singular analytic spaces[END_REF] in Lipschitz spaces and in the space B M (D) = {g, g B M (D) = sup z∈D |g(z)|d(z, bD) M < ∞}. In this later work, the generators f 1 , . . . , f k may have common zeroes but ∂f 1 ∧ . . . ∧ ∂f k can not vanish on bD ∩ {z, f 1 (z) = . . . = f k (z) = 0}.

The case of generators having common zeroes has also been investigated by Skoda in [START_REF] Skoda | Application des techniques L 2 à la théorie des idéaux d'une algèbre de fonctions holomorphes avec poids[END_REF] for weighted L 2 -spaces. Using and adapting the L 2 -techniques developed by Hörmander, for D pseudoconvex in C n , ψ a plurisubharmonic weight on D, f 1 , . . . , f k holomorphic in D, q = inf(n, k), α > 1 and g holomorphic in D such that D |g| 2 |f | 2αq+2 e -ψ < ∞, Skoda showed that there exist g 1 , . . . , g k ∈ O(D) such that (1) holds and such that for all i,

D |g i | 2 |f | 2αq e -ψ ≤ α α-1 D |g| 2
|f | 2αq+2 e -ψ . Moreover the result also holds when k is infinite and there is no restriction on ∂f 1 , . . . , ∂f k . However, if one takes g = f 1 for example, g does not satisfy the assumption of Skoda's theorem in general.

In this article we restrict ourself to a strictly convex domain D of C n and we consider the case of two generators f 1 and f 2 , holomorphic in a neighborhood of D. We denote by X 1 the set X 1 = {z, f 1 (z) = 0}, and by X 2 the set X 2 = {z, f 2 (z) = 0}. We assume that the intersections X 1 ∩ bD and X 2 ∩ bD are transverse in the sense of tangent cones and that X 1 ∩ X 2 is a complete intersection. Let us recall that an analytic subset A of pure co-dimension m in C n is said to be a complete intersection if there are m holomorphic functions h 1 , . . . , h m such that A = ∩ m i=1 {z, h i (z) = 0}; and that the intersection X l ∩ D, l = 1 or l = 2, is said to be transverse if for every p ∈ X l ∩ bD, the complex tangent space to bD at p and the tangent cone to X l at p span T p C n . Our goal here is to find assumptions on g, holomorphic in D, as close as possible to being necessary and sufficient, under which we can write g as g = g 1 f 1 + g 2 f 2 with g 1 and g 2 holomorphic and belonging to BM O(D) or L q (D), q ∈ [1, +∞).

Let us write D as D = {z ∈ C n , ρ(z) < 0} where ρ is a smooth strictly convex function defined on C n such that the gradient of ρ does not vanish in a neighborhood U of bD. We denote by D r , r ∈ R, the set D r = {z ∈ C n , ρ(z) < r}, by η ζ the outer unit normal to bD ρ(ζ) at a point ζ ∈ U and by v ζ a smooth unitary complex vector field tangent at ζ to bD ρ(ζ) . As a first result, we show: Theorem 1.1. Let D be a strictly convex domain of C 2 , f 1 and f 2 be two holomorphic functions defined on a neighborhood of D and set X l = {z, f l (z) = 0}, l = 1, 2. Suppose that X l ∩ bD is transverse for l = 1 and l = 2, and that X 1 ∩ X 2 is a complete intersection. Then there exist two integers k 1 , k 2 ≥ 1 depending only on f 1 and f 2 such that if g is any holomorphic function on D which belongs to the ideal generated by f 1 and f 2 and for which there exist two C ∞ smooth functions g1 and g2 such that (i) g = g1 f 1 + g2 f 2 on D, (ii) there exists N ∈ N such that |ρ| N g1 and |ρ| N g2 vanish to order k 2 on bD, (iii) there exists q ∈ [1, +∞] such that for l = 1, 2, ∂ α+β gl

∂η ζ α ∂v ζ β |ρ| α+ β 2 belongs to L q (D) for all non-negative integers α and β with α + β ≤ k 1 , then there exist two holomorphic functions g 1 , g 2 on D which belong to L q (D) if q < +∞ and to BM O(D) if q = +∞, such that g 1 f 1 + g 2 f 2 = g on D.
The number k 1 and k 2 are almost equal to the maximum of the multiplicities of the singularity of X 1 and X 2 . The functions g 1 and g 2 will be obtained via integral operators acting on g1 and g2 . These operators are a combination of a Berndtsson-Andersson kernel and of two (2,2)-currents T 1 and T 2 such that f 1 T 1 + f 2 T 2 = 1. So instead of first solving the Bezout equation [START_REF] Amar | On the corona problem[END_REF] in the sense of smooth functions, we solve it in the sense of currents and then, instead of solving a ∂-equation, we "holomorphy" the smooth solutions g1 and g2 of the equation g = g1 f 1 + g2 f 2 with integral operators using T 1 and T 2 . As we will see in Section 4, these operators can be constructed starting from any currents T1 and T2 such that f 1 T1 + f 2 T2 = 1. However, not all such currents will give operators such that g 1 and g 2 belongs to L q (D) or BM O(D); as we will see in Section 3, they have to be constructed taking into account the interplay between X 1 and X 2 . Moreover, if g1 and g2 are already holomorphic and satisfy the assumptions (i) -(iii) of Theorem 1.1, then g 1 = g1 and g 2 = g2 .

Observe that in Theorem 1.1, we do not make any assumption on f 1 or f 2 except that the intersection X 1 ∩ bD and X 2 ∩ bD are transverse in the sense of tangent cones, and that X 1 ∩ X 2 is a complete intersection. This later assumption can be removed provided we add a fourth assumption on g1 and g2 . If we moreover assume that

(iv) ∂ α+β g1 ∂η ζ α ∂v ζ β = 0 on X 2 ∩ D and ∂ α+β g2 ∂η ζ α ∂v ζ β = 0 on X 1 ∩ D for all non negative integers α and β with 0 < α + β ≤ k 1 ,
then Theorem 1.1 also holds whenever X 1 ∩ X 2 is not complete. However, it then becomes very difficult to find g1 and g2 which satisfy this fourth assumption, except if X 1 ∩ X 2 is actually complete. Indeed, the main difficulty in order to be able to apply Theorem 1.1 is to find the two functions g1 and g2 satisfying (i)-(iii). The canonical choice when |f | ≥ δ > 0 is to set g1 = gf 1 |f | -2 and g2 = gf 2 |f | -2 . If |f | ≥ δ > 0 and if g belongs to L q (D), then g1 and g2 will satisfy (i)-(iii) and we can then apply Theorem 1.1. However, if |f | does not admit a positive lower bound, this will not be necessarily the case. For example, when

D = {z ∈ C 2 , ρ(z) = |z 1 -1| 2 + |z 2 | 2 -1 < 0}, f 1 (z) = z 2 , f 2 (z) = z 2 -z 2
1 and g = f 1 , we can obviously find g1 and g2 which satisfy the assumption of Theorem 1.1 but if we make the canonical choices for g1 and g2 , they do not fulfill (iii) for q = ∞ because ∂g 1 ∂z 2 |ρ| 1 2 is not bounded near 0.

Therefore the question of the existence of g1 and g2 may itself become a problem that we have to solve. Using first Koranyi balls, we will reduce this global question to a local one and then, using divided differences, we will give numerical conditions under which there indeed exist functions satisfying the hypothesis of Theorem 1.1. We will also prove that these conditions are necessary in order to solve Equation (1) with the g i 's belonging to L q (D), q ∈ [1, +∞], even in C n . This leads us to an effective way of construction of the solutions of (1) belonging to L q (D) or BM O(D).

The Koranyi balls in C 2 are defined as follows. We call the coordinates system centered at

ζ of basis η ζ , v ζ the Koranyi coordinates at ζ. We denote by (z * 1 , z * 2 ) the coordinates of a point z in the Koranyi coordinates at ζ. The Koranyi ball centered in ζ of radius r is the set P r (ζ) := {ζ + λη ζ + µv ζ , |λ| < r, |µ| < r 1 2 }. We observe that, by convexity, P ακ|ρ(ζ)| (ζ) is included in D if α is small enough.
The following theorem enables us to go from a local division formula in L ∞ to a global division formula in BM O.

Theorem 1.2. Let D be a strictly convex domain of C 2 , f 1 and f 2 be two holomorphic functions defined on a neighborhood of D and set X l = {z, f l (z) = 0}, l = 1, 2. Suppose that X 1 ∩ bD and X 2 ∩ bD are transverse, and that X 1 ∩ X 2 is a complete intersection. Let g be a function holomorphic on D and assume that there exists κ > 0 such that for all z ∈ D, there exist two functions ĝ1 and ĝ2 , depending on z, C ∞ -smooth on P κ|ρ(z)| (z), such that (a) g = ĝ1 f 1 + ĝ2 f 2 on P κ|ρ(z)| (z); (b) for all non negative integers α, β, α and β, there exist c > 0, not depending on z,

such that sup P κ|ρ(z)| (z) ∂ α+α+β+β ĝl ∂z * 1 α ∂z * 2 β ∂z * 1 α ∂ζ * 2 β ≤ c|ρ(z)| -α-β 2 for l = 1 and l = 2.
Then there exist two smooth functions g1 and g2 which satisfy the assumptions (i)-(iii) of Theorem 1.1 for q = +∞.

An analogous theorem holds true in the L q -case (see Theorem 6.1). We observe that if, for all z ∈ D, there exist two functions ĝ1 and ĝ2 , holomorphic and bounded on P 2κ|ρ(z)| (z) by a constant c which does not depend on z, and such that g = ĝ1 f 1 + ĝ2 f 2 on P 2κ|ρ(z)| (z), then Cauchy's inequalities implies that ĝ1 and ĝ2 satisfy the assumption of Theorem 1.2 on P κ|ρ(z)| (z) for all z. Therefore Theorem 1.2 implies that the global solvability of (1) in the BM O space of D is nearly equivalent to its uniform local solvability. In order to prove Theorem 1.2, we will cover D with Koranyi balls and using a suitable partition of unity, we will glue together the ĝ1 and ĝ2 which we got on each ball. We point out that when we glue together the local ĝ1 's, except if X 1 ∩ X 2 is a complete intersection, in general the "fourth" assumption (iv) of Theorem 1.1 is not satisfied. This is why we chose to present Theorem 1.1 as we did.

When looking for necessary conditions in order to solve Equation (1) with g 1 and g 2 bounded, we first observe that g is trivially bounded by max(

g 1 L ∞ , g 2 L ∞ )(|f 1 | + |f 2 |).
Therefore, in order for g to be written as g = g 1 f 1 + g 2 f 2 with g 1 and g 2 bounded, it is necessary that |g| |f 1 |+|f 2 | be bounded. However this condition alone does not suffice in general. Consider for example the ball

D := {z ∈ C 2 , ρ(z) = |z 1 -1| 2 + |z 2 | 2 -1 < 0}, f 1 (z) = z 2 2 , f 2 (z) = z 2 2 -z q 1 and g(z) = z q 2
1 z 2 where q ≥ 3 is an odd integer. Then

g(z) = z 2 z -q 2 1 f 1 (z) -z 2 z -q 2
1 f 2 (z), so g belongs to the ideal generated by f 1 and f 2 , and |g| |f 1 |+|f 2 | is bounded on D by 3 2 ; in particular, the classical choice g1 =

gf 1 |f 1 | 2 +|f 2 | 2 and g2 = gf 2 |f 1 | 2 +|f 2 | 2
gives two functions g1 and g2 which are smooth and bounded on D. However, (1) can not be solved with g 1 and g 2 bounded on D. In order to see this, a good tool is divided differences. On the one hand, if

g = g 1 f 1 + g 2 f 2 , then g 1 = g • f -1 1 on X 2 \ X 1 .
On the other hand, for all z ∈ D, all unit vector v tangent to bD -ρ(z) at z, all complex numbers λ 1 and λ 2 with ρ(z + λ 1 v) < ρ(z) and ρ(z + λ 2 v) < ρ(z), Montel [START_REF] Montel | Sur une formule de Darboux et les polynômes d'interpolation[END_REF] asserts that the modulus of the divided difference g 1 (z+λ 1 v)-g 1 (z+λ 2 v) λ 1 -λ 2 behaves like ∂g 1 ∂v at some point z + µv where µ is an element of the segment [λ 1 , λ 2 ]. Cauchy's inequalities then imply that, up to a uniform multiplicative constant,

g 1 (z+λ 1 v)-g 1 (z+λ 2 v) λ 1 -λ 2 is bounded by g 1 L ∞ (D) |ρ(z)| -1
2 . So when we compute the divided differences of g 1 at points z + λ 1 v and z + λ 2 v which belong to X 2 \ X 1 , whatever g 1 and g 2 may be, we actually compute the divided difference of g • f -1

1 . And if g 1 is bounded, this divided difference times |ρ(z)| 1 2 must be bounded by some uniform constant. But in our example, this is not the case because for small ε > 0, setting z = (ε, 0), v = (0, 1), λ 1 = ε q 2 and λ 2 = -ε q 2 , we have that

(g•f -1 1 )(z+λ 1 v)-(g•f -1 1 )(z+λ 2 v) λ 1 -λ 2 |ρ(z)| 1 2 = ε 1-q 2
which is unbounded when ε goes to zero. In C n , we will prove that the divided differences of any order of g

• f 1 -1 and g • f 2 -1
must satisfy some boundedness properties when (1) is solvable with g 1 and g 2 in L q (D), q ∈ [1, +∞] (see Theorems 6.3 and 6.5 for precise statements). Conversely, in C 2 , if those boundedness properties are satisfied, by polynomial interpolation and on any Koranyi balls, we construct two functions g1 and g2 which satisfy the assumptions of Theorem 1.2.

It must be mentioned that the error term we will get during the interpolation process will be very difficult to handle. Although the interpolation procedure is a holomorphic one, we will not get two holomorphic functions g1 and g2 because we will have to split the error term in a appropriate way in two parts, which will lead to C ∞ -smooth but not holomorphic functions. Then it will follow from Theorem 1.1 that there exist two functions g 1 and g 2 holomorphic on D, belonging to BM O(D) such that g = g 1 f 1 + g 2 f 2 . An analogue result for holomorphic functions in L q (D), q ∈ [1, +∞), will be also proved. These two results are precisely stated in Theorem 6.4 and 6.6.

The article is organized as follows. In Section 2, we recall some tools needed for the construction and the estimation of the division formula. In Section 3, we construct the currents which enable us to construct our division formula in Section 4. In Section 5 we establish Theorem 1.1 and finally, in Section 6, we prove the theorems related to local division in the L ∞ and L q case.

Notations and tools

2.1. Koranyi balls. The Koranyi balls centered at a point z in D have properties linked with distance from z to the boundary of D in a direction v. They were generalized in the case of convex domains of finite type by McNeal in [START_REF] Mcneal | Convex domains of finite type[END_REF] and [START_REF] Mcneal | Estimates on the Bergman kernels of convex domains[END_REF]. A strictly convex domain being in particular a convex domain of finite 2, we will adopt the formalism of convex domain of finite type. For z ∈ C n , v a unit vector in C n , and ε > 0, the distance from z to bD ρ(z)+ε in the direction v is defined by

τ (z, v, ε) = sup{τ > 0, ρ(z + λv) -ρ(z) < ε for all λ ∈ C, |λ| < τ }. Thus τ (z, v, ε) is the maximal radius r > 0 such that the disc ∆ z,v (r) = {z + λv, |λ| < r} is included in D ρ(z)+ε ; if v is a tangent vector to bD ρ(z) at z, then τ (z, v, ε) is comparable to ε 1 2 and τ (z, η z , ε) is comparable to ε.
Before we recall the properties of the Koranyi balls we will need, we adopt the following notation. We write A B if there exists some constant c > 0 such that A ≤ cB. Each time we will mention on which parameters c depends. We will write A B if A B and B

A both holds. The following propositions are part of well known properties of Koranyi balls and McNeal polydiscs. The interested reader can find a proof of each statements in [START_REF] Mcneal | Convex domains of finite type[END_REF] in the case of convex domains of finite type, keeping in mind that a strictly convex domain is a convex domain of type 2.

Proposition 2.1. There exists a neighborhood U of bD and positive real numbers κ and c 1 such that

(i) for all ζ ∈ U ∩ D, P 4κ|ρ(ζ)| (ζ) is included in D. (ii) for all ε > 0, all ζ, z ∈ U, P ε (ζ) ∩ P ε (z) = ∅ implies P ε (z) ⊂ P c 1 ε (ζ). (iii) for all ε > 0 sufficiently small, all z ∈ U, all ζ ∈ P ε (z) we have |ρ(z) -ρ(ζ)| ≤ c 1 ε. (iv) for all ε > 0, all unit vectors v ∈ C n , all z ∈ U and all ζ ∈ P ε (z), τ (z, v, ε) τ (ζ, v, ε)
uniformly with respect to ε, z and ζ.

For U given by Proposition 2.1 and z and ζ belonging to U, we set δ(z, ζ) = inf{ε > 0, ζ ∈ P ε (z)}. Proposition 2.1 implies that δ is a pseudo-distance in the following sense: Proposition 2.2. For U and c 1 given by Proposition 2.1 and for all z, ζ and ξ belonging to U we have

1 c 1 δ(ζ, z) ≤ δ(z, ζ) ≤ c 1 δ(ζ, z) and δ(z, ζ) ≤ c 1 (δ(z, ξ) + δ(ξ, ζ)) 2.2. Berndtsson-Andersson reproducing kernel in C 2 .
Berndtsson-Andersson's kernel will be one of our most important ingredients in the construction of the functions g 1 and g 2 of Theorem 1.1. We now recall its definition for D a strictly convex domain of

C 2 . We set h 1 (ζ, z) = -1 2 ∂ρ ∂ζ 1 (ζ), h 2 (ζ, z) = -1 2 ∂ρ ∂ζ 2 (ζ), h = i=1,2 h i dζ i and h = 1 ρ h. For a (1, 0)-form β(ζ, z) = i=1,2 β i (ζ, z)dζ i we set β(ζ, z), ζ -z = i=1,2 β i (ζ, z)(ζ i -z i ).
Then we define the Berndtsson-Andersson reproducing kernel by setting for an arbitrary positive integer N , n = 1, 2 and all ζ, z ∈ D:

P N,n (ζ, z) = C N,n 1 1 + h(ζ, z), ζ -z N +n ∂ h n ,
where C N,n ∈ C is a suitable constant. We also set P N,n (ζ, z) = 0 for all z ∈ D and all ζ / ∈ D. Then the following theorem holds true (see [START_REF] Berndtsson | Henkin-Ramirez formulas with weight factors[END_REF]):

Theorem 2.3. For all g ∈ O(D) ∩ C ∞ (D) we have g(z) = D g(ζ)P N,2 (ζ, z).
In order to find an upper bound for this kernel, we will need lower bound for 1 + h(ζ, z), ζ -z . This classical bound in the field is given by the following proposition. We include its proof for the reader convenience.

(z) -ρ(ζ) ≥ 2Re (∂ρ(ζ) • (z -ζ)) + c|ζ -z| 2 = 4Re h(ζ, z), ζ -z + c|ζ -z| 2 . ( 4
)
If Re λ < 0, we get from ( 4)

|ρ(ζ) + h(ζ, z), ζ -z | ≥ -ρ(ζ) -Re h(ζ, z), ζ -z + |Im h(ζ, z), ζ -z | -ρ(z) -ρ(ζ) + c|ζ -z| 2 + |λ| δ(ζ, z) + |ρ(ζ)| + |ρ(z)|. If Re λ > 0, (4) now yields |ρ(ζ) + h(ζ, z), ζ -z | -ρ(ζ) -2Re h(ζ, z), ζ -z + Re h(ζ, z), ζ -z + |Im h(ζ, z), ζ -z | -ρ(z) -ρ(ζ) + c|ζ -z| 2 + |λ| δ(ζ, z) + |ρ(ζ)| + |ρ(z)|.
We will also need an upper bound for h and thus for h. In order to get this bound, for a fixed z ∈ D, we write h in the Koranyi coordinates at z. We denote by

(ζ * 1 , ζ * 2 ) the Koranyi coordinates of ζ at z. We set h * 1 = -1 2 ∂ρ ∂ζ * 1 (ζ) and h * 2 = -1 2 ∂ρ ∂ζ * 2 (ζ) so that h(ζ, z) = i=1,2 h * i (ζ, z)dζ * i .
The following Proposition is then a direct consequence of the smoothness of ρ. Proposition 2.5. For all ζ ∈ P ε (z) we have uniformly with respect to z, ζ and ε

(i) |h * 1 (ζ, z)| 1, |h * 2 (ζ, z)| ε 1 2 , (ii) ∂h * k ∂ζ * l (ζ, z) , ∂h * k ∂ζ * l (ζ, z) 1 for k, l ∈ {1, 2}.

Construction of the currents

If f 1 and f 2 are two holomorphic functions near the origin in C n , Mazzilli constructed in [START_REF] Mazzilli | Courants du type résiduel attachés à une intersection complète[END_REF] two currents T and S such that f 1 T = 1, f 2 S = ∂T and f 1 S = 0 on a sufficiently small neighborhood U of 0. He also proved that if T and S are any currents satisfying these three hypothesis, then any function g holomorphic on U can be written as g = f 1 g 1 + f 2 g 2 on U if and only if g∂S = 0. Moreover, g 1 and g 2 can be explicitly written down using T and S.

Here, when f 1 and f 2 are holomorphic on a domain D, we first want to obtain a decomposition g = g 1 f 1 + g 2 f 2 on the whole domain D and then secondly we want to obtain growth estimates on g 1 and g 2 . As a first approach, we could try to globalize the currents T and S of [START_REF] Mazzilli | Courants du type résiduel attachés à une intersection complète[END_REF] in order to have a global decomposition. However, such an approach would fail to give the growth estimates we want.

In [START_REF] Mazzilli | Courants du type résiduel attachés à une intersection complète[END_REF], f 1 plays a leading role and T is constructed independently of f 2 , using only f 1 . Then S is constructed using f 1 and f 2 . If we assume for example that f 1 vanishes at a point ζ 0 near bD, because T is constructed independently of f 2 , it seems difficult to prove that g 1 obtained using T is bounded except if we require that g vanishes at ζ 0 too; but considering g = f 2 , we easily see that in general this condition is not necessary when one wants to write g as g = g 1 f 1 + g 2 f 2 with g 1 and g 2 bounded for example. So the currents in [START_REF] Mazzilli | Courants du type résiduel attachés à une intersection complète[END_REF] probably do not give a good decomposition.

Actually, it appears that the role of f 2 must be emphasized in the construction of the currents near a boundary point ζ 0 such that f 1 (ζ 0 ) = 0 and f 2 (ζ 0 ) = 0, or more generally when f 2 is in some sense greater than f 1 and conversely. Following this idea, we construct two currents T 1 and T 2 such that f 1 T 1 + f 2 T 2 = 1 on D. These currents are defined locally and using a suitable partition of unity we glue together the local currents and get a global current. We now define these local currents.

Let ε 0 be a small positive real number to be chosen later and let ζ 0 be a point in D. We distinguish three cases.

First case: If ζ 0 belongs to D -ε 0 , i.e. if ζ 0 is far from the boundary, we do not need to be careful. Using Weierstrass' preparation theorem when ζ 0 belongs to X 1 , we write

f 1 = u 0,1 P 0,1 where u 0,1 is a non vanishing holomorphic function in a neighborhood U 0 ⊂ D -ε 0 2 of ζ 0 and P 0,1 (ζ) = ζ i 0,1 2 + ζ i 0,1 -1 2 a (1) 0,1 (ζ 1 ) + . . . + a (i 0,1 ) 0,1 (ζ 1 ), a (k)
0,1 holomorphic on U 0 for all k. If ζ 0 does not belong to X 1 , we set P 0,1 = 1, i 0,1 = 0, u 0,1 = f 1 and we still have f 1 = u 0,1 P 0,1 with u 0,1 which does not vanish on some neighborhood U 0 of ζ 0 .

For a smooth (2, 2)-form ϕ compactly supported in U 0 we set

T 0,1 , ϕ = 1 c 0 U 0 P 0,1 (ζ) f 1 (ζ) ∂ i 0,1 ϕ ∂ζ i 0,1 2 (ζ), T 0,2 , ϕ = 0,
where c 0 is a suitable constant (see [START_REF] Mazzilli | Courants du type résiduel attachés à une intersection complète[END_REF]). Integrating by parts we get

f 1 T 0,1 + f 2 T 0,2 = 1 on U 0 . Second case: If ζ 0 belongs to bD \ (X 1 ∩ X 2 ), i.e. if ζ 0 is "far" from X 1 ∩ X 2 , without restriction we assume that f 1 (ζ 0 ) = 0. Let U 0 be a neighborhood of ζ 0 such that f 1 does not vanish in U 0 . As in the first case when f 1 (ζ 0 ) = 0, we set P 0,1 = 1, i 0,1 = 0, u 0,1 = f 1 and for any smooth (2, 2)-form ϕ compactly supported in D ∩ U 0 we put T 0,1 , ϕ = 1 c 0 U 0 P 0,1 (ζ) f 1 (ζ) ∂ i 0,1 ϕ ∂ζ i 0,1 2 (ζ), T 0,2 , ϕ = 0.
where as previously c 0 is a suitable constant. Again, we have

f 1 T 0,1 +f 2 T 0,2 = 1 on U 0 ∩D. Third case: If ζ 0 belongs to X 1 ∩ X 2 ∩
bD, the situation is more intricate. As in [START_REF] Alexandre | Extension with growth estimates of holomorphic functions defined on singular analytic spaces[END_REF], we cover a neighborhood U 0 of ζ 0 by a family of polydiscs P κ|ρ(z j,k )| (z j,k ), j ∈ N and k ∈ {1, . . . , n j } such that:

(i) For all j ∈ N, and all k ∈ {1, . . . , n j }, z j,k belongs to bD -(1-cκ) j ε 0 where c is small positive real constant. (ii) For all j ∈ N, all k, l ∈ {1, . . . , n j }, k = l, we have δ(z j,k , z j,l ) ≥ cκ(1 -cκ) j ε 0 . (iii) For all j ∈ N, all z ∈ bD -(1-cκ) j ε 0 , there exists k ∈ {1, . . . , n j } such that δ(z, z j,k

) < cκ(1 -cκ) j ε 0 , (iv) D ∩ U 0 is included in ∪ +∞ j=0 ∪ n j k=1 P κ|ρ(z j,k )| (z j,k ), (v) there exists M ∈ N such that for z ∈ D \ D -ε 0 , P 4κ|ρ(z)| (z) intersect at most M Koranyi balls P 4κ|ρ(z j,k )| (z j,k ).
Such a family of polydiscs will be called a κ-covering. We define on each polydisc

P κ|ρ(z j,k )| (z j,k ) two currents T (j,k) 0,1 and
T (j,k) 0,2 such that f 1 T (j,k) 0,1 + f 2 T (j,k) 0,2
= 1 as follows. We denote by ∆ ξ (ε) the disc of center ξ and radius ε and by (ζ * 0,1 , ζ * 0,2 ) the coordinates of ζ 0 in the Koranyi basis at z j,k . In [START_REF] Alexandre | Extension with growth estimates of holomorphic functions defined on singular analytic spaces[END_REF] were proved the next two propositions:

Proposition 3.1. If κ > 0 is small enough and if P 4κ|ρ(z j,k )| (z j,k ) ∩ X l = ∅ then |ζ * 0,1 | ≥ 4κ|ρ(z j,k )|.
We assume κ so small that Proposition 3.1 holds for both X 1 and X 2 with the same κ. When |ζ * 0,1 | ≥ 4κ|ρ(z j,k )| then X l can be parametrized as follows (see [START_REF] Alexandre | Extension with growth estimates of holomorphic functions defined on singular analytic spaces[END_REF]):

Proposition 3.2. If |ζ * 0,1 | ≥ 4κ|ρ(z j,k )|, for l = 1 and l = 2, there exists p l functions α (j,k) l,1 , . . . , α (j,k)
l,p l holomorphic on ∆ 0 (4κ|ρ(z j,k )|), there exists r > 0, depending neither on j nor on k, and there exists u (j,k) l holomorphic on the ball of center ζ 0 and radius r, bounded and bounded away from 0, such that:

(i) ∂α (j,k) l,i ∂ζ * 1 is bounded on ∆ 0 (4κ|ρ(z j,k )|) uniformly with respect to j and k, (ii) for all ζ ∈ P 4κ|ρ(z j,k )| (z j,k ), f l (ζ) = u (j,k) l (ζ) p l i=1 (ζ * 2 -α (j,k) l,i (ζ * 1 )).
Now we define T (j,k) 0,1 and T (j,k) 0,2 with the following settings. If |ζ * 0,1 | < 4κ|ρ(z j,k )|, then for l = 1 or l = 2, P 4κ|ρ(z j,k | (z j,k ) ∩ X l = ∅, which means that z j,k is "far" from X 1 and X 2 . In this case we set for l = 1 and l = 2:

I (j,k) l := ∅, i (j,k) l := 0, P (j,k) l (ζ) := 1. If |ζ * 0,1 | ≥ 4κ|ρ(z j,k )|, then we may have P 4κ|ρ(z j,k | (z j,k ) ∩ X l = ∅ for l = 1 or l = 2.
In that case we set for l = 1 and l = 2:

I (j,k) l := {i, ∃z * 1 ∈ C, |z * 1 | < 2κ|ρ(z j,k )| and |α (j,k) l,i (z * 1 )| < ( 5 2 κ|ρ(z j,k )|) 1 2 }, i (j,k) l := #I (j,k) l , the cardinal of I (j,k) l , P (j,k) l (ζ) := i∈I (j,k) l ζ * 2 -α (j,k) i,l (ζ * 1 )
.

In both case we set

U (j,k) 1 :=      ζ ∈ P κ|ρ(z j,k )| (z j,k ), f 1 (ζ)|ρ(z j,k )| i (j,k) 1 2 P (j,k) 1 (ζ) > 1 3 f 2 (ζ)|ρ(z j,k )| i (j,k) 2 2 P (j,k) 2 (ζ)      , U (j,k) 2 :=      ζ ∈ P κ|ρ(z j,k )| (z j,k ), 2 3
f 2 (ζ)|ρ(z j,k )| i (j,k) 2 2 P (j,k) 2 (ζ) > f 1 (ζ)|ρ(z j,k )| i (j,k) 1 2 P (j,k) 1 (ζ)      , so that P κ|ρ(z j,k )| (z j,k ) = U (j,k) 1 ∪ U (j,k) 2
. These open sets are designed in order to quantify where f 1 is "bigger" than f 2 and conversely. The idea is the following. If i belongs to

I (j,k) l then |ζ * 2 -α (j,k) l,i (ζ * 1 )| |ρ(z j,k )| 1 2 for all ζ ∈ P κ|ρ(z j,k )| (z j,k ). Thus each zero of f l in P κ|ρ(z j,k )| (z j,k ) brings in some sense a factor |ρ(z j,k )| 1 2 in f l (ζ). In the definition of U (j,k) l
, we take into account the zeros of f 1 and f 2 which are in the polydisc

P κ|ρ(z j,k )| (z j,k ) with the term |ρ(z j,k )| i (j,k) 1 2 and |ρ(z j,k )| i (j,k) 2 2
. This means in particular that all the zeros in the polydisc are treated in the same way, we don't care if they are close from each others, from the boundary of the polydisc or not. The zeros which are outside the polydisc are taken into account by f l (ζ)

P (j,k) l (ζ)
, which will also measure how far they are from the polydisc. Therefore, U

(j,k) 1
is the open set where f 1 is bigger than f 2 for an order such that the zeros which are outside of the polydisc are taken into account with the term f l (ζ)

P (j,k) l (ζ)
and the zeros which are inside with the term |ρ(z j,k )| i (j,k) l 2

. For l = 1, 2 and for a smooth (2, 2)-form ϕ compactly supported in U (j,k) l we set [START_REF] Mazzilli | Courants du type résiduel attachés à une intersection complète[END_REF]). Now we glue together the currents T (j,k) 0,l in order to define the current T 0,l , l = 1, 2, such that

T (j,k) 0,l , ϕ := C 2 P (j,k) l (ζ) f l (ζ) ∂ i (j,k) l ϕ ∂ζ * 2 i (j,k) l (ζ). Integrating i (j,k) l -times by parts, we get f l T (j,k) 0,l = c (j,k) l on U (j,k) l where c (j,k) l is an integer bounded by i (j,k) l ! (see
f 1 T 0,1 + f 2 T 0,2 = 1 on D ∩ U 0 . Let ( χj,k ) j∈N k∈{1,...,n j }
be a partition of unity subordinated to the covering (P κ|ρ(z j,k )| (z j,k )) j∈N k∈{1,...,n j } of U 0 . Without restriction, we assume that

∂ α+β+α+β χj,k ∂ζ * 1 α ∂ζ * 2 β ∂ζ * 1 α ∂ζ * 2 β (ζ) 1 |ρ(z j,k )| α+α+ β+β 2
. Let also χ be a smooth function on

C 2 \ {0} such that χ(z 1 , z 2 ) = 1 if |z 1 | > 2 3 |z 2 | and χ(z 1 , z 2 ) = 0 if |z 1 | < 1 3 |z 2 | and let us define χ (j,k) 1 (ζ) = χj,k (ζ) • χ    f 1 (ζ)|ρ(z j,k )| i (j,k) 1 2 P (j,k) 1 (ζ) , f 2 (ζ)|ρ(z j,k )| i (j,k) 2 2 P (j,k) 2 (ζ)
   , χ (j,k) 2 (ζ) = χj,k (ζ) •   1 -χ    f 1 (ζ)|ρ(z j,k )| i (j,k) 1 2 P (j,k) 1 (ζ) , f 2 (ζ)|ρ(z j,k )| i (j,k) 2 2 P (j,k) 2 (ζ)       .
For l = 1 and l = 2, the support of χ

(j,k) l is included in U (j,k) l
so we can put

T 0,l = j∈N k∈{1,...,n j } 1 c (j,k) l χ (j,k) l T (j,k) 0,l and we have f 1 T 0,1 + f 2 T 0,2 = 1 on U 0 ∩ D.
Now for all ζ 0 ∈ bD ∪ D -ε 0 we have constructed a neighborhood U 0 of ζ 0 and two currents T 0,1 and T 0,2 such that f 1 T 0,1 + f 2 T 0,2 = 1 on U 0 ∩ D. If ε 0 > 0 is sufficiently small, we can cover D by finitely many open sets U 1 , . . . , U n . Let χ 1 , . . . , χ n be a partition of unity subordinated to this family of open sets and T 1,1 , . . . , T n,1 and T 1,2 , . . . , T n,2 be the corresponding currents defined on U 1 , . . . , U n . We glue together this current and we set

T 1 = n j=1 χ j T j,1 and T 2 = n j=1 χ j T j,2 , so that f 1 T 1 + f 2 T 2 = 1 on D.
Moreover T 1 and T 2 are currents supported in D thus they have a finite order k 2 and we can apply T 1 and T 2 to functions of class C k 2 with support in D. This gives k 2 from Theorem 1.1.

The division formula

In this part, given any two currents T 1 and T 2 of order k 2 such that f 1 T 1 + f 2 T 2 = 1, assuming that g is a holomorphic function on D which belongs to the ideal generated by f 1 and f 2 , and which can be written as g = g1 f 1 + g2 f 2 , where g1 and g2 are two C ∞ -smooth functions on D such that |ρ| N g1 and |ρ| N g2 vanish to order k 2 on bD for some N ∈ N sufficiently big, we write g as g = g 1 f 1 + g 2 f 2 with g 1 and g 2 holomorphic on D. We point out that the formula we get is valid for any T 1 and T 2 of order k 2 such that

f 1 T 1 + f 2 T 2 = 1.
Under our assumptions, for k = 1 and k = 2 and all fixed z ∈ D, g1 P N,k (•, z) and g2 P N,k (•, z) can be extended by zero outside D and are of class C k 2 on C 2 . So we can apply T 1 and T 2 to g1 P N,k (•, z) and g2 P N,k (•, z).

For l = 1, 2, we denote by

b l = b l,1 dζ 1 + b l,2 dζ 2 a (1, 0)-form such that f l (z) -f l (ζ) = i=1,2 b l,i (ζ, z)(z i -ζ i ).
For the estimates, we will take b l,i (ζ, z) =

1 0 ∂f l ∂ζ i (ζ + t(z -ζ))
dt, but this is not necessary to get a division formula.

In order to construct the formula, we will need the following lemma which was proved in [START_REF] Mazzilli | Division des distributions et applications à l'étude d'idéaux de fonctions holomorphes[END_REF], Lemma 3.1:

Lemma 4.1. Let Q = n i=1 Q i dζ i be a (1, 0) form of C n , let H 1 , .
. . , H p be p (1, 0)-forms in C n and let W 1 , . . . , W p-1 be p -1 (0, 1)-forms in C n . Then the following equality holds

∂( Q, z -ζ )(∂Q) n-p ∧ H p ∧ p-1 k=1 W k ∧ H k = 1 n -p + 1 H p , z -ζ (∂Q) n-p+1 ∧ p-1 k=1 W k ∧ H k + 1 n -p + 1 p-1 l=1 H l , z -ζ (∂Q) n-p+1 H p ∧ W l ∧ p-1 k=1 k =l W k ∧ H k .
We now establish the division formula. From Theorem 2.3, we have for all z ∈ D:

g(z) = D g(ζ)P N,2 (ζ, z) and since g = g1 f 1 + g2 f 2 g(z) = f 1 (z) D g1 (ζ)P N,2 (ζ, z) + f 2 (z) D g2 (ζ)P N,2 (ζ, z) + D g1 (ζ) (f 1 (ζ) -f 1 (z)) P N,2 (ζ, z) + D g2 (ζ) (f 2 (ζ) -f 2 (z)) P N,2 (ζ, z). ( 5
)
Now from Lemma 4.1, there exists cN,2 such that

(f 1 (ζ) -f 1 (z)) P N,2 (ζ, z) = cN,2 b 1 (ζ, z) ∧ ∂P N,1 (ζ, z)
and since by assumption g1 P N,1 vanishes on bD, Stokes' Theorem yields

D g1 (ζ) (f 1 (ζ) -f 1 (z)) P N,2 (ζ, z) = cN,2 D ∂g 1 (ζ) ∧ b 1 (ζ, z) ∧ P N,1 (ζ, z). ( 6
)
We now use the fact that f 1 T 1 + f 2 T 2 = 1 in order to rewrite this former integral:

D ∂g 1 (ζ) ∧ b 1 (ζ, z) ∧ P N,1 (ζ, z) = f 1 T 1 + f 2 T 2 , ∂g 1 ∧ b 1 (•, z) ∧ P N,1 (•, z) = f 1 T 1 , ∂g 1 ∧ b 1 (•, z) ∧ P N,1 (•, z) + f 2 (z) T 2 , ∂g 1 ∧ b 1 (•, z) ∧ P N,1 (•, z) + T 2 , (f 2 -f 2 (z)) ∂g 1 ∧ b 1 (•, z) ∧ P N,1 (•, z) . (7)
Again from Lemma 4.1, there exists cN,1 such that

(f 2 (ζ) -f 2 (z)) b 1 (ζ, z) ∧ ∂g 1 ∧ P N,1 (ζ, z) -(f 1 (ζ) -f 1 (z)) b 2 (ζ, z) ∧ ∂g 1 ∧ P N,1 (ζ, z) = cN,1 b 1 (ζ, z) ∧ b 2 (ζ, z) ∧ ∂g 1 ∧ ∂P N,0 (ζ, z). So T 2 , (f 2 -f 2 (z)) ∂g 1 ∧ b 1 (•, z) ∧ P N,1 (•, z) = -f 1 (z) T 2 , ∂g 1 ∧ b 2 (•, z) ∧ P N,1 (•, z) + T 2 , f 1 ∂g 1 ∧ b 2 (•, z) ∧ P N,1 (•, z) +c N,1 T 2 , ∂g 1 ∧ b 1 (•, z) ∧ b 2 (•, z) ∧ ∂P N,0 (•, z) (8)
We plug together ( 6), ( 7) and ( 8) and their analogue for 5) and we get

D g 2 (ζ) (f 2 (ζ) -f 2 (z)) P N,2 (ζ, z) in (
g(z) = f 1 (z) D g1 (ζ)P N,2 (ζ, z) -cN,2 f 1 (z) T 2 , ∂g 1 ∧ b 2 (•, z) ∧ P N,1 (•, z) +c N,2 f 2 (z) T 2 , ∂g 1 ∧ b 1 (•, z) ∧ P N,1 (•, z) +f 2 (z) D g2 (ζ)P N,2 (ζ, z) -cN,2 f 2 (z) T 1 , ∂g 2 ∧ b 1 (•, z) ∧ P N,1 (•, z) +c N,2 f 1 (z) T 1 , ∂g 2 ∧ b 2 (•, z) ∧ P N,1 (•, z) +c N,2 T 1 , f 1 ∂g 1 ∧ b 1 (•, z) ∧ P N,1 (•, z) + cN,2 T 2 , f 1 ∂g 1 ∧ b 2 (•, z) ∧ P N,1 (•, z) (9) +c N,2 T 2 , f 2 ∂g 2 ∧ b 2 (•, z) ∧ P N,1 (•, z) + cN,2 T 1 , f 2 ∂g 2 ∧ b 1 (•, z) ∧ P N,1 (•, z) (10) +c N,2 cN,1 ∂g 1 ∧ T 2 -∂g 2 ∧ T 1 , b 1 (•, z) ∧ b 2 (•, z) ∧ ∂P N,0 (•, z)
Now since ∂g = f 1 ∂g 1 + f 2 ∂g 2 = 0, the line ( 9) and (10) vanish. Therefore in order to get our division formula, it suffices to prove that ∂(∂g 1 ∧ T 2 -∂g 2 ∧ T 1 ) = 0. When X 1 ∩X 2 is not a complete intersection and when assumption (iv) in the introduction is satisfied by g1 and g2 , one can prove that ∂g 1 ∧ ∂T 2 = 0 and ∂g 2 ∧ ∂T 1 = 0. When X 1 ∩ X 2 is a complete intersection, we prove that for any ζ 0 ∈ D there exists a neighborhood U 0 of ζ 0 such that for all (2, 1)-form ϕ, smooth and supported in U 0 , we have

∂g 1 ∧ T 2 -∂g 2 ∧ T 1 , ∂ϕ = 0. Let ζ 0 be a point in D.
By assumption on g, there exists a neighborhood U 0 of ζ 0 and two holomorphic functions γ 1 and γ 2 such that g = γ 1 f 1 + γ 2 f 2 on U 0 . We now use the following lemma whose proof is postponed to the end of this section: Lemma 4.2. Let f 1 and f 2 be two holomorphic functions defined in a neighborhood of 0 in C 2 , X 1 = {z, f 1 (z) = 0} and X 2 = {z, f 2 (z) = 0}. We assume that X 1 ∩ X 2 is a complete intersection and that 0 belongs to X 1 ∩ X 2 . Let ϕ 1 and ϕ 2 be two C ∞ -smooth

functions such that f 1 ϕ 1 = f 2 ϕ 2 . Then, ϕ 1 f 2 and ϕ 2 f 1 are C ∞ -smooth in a neighborhood of 0. Lemma 4.2 implies that the function ψ = g1 -γ 1 f 2 = γ 2 -g 2 f 1
is smooth on a perhaps smaller neighborhood of ζ 0 still denoted by U 0 . Thus

∂g 1 ∧ T 2 -∂g 2 ∧ T 1 , ∂ϕ = ∂(g 1 -γ 1 ) ∧ T 2 + ∂(γ 2 -g2 ) ∧ T 1 , ∂ϕ = ∂(f 2 ψ) ∧ T 2 + ∂(f 1 ψ) ∧ T 1 , ∂ϕ = f 2 T 2 + f 1 T 1 , ∂ψ ∧ ∂ϕ = U 0 ∂ψ ∧ ∂ϕ
and since ϕ is supported in U 0 we have U 0 ∂ψ ∧ ∂ϕ = -U 0 d(ϕ∂ψ) = 0 and so

∂g 1 ∧ T 2 -∂g 2 ∧ T 1 , ∂ϕ = 0. Now we set g 1 (z) = D g1 (ζ)P N,2 (ζ, z) +c N,2 T 1 , ∂g 2 ∧ b 2 (•, z) ∧ P N,1 (•, z) -T 2 , ∂g 1 ∧ b 2 (•, z) ∧ P N,1 (•, z) g 2 (z) = D g2 (ζ)P N,2 (ζ, z) +c N,2 T 2 , ∂g 1 ∧ b 1 (•, z) ∧ P N,1 (•, z) -T 1 , ∂g 2 ∧ b 1 (•, z) ∧ P N,1 (•, z) and we have g = g 1 f 1 + g 2 f 2
with g 1 and g 2 holomorphic on D. We notice that if g1 and g2 are already holomorphic functions then g 1 = g1 and g 2 = g2 .

Proof of Lemma 4.2: Maybe after a unitary change of coordinates if needed, using Weierstrass' preparation Theorem, we can assume that for l = 1, 2, the function f l is given by

f l (z, w) = z k l + a (l) 1 (w)z k l -1 + . . . + a (l) k l (w) where a (l) 1 , . . . , a (l)
k l are holomorphic near 0 and vanish at 0. Moreover, since the intersection X 1 ∩X 2 is transverse, P 1 and P 2 are relatively prime. Thus there exists two polynomials α 1 and α 2 with holomorphic coefficients in w and a function β of w not identically zero such that

α 1 (z, w)f 1 (z, w) + α 2 (z, w)f 2 (z, w) = β(w).
Multiplying this equality by ϕ 1 we get

f 2 (α 1 ϕ 2 + α 2 ϕ 1 ) = βϕ 1 .
We now prove that β divides the function ψ := α 1 ϕ 2 + α 2 ϕ 1 . If β(0) = 0, there is nothing to do. Otherwise, since β is not identically zero, there exists k ∈ N such that β(w) = w k γ(w) where γ(0) = 0. For all j ∈ N we have

f 2 (z, w) ∂ j ψ ∂w j (z, w) = β(w) ∂ϕ 1 ∂w j (z, w) (11)
and for w = 0 and all z we thus get ∂ j ψ ∂w j (z, 0) = 0. By induction we then deduce from (11) that ∂ i+j ψ ∂w i ∂w j (z, 0) = 0 for all i ∈ {0, . . . , k -1} and all j ∈ N. For any integer n ≥ k we therefore can write for all z and all w ψ(z, w)

w k = k≤i+j≤n i≥k w i-k w j ∂ i+j ψ ∂w i ∂w j (z, 0) + i+j=n+1 w i-k w j 1 0 ∂ n+1 ψ ∂w i ∂w j (z, tw)dt.
Now, it is easy to check by induction that the function w → w i+j w i is of class C j-1 for all positive integer j and all non negative integer i. This implies that ψ(z,w) w k is of class C n for all positive integer n and therefore

ϕ 1 f 2 = ψ β is of class C ∞ .

Proof of the main result

In order to prove Theorem 1.1, for any k and l in {1, 2} and any q ∈ [1, +∞], we have to prove that if h is a smooth function such that, for all non-negative integers α and β,

∂ α+β h ∂η ζ α ∂v ζ β |ρ| α+ β 2 belongs to L q (D), then the function z → T l , ∂h ∧ b k (•, z) ∧ P N,1 (•, z) belongs to L q (D) if q < ∞ and to BM O(D) if q = +∞.
As usually, since the modulus of the denominator in P N,1 is greater than |ρ(z)| + |ρ(ζ)| + δ(z, ζ), the difficulties occurs when we integrate for ζ near z and when z is near bD. Moreover, by construction of T 1 and T 2 , the main difficulty is when, in addition, z is near a point ζ 0 which belongs to bD ∩ X 1 ∩ X 2 and we only consider that case.

We assume that z belongs to the neighborhood U 0 of a point ζ 0 ∈ bD ∩ X 1 ∩ X 2 and we use the same notations as in Section 3 for the construction of the currents. Moreover, we assume that the Koranyi basis at ζ 0 is the canonical basis of C 2 and that ζ 0 is the origin of C 2 .

We will need an upper bound of . We set Q (j,k) l = f l P (j,k) l and we begin with the following lemma: Lemma 5.1. For all j ∈ N, all k ∈ {1, . . . , n j }, all α and β in N, l = 1, 2, and all ζ in P 2κ|ρ(z j,k )| (z j,k ), we have uniformly with respect to j, k, l, and

P (j,k) l f l ∂ α+β f l ∂ζ * 1 α ∂ζ *
ζ 1 Q (j,k) l (ζ) ∂ α+β ∂ζ * 1 α ∂ζ * 2 β Q (j,k) l (ζ) |ρ(z j,k )| -α-β 2 .
Proof: We denote by (ζ * 0,1 , ζ * 0,2 ) the coordinates of ζ 0 in the Koranyi coordinates at z j,k . The definition of P (j,k) l forces us to distinguish three cases:

First case: If |ζ * 0,1 | > 4κ|ρ(z j,k )|, let α (j,k)
l,i , i = 1, . . . , p l , be the family of parametrization given by Proposition 3.2. In this case, we actually seek an upper bound for

1 i / ∈I (j,k) l ζ * 2 -α (j,k) l,i (ζ * 1 ) ∂ α+β ∂ζ * 1 α ∂ζ * 2 β    i / ∈I (j,k) l ζ * 2 -α (j,k) l,i (ζ * 1 )    ,
and it suffices to prove for all i / ∈ I (j,k) l and all α and β that 1

ζ * 2 -α (j,k) l,i (ζ * 1 ) ∂ α+β ∂ζ * 1 α ∂ζ * 2 β (ζ * 2 -α (j,k) l,i (ζ * 1 )) |ρ(z j,k )| -α-β 2 . ( 12
)
By definition of I (j,k) l , we have |α

(j,k) l,i (ζ * 1 )| ≥ ( 5 2 κ|ρ(z j,k )|) 1 2 for all ζ * 1 ∈ ∆ 0 (2κ|ρ(z j,k )|) so |ζ * 2 -α (j,k) l,i (ζ * 1 )| |ρ(z j,k )| 1
2 and ( 12) holds true for α = 0 and β = 1.

According to Proposition 3.2,

∂α (j,k) l,i ∂ζ *
is uniformly bounded on ∆ 0 (4κ|ρ(z j,k )|). Cauchy's inequalities then yields

∂ α α (j,k) l,i ∂ζ * 1 α (ζ * 1 ) |ρ(z j,k )| 1-α . Since |ζ * 2 -α (j,k) l,i (ζ * 1 )| |ρ(z j,k )| 1 2 , ( 12
)
holds true for α > 0 and β = 0. Since the other cases are trivial, we are done in this case.

When |ζ * 0,1 | < 4κ|ρ(z j,k )|, we do not have a parametrization of X l but according to proposition 3.1, P 4κ|ρ(z j,k )| (z j,k ) ∩ X l is empty, which means that any ζ ∈ P 2κ|ρ(z j,k )| (z j,k ) is far from X l . We then have to distinguish two cases, depending on what "far" means. Before, we notice that, since P 4κ|ρ(z j,k )| (z j,k ) ∩ X l = ∅, I (j,k) l is also empty and P

(j,k) l = 1. Second case: If |ζ * 0,1 | < 4κ|ρ(z j,k )| and |ζ * 0,2 | < (4κ|ρ(z j,k )|) 1 2 , then δ(z j,k , ζ 0 ) |ρ(z j,k )| and thus for all ζ ∈ P 2κ|ρ(z j,k )| (z j,k ), δ( ζ, ζ 0 ) |ρ(z j,k )|.
In particular, all ζ belonging to P 2κ|ρ(z j,k )| (z j,k ) is almost at the same (pseudo-)distance from z j,k as from X l . For all ε > 0 and all

ζ ∈ P ε (ζ 0 ), it is then easy to see that |f l (ζ)| ε p l 2 .
Therefore, Cauchy's inequalities give

∂ α+β f l ∂ζ * 1 α ∂ζ * 2 β (ζ) |ρ(z j,k )| p l 2 -α-β 2 for all ζ ∈ P 2κ|ρ(z j,k )| (z j,k ). Moreover, since |ζ * 0,1 | < 4κ|ρ(z j,k )|, on the one hand f l = Q (j,k) l
. On the other hand it follows from Proposition 3.

1 that P 4κ|ρ(z j,k )| (z j,k ) ∩ X l = ∅. This yields |f l (ζ)| |ρ(z j,k )| p l 2 for all ζ ∈ P 2κ|ρ(z j,k )| (z j,k ), thus 1 Q (j,k) l (ζ) ∂ α+β ∂ζ * 1 α ∂ζ * 2 β Q (j,k) l (ζ) |ρ(z j,k )| -α-β 2 .
Third case:

If |ζ * 0,1 | < 4κ|ρ(z j,k )| and |ζ * 0,2 | ≥ (4κ|ρ(z j,k )|) 1 2 , then all ζ ∈ P 3κ|ρ(z j,k )| (z j,k ) is far from ζ * 0 and Q (j,k) l = f l . We will see that |f l (ζ)| is comparable to |ζ * 0,2 | p l for all ζ ∈ P 3κ|ρ(z j,k )| (z j,k ). We set a(z j,k ) = ∂ρ ∂ζ 1 (z j,k ), b(z j,k ) = ∂ρ ∂ζ 2 (z j,k
) and

P (z j,k ) = 1 |a(z j,k )| 2 + |b(z j,k )| 2 a(z j,k ) b(z j,k ) -b(z j,k ) a(z j,k )
.

Then we have ζ * = P (z j,k )(ζ -z j,k ) and moreover |a(z j,k )| 1 and b(z j,k ) tends to 0 when z j,k goes to ζ 0 , hence, b(z j,k ) is arbitrary small provided U 0 is sufficiently small. Therefore, if U 0 is sufficiently small, for all ζ ∈ P 3κ|ρ(z j,k )| (z j,k ),

|ζ 2 | ≥ 1 |a(z j,k )| 2 + |b(z j,k )| 2 (|a(z j,k )||ζ * 0,2 | -|b(z j,k )||ζ * 0,1 | -|b(z j,k )||ζ * 1 | -|a(z j,k )||ζ * 2 |) |ζ * 0,2 |. We also trivially have |ζ 2 | |ζ * 0,2 | and so |ζ 2 | |ζ * 0,2 |.
On the other hand

|ζ 1 | ≤ 1 |a(z j,k )| 2 + |b(z j,k )| 2 |a(z j,k )|(|ζ * 0,1 | + |ζ * 1 |) + |b(z j,k )|(|ζ * 0,2 | + |ζ * 2 |) ≤ 6κ|ρ(z j,k )| + |b(z j,k )|(|ζ * 0,2 | + (2κ|ρ(z j,k )|) 1
2 ) ≤ c|ζ * 0,2 | where c depends neither on z j,k nor on ζ and is arbitrarily small provided U 0 is small enough. Now let α ∈ C be such that f l (ζ 1 , α) = 0. Since the intersection X l ∩ bD is transverse, there exists a positive constant C depending neither on ζ, nor on α, nor on j and nor on

k such that |α| ≤ C|ζ 1 |. Therefore if U 0 is small enough, |α| ≤ 1 2 |ζ 2 |. This yields, for all ζ ∈ P 3κ|ρ(z j,k )| (z j,k ), |f l (ζ)| α/f l (ζ 1 ,α)=0 |ζ 2 -α| |ζ * 0,2 | p l . Cauchy's inequalities then give for all ζ ∈ P 2κ|ρ(z j,k )| (z j,k ) ∂ α+β f l ∂ζ * 1 α ∂ζ * 2 β (ζ) |ζ * 0,2 | p l |ρ(z j,k )| -α-β 2 ,
and since Q (j,k) l = f l , we are done in this case and the lemma is shown.

Lemma 5.1 gives us an upper bound for the derivatives of χ (j,k) l : Corollary 5.2. For all j ∈ N, all k ∈ {1, . . . , n j }, all α and β in N, l = 1, 2 and all ζ ∈ P κ|ρ(z j,k )| (z j,k ), we have uniformly with respect to j, k, l and ζ

∂ α+β χ (j,k) l ∂ζ * 1 α ∂ζ * 2 β (ζ) |ρ(z j,k )| -α-β 2 .
Proof: Since by construction

∂ α+β χj,k ∂ζ * 1 α ∂ζ * 2 β (ζ) |ρ(z j,k )| -α-β 2 ,
we only have to consider

∂ α+β ∂ζ * 1 α ∂ζ * 2 β χ f 1 (ζ) P (j,k) 1 (ζ) |ρ(z j,k )| i (j,k) 1 , f 2 (ζ) P (j,k) 2 (ζ) |ρ(z j,k )| i (j,k) 2 . The derivative ∂ γ+δ χ ∂z γ 1 ∂z δ 2 (z 1 , z 2
) is bounded up to a uniform multiplicative constant by

1 |z 1 | γ |z 2 | δ when 1 3 |z 2 | < |z 1 | < 2 3 |z 2 |
and is zero otherwise. Therefore, we can estimate

∂ α+β χ (j,k) l ∂ζ * 1 α ∂ζ * 2 β by a sum of products of 1 Q (j,k) l ∂ γ+ δ ∂ζ * 1 γ ∂ζ * 2 δ Q (j,k) l
where the sum of the γ's equals α and the sum of the δ's equals β. Lemma 5.1 then gives the wanted estimates.

Corollary 5.3. For any smooth function h, we can write

∂ i (j,k) l ∂ζ * 2 i (j,k) l χ (j,k) l (ζ)∂h(ζ) ∧ P N,1 (ζ, z) = ψ (j,k,l) 1 (ζ, z)dζ * 1 + ψ (j,k,l) 2 (ζ, z)dζ * 2 with ψ (j,k,l) 1
and ψ (j,k,l) 2 two (0,2)-forms supported in U (j,k) l satisfying uniformly with respect to j, k, z and ζ ∈ U

(j,k) l : ψ (j,k,l) 1 (ζ, z) |ρ(z j,k )| - i (j,k) l 2 -5 2 |ρ(z j,k )| |ρ(z j,k )| + |ρ(z)| + δ(z j,k , z) N h(ζ), ψ (j,k,l) 2 (ζ, z) |ρ(z j,k )| - i (j,k) l 2 -2 |ρ(z j,k )| |ρ(z j,k )| + |ρ(z)| + δ(z j,k , z) N h(ζ),
and, for ∇ z a differential operators of order 1 acting on z,

∇ z ψ (j,k,l) 1 (ζ, z) |ρ(z j,k )| - i (j,k) l 2 -7 2 |ρ(z j,k )| |ρ(z j,k )| + |ρ(z)| + δ(z j,k , z) N h(ζ), ∇ z ψ (j,k,l) 2 (ζ, z) |ρ(z j,k )| - i (j,k) l 2 -3 |ρ(z j,k )| |ρ(z j,k )| + |ρ(z)| + δ(z j,k , z) N h(ζ), where h(ζ) = max n∈{0,...,i (j,k) l } ∂ n+1 h ∂ζ * 2 n+1 (ζ)|ρ(ζ)| n+1 2 , ∂ n+1 h ∂ζ * 1 ∂ζ * 2 n (ζ)|ρ(ζ)| n 2 +1
.

Proof: Propositions 2.4 and 2.5 imply that

∂ n ∂ζ * 2 n P N,1 (ζ, z) = p,q=1,2 ψ(n,N) p,q (ζ, z)dζ * p ∧ dζ * q where | ψn,N p,q (ζ, z)| |ρ(ζ)| |ρ(ζ)| + |ρ(z)| + δ(ζ, z) N |ρ(ζ)| -1 p -1 q -n 2 .
From proposition 2.1, if κ is small enough, we have for all ζ ∈ P κ|ρ(z j,k )| (z j,k ), 1 2 |ρ(z j,k )| ≤ |ρ(ζ)| and thus, provided κ is small enough:

|ρ(ζ)| + δ(ζ, z) ≥ 1 2 |ρ(z j,k )| + 1 c 1 δ(z, z j,k ) -δ(z j,k , ζ) |ρ(z j,k )| + δ(z, z j,k ) and so | ψn,N p,q (ζ, z)| |ρ(z j,k )| |ρ(z j,k )|+|ρ(z)|+δ(z j,k ,z) N |ρ(z j,k )| -1 p -1 q -n 2 .
This inequality and Corollary 5.2 now yield the two first estimates. The two others can be shown in the same way.

In order to estimate P (j,k) l f l b m , we need the following lemma: Lemma 5.4. For all j ∈ N, all k ∈ {1, . . . , n j }, all α and β in N, l = 1, 2 and all ζ ∈ P 2κ|ρ(z j,k )| (z j,k ) we have uniformly with respect to j, k, l and ζ

∂ α+β ∂ζ * 1 α ∂ζ * 2 β    i∈I (j,k) l (ζ * 2 -α (j,k) l,i (ζ * 1 ))    |ρ(z j,k )| i (j,k) l 2 -α-β 2 .
Proof:

For i ∈ I (j,k) l , there exists z * 1 ∈ ∆ 0 (2κ|ρ(z j,k )|) such that |α (j,k) l,i (z * 1 )| < 5 2 κ|ρ(z j,k )| 1 2 . Since ∂α (j,k) l,i ∂ζ * 1 is uniformly bounded on ∆ 0 (4κ|ρ(z j,k )|), for all ζ ∈ P 4κ|ρ(z j,k )| (z j,k ), we have i∈I (j,k) l ζ * 2 -α (j,k) l,i (ζ * 1 ) |ρ(z j,k )| i (j,k) l 2
. Cauchy's inequalities then give the results.

As a direct corollary of Lemma 5.1 and 5.4 we get Corollary 5.5. For all j ∈ N, all k ∈ {1, . . . , n j }, all α and β in N, l = 1, 2 and all ζ ∈ P 2κ|ρ(z j,k )| (z j,k ) we have uniformly with respect to j, k, l and ζ

P (j,k) l (ζ) f l (ζ) ∂ α+β f l ∂ζ * 1 α ∂ζ * 2 β (ζ) |ρ(z j,k )| i (j,k) l 2 -α-β 2 .
In the following corollary, we give estimates for l, m ∈ {1, 2} of

P (j,k) l f l b m , which do not depend on m thanks to the covering U (j,k) 1 , U (j,k) 2 of P κ|ρ(z j,k )| (z j,k ).
Corollary 5.6. For l, m ∈ {1, 2}, we can write

P (j,k) l f l b m = ϕ (j,k,l,m) 1 dζ * 1 + ϕ (j,k,l,m) 2 dζ * 2 with ϕ (j,k,l,m) 1 and ϕ (j,k,l,m) 2 satisfying for all ζ ∈ U (j,k) l ϕ (j,k,l,m) 1 (ζ, z) 0≤α+β≤max(p 1 ,p 2 ) |ρ(z j,k )| i (j,k) l 2 -1 δ(ζ, z) ρ(z j,k ) α+ β 2 , ϕ (j,k,l,m) 2 (ζ, z) 0≤α+β≤max(p 1 ,p 2 ) |ρ(z j,k )| i (j,k) l 2 -1 2 δ(ζ, z) ρ(z j,k ) α+ β 2 ,
and for all differential operators ∇ z of order 1 acting on z,

∇ z ϕ (j,k,l,m) 1 (ζ, z) 0≤α+β≤max(p 1 ,p 2 ) |ρ(z j,k )| i (j,k) l 2 -2 δ(ζ, z) ρ(z j,k ) α+ β 2 , ∇ z ϕ (j,k,l,m) 2 (ζ, z) 0≤α+β≤max(p 1 ,p 2 ) |ρ(z j,k )| i (j,k) l 2 -3 2 δ(ζ, z) ρ(z j,k ) α+ β 2 ,
uniformly with respect to ζ, z, j and k.

Proof: Without restriction we assume l = 1 and for m = 1, 2, we write b

m (ζ, z) = b * m,1 (ζ, z)dζ * 1 + b * m,2 (ζ, z)dζ * 2 where b * m,n = 1 0 ∂fm ∂ζ * n (ζ + t(z -ζ))dt. So b * m,n (ζ, z) = 0≤α+β≤max(p 1 ,p 2 ) 1 α + β + 1 ∂ α+β+1 f m ∂ζ * n ∂ζ * 1 α ∂ζ * 2 β (ζ)(z * 1 -ζ * 1 ) α (z * 2 -ζ * 2 ) β + o |z -ζ| max(p 1 ,p 2 )
and Corollary 5.5 yields for all ζ ∈ P κ|ρ(z j,k )| (z j,k ):

P (j,k) 1 (ζ) f 1 (ζ) b 1,1 (ζ, z) 0≤α+β≤max(p 1 ,p 2 ) |ρ(z j,k )| i (j,k) 1 2 -1 δ(ζ, z) ρ(z j,k ) α+ β 2
uniformly with respect to z, ζ, j and k. The proof of the inequality for

P (j,k) 1 (ζ) f 1 (ζ) b 1,2 (ζ, z)
is exactly the same. The one for

P (j,k) 1 (ζ) f 1 (ζ) b 2,1 (ζ, z) uses the definition of U (j,k) 1 . On U (j,k) 1
, we have

P (j,k) 1 f 1 P (j,k) 2 f 2 |ρ(z j,k )| i (j,k) 1 -i (j,k) 2 2
and again Corollary 5.5 yields

P (j,k) 1 (ζ) f 1 (ζ) b 2,1 (ζ, z) P (j,k) 2 (ζ) f 2 (ζ) b 2,1 (ζ, z) |ρ(z j,k )| i (j,k) 1 -i (j,k) 2 2 0≤α+β≤max(p 1 ,p 2 ) |ρ(z j,k )| i (j,k) 1 2 -1 δ(ζ, z) ρ(z j,k ) α+ β 2
uniformly with respect to z, ζ, j and k. Again, the inequality for

P (j,k) 1 (ζ) f 1 (ζ) b 2,2 (ζ, z
) can be obtained in the same way. Corollary 5.3 and 5.6 imply for some N arbitrarily large, provided N is large enough, and for all ζ ∈ P κ|ρ(z j,k )| (z j,k ) that

P (j,k) l (ζ) f l (ζ) b m (ζ, z) ∧ ∂ i (j, k) l ∂ζ * 2 i (j,k) l χ (j,k) l (ζ)∂h(ζ)
∧ P N,1 (ζ, z) ≤ |ρ(z j,k )| -3 |ρ(z j,k )| |ρ(z j,k )| + |ρ(z)| + δ(z j,k , z) N h(ζ)
and for ∇ z a differential of order 1

∇ z   P (j,k) l (ζ) f l (ζ) b m (ζ, z) ∧ ∂ i (j,k) l ∂ζ * 2 i (j,k) l χ (j,k) l (ζ)∂h(ζ) ∧ P N,1 (ζ, z)   ≤ |ρ(z)| -1 |ρ(z j,k )| -3 |ρ(z j,k )| |ρ(z j,k )| + |ρ(z)| + δ(z j,k , z) N h(ζ) where h(ζ) = max n∈{0,...,i (j,k) l } ∂ n+1 h ∂ζ * 2 n+1 (ζ)|ρ(ζ)| n+1 2 , ∂ n+1 h ∂ζ * 1 ∂ζ * 2 n (ζ)|ρ(ζ)| n 2 +1
. We conclude as in the proof of Theorem 1.1 of [START_REF] Alexandre | Extension with growth estimates of holomorphic functions defined on singular analytic spaces[END_REF] that Theorem 1.1 holds true.

Local division

6.1. Local holomorphic division. In this subsection we will prove Theorem 1.2 and his analogue in the L q case, the following theorem.

Theorem 6.1. When n = 2, let g be a holomorphic function defined on D. Assume that X 1 ∩ X 2 is a complete intersection and that there exist κ > 0, a real number q ≥ 1 and a locally finite covering

P κ|ρ(ζ j )| (ζ j )
j∈I of D such that for all j ∈ I, there exist two function ĝ(j)

1 and ĝ(j) 2 , C ∞ -smooth on P κ|ρ(ζ j )| (ζ j ), which satisfy (a) g = ĝ(j) 1 f 1 + ĝ(j) 2 f 2 on P κ|ρ(ζ j )| (ζ j ); (b) j∈I P κ|ρ(ζ j )| (ζ j ) ∂ α+β ĝ(j) l ∂ζ * 1 α ∂ζ * 2 β (z)|ρ(ζ j )| α+ β 2 q
dV (z) < ∞ for l = 1 and l = 2 and all integers α and β; (c) for l = 1 and l = 2, for all non negatives integers α, α, β and β, there exist N ∈ N and c > 0 such that for all j, |ρ(

ζ j )| N sup P κ|ρ(ζ j )| (ζ j ) ∂ α+α+β+β ĝ(j) l ∂ζ * 1 α ∂ζ * 2 β ∂ζ * 1 α ∂ζ * 2 β ≤ c.
Then there exist two smooth functions g1 and g2 which satisfy (i)-(iii) of Theorem 1.1 with q.

Proof: It suffices to glue together all the ĝ(j) 1 and ĝ(j) 2 using a suitable partition of unity. Let (χ j ) j∈N be a partition of unity subordinated to P κ|ρ(ζ j )| (ζ j ) j∈N such that for all j and all ζ ∈ P κ|ρ(ζ j )| (ζ j ), we have

∂ α+α+β+β χ j ∂z * 1 α ∂z * 2 β ∂z * 1 α ∂z * 2 β (ζ) 1 |ρ(ζ j )| α+α+ β+β 2
, uniformly with respect to ζ j and ζ. We set g1 = j χ j ĝ(j) 1 and g2 = j χ j ĝ(j) 2 and thus we get the two functions defined on D which satisfy (i), (ii) and (iii) by construction.

The proof of Theorem 1.2 is exactly the same so we omit it. 6.2. Divided differences and division. In order to apply Theorem 1.2 and 6.1, we will use divided differences and find numerical conditions on g which ensure the existence of local smooth division formula in L ∞ and in L q . We define the divided differences using the following settings.

We set

Λ (1) z,v = {λ ∈ C, |λ| < τ (z, v, 3κ|ρ(z)|) and z + λv ∈ X 2 \ X 1 } Thus the points z + λv, λ ∈ Λ (1)
z,v , are the points of X 2 \ X 1 which belong to the disc ∆ z,v (τ (z, v, 3κ|ρ(z)|)), so they all belong to D as soon as κ < 1 3 . We analogously define Λ (2) z,v = {λ ∈ C, |λ| < τ (z, v, 3κ|ρ(z)|) and z + λv ∈ X 1 \ X 2 }. For a function h defined on a subset U of C n , z ∈ C n , v a unit vector of C n and λ ∈ C such that z + λv belongs to U, we set h z,v [λ] = h(z + λv).

If for µ 1 , . . . , µ k pairewise distinct h z,v [µ 1 , . . . , µ k ] is defined, for λ 1 , . . . , λ k+1 ∈ C pairwise distinct such that z + λ i v belongs to U for all i, we set

h z,v [λ 1 , . . . , λ k+1 ] := h z,v [λ 1 , . . . , λ k ] -h z,v [λ 2 , . . . , λ k+1 ] λ 1 -λ k+1 . Now, for z ∈ X 2 \X 1 (resp. z ∈ X 1 \X 2 ) let us define g (2) (z) = g(z)
f 2 (z) (resp. g (1) (z) = g(z) f 1 (z) ). For l = 1 or l = 2, the quantity g

(l) z,v [λ 1 , . . . , λ k ] make sense for all λ 1 , . . . , λ k ∈ Λ (l) z,v pairwise distinct.
We first prove a lemma we will need in this section.

Lemma 6.2. Let α and β be two functions defined on a subset U of C. Then, for all z 1 , . . . , z n pairwise distinct points of U we have

(α • β)[z 1 , . . . , z n ] = n k=1 α[z 1 , . . . , z k ] • β[z k , • • • , z n ].
Proof: We prove the lemma by induction on n, the case n = 1 being trivial. We assume the lemma proved for n points, n ≥ 1. Let z 1 , . . . , z n+1 be n + 1 points of U. Then

(α • β)[z 1 , . . . , z n+1 ] = (α • β)[z 1 , z 3 , . . . , z n+1 ] -(α • β)[z 2 , . . . , z n+1 ] z 1 -z 2 = 1 z 1 -z 2 n+1 k=3 α[z 1 , z 3 , . . . , z k ]β[z k , . . . , z n+1 ] + α[z 1 ]β[z 3 , . . . , z n+1 ] - 1 z 1 -z 2 n+1 k=2 α[z 2 , . . . , z k ]β[z k , . . . , z n+1 ] = n+1 k=3 α[z 1 , z 3 , . . . , z k ] -α[z 2 , . . . , z k ] z 1 -z 2 β[z k , . . . , z n+1 ] + α[z 1 ] -α[z 2 ] z 1 -z 2 β[z 2 , . . . , z n+1 ] + α[z 1 ] β[z 1 , z 3 , . . . , z n+1 ] -β[z 2 , . . . , z n+1 ] z 1 -z 2 .
6.2.1. The L ∞ -BM O-case. In this subsection, we establish the necessary conditions in C n and the sufficient conditions in C 2 for a function g to be written as g = g 1 f 1 + g 2 f 2 , with g 1 and g 2 smooth functions satisfying the hypothesis of Theorem 1.1. For l = 1 and l = 2 let us define the numbers

c (l) ∞ (g) = sup |g (l) z,v [λ 1 , . . . , λ k ]|τ (z, v, |ρ(z)|) k-1
where the supremum is taken over all z ∈ D, all v ∈ C n with |v| = 1, all k ∈ N * and λ 1 , . . . , λ k ∈ Λ (l) z,v pairwise distinct. We have the following necessary conditions in C n , n ≥ 2. Theorem 6.3. In C n , n ≥ 2, let g 1 , g 2 be two bounded holomorphic functions on D and set g = g 1 f 1 + g 2 f 2 . Then

g max(|f 1 |, |f 2 |) L ∞ (D) max( g 1 L ∞ (D) , g 2 L ∞ (D) )
and for l = 1, 2:

c (l) ∞ (g) sup b∆z,v(τ (z,v,4κ|ρ(z)|)) |g l |.
Proof: The first point is trivial and we only prove the second one for l = 1. Let λ 1 , . . . , λ k be k pairwise distinct elements of Λ

z,v . For all i we have g

(1) z,v [λ i ] = g 1 (z + λ i v) because f 2 (z + λ i v) = 0. Therefore, g (1)
z,v [λ 1 , . . . , λ k ] = (g 1 ) z,v [λ 1 , . . . , λ k ]. Since [19] g (1) z,v [λ 1 , . . . , λ k ]| = 1 2iπ |λ|=τ (z,v,4κ|ρ(z)|) g 1 (z+λv) k i=1 (λ-λ i ) dλ, it follows that |g (1) z,v [λ 1 , . . . , λ k ]| τ (z, v, |ρ(z)|) -k+1 sup b∆z,v(τ (z,v,4κ|ρ(z)|)) |g 1 |.
Now we prove that these conditions are sufficient in C 2 in order to get a BM O division.

Theorem 6.4. In C 2 , let g be a holomorphic function on D which belongs to the ideal of O(D) generated by f 1 and f 2 and such that

(i) c(g) = sup z∈D |g(z)| max(|f 1 (z)|,|f 2 (z)|) < ∞, (ii) c (1)
∞ (g) and c

(2) ∞ (g) are finite. There exist two holomorphic functions g 1 and g 2 which beblong to BM O(D) and such that g 1 f 1 + g 2 f 2 = g. If ζ 0 belongs to X 1 ∩ X 2 ∩ bD, since the intersection X 1 ∩ X 2 is complete, without restriction we can choose a neighborhood U 0 of ζ 0 such that X 1 ∩ X 2 ∩ U 0 = {ζ 0 }. Then we fix some point z in U 0 and we construct ĝ1 and ĝ2 on P κ|ρ(z)| (z) which satisfy (a) and (b) of Theorem 1.2. We denote by p 1 and p 2 the multiplicity of ζ 0 as singularity of f 1 and f 2 respectively. We also denote by

Q l (ζ) = f l (ζ).
Otherwise, we use the parametrization α 1,i , i ∈ {1, . . . , p 1 }, of X 1 and α 2,i , i ∈ {1, . . . , p 2 }, of X 2 given by Proposition 3.2. We denote by I l the set

I l = {i, ∃z * 1 ∈ ∆ 0 (2κ|ρ(z)|) such that |α l,i (z * 1 )| ≤ ( 5 2 κ|ρ(z)|) 1 2 }, i l = #I l , P l (ζ) = i∈I l (ζ * 2 -α l,i (ζ * 1
)) and Q l (ζ) = f l P l . Our first goal is to find h1 and h2 in C ∞ (P κ|ρ(z)| (z)) such that g = h1 P 1 + h2 P 2 on P κ|ρ(z)| (z) and which moreover satisfy good estimates. The function g belong to the ideal of O(P 4κ|ρ(z)| (z)) generated by f 1 and f 2 and so there exist h 1 and h 2 holomorphic in P 4κ|ρ(z)| (z) such that g = P 1 h 1 + P 2 h 2 . Moreover, we observe that necessarily h2 (ζ) =

h 2 (ζ) = g(ζ)
P 2 (ζ) for all ζ such that P 1 (ζ) = 0 and P 2 (ζ) = 0, but we also notice that h 2 may not satisfy good estimates like uniform boundedness for example. Thus, we already know h2 (ζ) for such ζ and by interpolation, we will reconstruct a "good" h2 in the whole polydisc P κ|ρ(z)| (z). We point out that we do not directly divide by f 1 and f 2 because if we do so, we are not able to handle the error term we get during the interpolation procedure.

If i 1 = 0 we set ĥ2 = 0. Otherwise, without restriction we assume that I 1 = {1, . . . , i 1 } and for k ≤ i 1 and ζ * 1 such that

P 2 (z + ζ * 1 η z + α 1,i (ζ * 1 )v z ) = 0, we introduce h (2) 1,...,k (ζ * 1 ) := g P 2 z+ζ * 1 ηz,vz [α 1,1 (ζ * 1 ), . . . , α 1,k (ζ * 1 )]. ( 13
) and ĥ2 (ζ) = i 2 k=1 h (2) 1,...,k (ζ * 1 ) k-1 i=1 (ζ * 2 -α 1,i (ζ * 1
)).

We define ĥ1 analogously. Since X 1 ∩ X 2 ∩ U 0 = {ζ 0 }, ĥ1 and ĥ2 are defined on P 4κ|ρ(z)| (z). Moreover, ĥ2 (ζ

(ζ) = 1 2iπ |ξ|=(4κ|ρ(z)|) 1 2 h 2 (ζ * 1 , ξ) P 1 (ζ * 1 , ξ) • (ξ -ζ * 2 )
dξ.

We have an analogous expression for h 1 and we point out that ( 14), [START_REF] Mazzilli | Division des distributions et applications à l'étude d'idéaux de fonctions holomorphes[END_REF] and theirs analogue for g 1 also holds if i 1 = 0 or i 2 = 0. This yields

g(ζ) = P 1 (ζ) ĥ1 (ζ) + P 2 (ζ) ĥ2 (ζ) + P 1 (ζ)P 2 (ζ)e(ζ) (16)
where

e(ζ) = e 1 (ζ) + e 2 (ζ) = 1 2iπ |ξ|=(4κ|ρ(z)|) 1 2 g(ζ * 1 , ξ) P 1 (ζ * 1 , ξ) • P 2 (ζ * 1 , ξ) • (ξ -ζ * 2 )
dξ.

If we were trying to divide by f 1 and f 2 directly, in the error term above, we wouldn't get g but h 1 P 1 + h 2 P 2 that we can not handle. Of course, ĥ2 will be a part of the function h2 we are looking for. We first look for an upper bound for ĥ2 using our assumption on the divided differences of g (2) = g f 2 . Fact 1: ĥ2 satisfies for all ζ ∈ P 2κ|ρ(z)| (z), uniformly with respect to z and ζ

| ĥ2 (ζ)| c (2) ∞ (g) sup |ξ|=(4κ|ρ(z)|) 1 2 |Q 2 (z + ζ * 1 η z + ξv z )| (17)
Indeed: We have by Lemma 6.2

h (2) 1,...,k (ζ * 1 ) = g P 2 z+ζ * 1 ηz,vz [α 1,1 (ζ * 1 ), . . . , α 1,k (ζ * 1 )] = g (2) Q 2 z+ζ * 1 ηz,vz [α 1,1 (ζ * 1 ), . . . , α 1,k (ζ * 1 )] = k j=1 g (2) z+ζ * 1 ηz,vz [α 1,1 (ζ * 1 ), . . . , α 1,j (ζ * 1 )] (Q 2 ) z+ζ * 1 ηz,vz [α 1,j (ζ * 1 ), . . . , α 1,k (ζ * 1 )].
From Montel's theorem [START_REF] Montel | Sur une formule de Darboux et les polynômes d'interpolation[END_REF] on divided differences in C and from Cauchy's inequalities, since τ

(z, v z , 4κ|ρ(z)|) (4κ|ρ(z)|) 1 2 , it follows that | (Q 2 ) z+ζ * 1 ηz,vz [α 1,j (ζ * 1 ), . . . , α 1,k (ζ * 1 )]| |ρ(z)| j-k 2 sup |ξ|=(4κ|ρ(z)|) 1 2 |Q 2 (z + ζ * 1 η z + ξv z )|.
With the assumption c

(2)

∞ (g) < ∞, this gives for all ζ * 1 ∈ ∆ 0 (2κ|ρ(z)|): |h (2) 1,...,k (ζ * 1 )| c (2) ∞ (g)|ρ(z)| 1-k 2 sup |ξ|=(4κ|ρ(z)|) 1 2 |Q 2 (z + ζ * 1 η z + ξv z )| (18)
and so [START_REF] Mcneal | Convex domains of finite type[END_REF] holds true.

Of course we have the analogous estimate for ĥ1 . Now we have to handle the error term in [START_REF] Mazzilli | Courants du type résiduel attachés à une intersection complète[END_REF]. Since there is a factor P 1 P 2 in front of e in (16), we can put P 2 e either with ĥ1 in h1 or we can put P 1 e with ĥ2 in h2 . But in order to have a good upper bound for h1 and h2 , we have to cut it in two pieces in a suitable way. This will be done analogously to the construction of the currents. Let

U 1 := ζ ∈ P κ|ρ(z)| (z), f 1 (ζ)|ρ(z)| i 1 2 P 1 (ζ) > 1 3 f 2 (ζ)|ρ(z)| i 2 2 P 2 (ζ) , U 2 := ζ ∈ P κ|ρ(z)| (z), 2 3
f 2 (ζ)|ρ(z)| i 2 2 P 2 (ζ) > f 1 (ζ)|ρ(z)| i 1 2 P 1 (ζ) .
Let also χ be a smooth function on

C 2 \ {0} such that χ(z 1 , z 2 ) = 1 if |z 1 | > 2 3 |z 2 | and χ(z 1 , z 2 ) = 0 if |z 1 | < 1 3 |z 2 |. We set χ 1 (ζ) = χ f 1 (ζ)|ρ(z)| i 1 2 P 1 (ζ) , f 2 (ζ)|ρ(z)| i 2 2 P 2 (ζ) , χ 2 (ζ) = 1-χ 1 (ζ)
and at last we define h1

(ζ) = ĥ1 (ζ) + χ 1 (ζ)P 2 (ζ)e(ζ), h2 (ζ) = ĥ2 (ζ) + χ 2 (ζ)P 1 (ζ)e(ζ).
And we now look for an upper bound for P 1 (ζ)e(ζ) on U 1 .

Fact 2: For all ζ belonging to P 4κ|ρ(z)| (z), we have uniformly with respect to ζ and z

|P 1 (ζ)e(ζ)| c(g) |ρ(z)| i 1 -i 2 2 sup P 4κ|ρ(z)| (z) |Q 1 | + sup P 4κ|ρ(z)| (z) |Q 2 | . (19)
|g(ζ)| ≤ c(g)(|f 1 (ζ)| + |f 2 (ζ)|) c(g) |ρ(z)| i 1 2 |Q 1 (ζ)| + |ρ(z)| i 2 2 |Q 2 (ζ)| .

This yields for all ζ

∈ P κ|ρ(z)| (z) |e(ζ)| c(g) |ρ(z)| -i 2 2 sup P 4κ|ρ(z)| (z) |Q 1 | + |ρ(z)| -i 1 2 sup P 4κ|ρ(z)| (z) |Q 2 |
from which [START_REF] Montel | Sur une formule de Darboux et les polynômes d'interpolation[END_REF] follows.

Therefore we have the identity g = P 1 h1 + P 2 h2 and upper bounds for h2 using ( 17) and ( 19), the corresponding one for h1 being also true of course. But our final goal is to write g as g = ĝ1 f 1 + ĝ2 f 2 . So we put ĝ1 = h1 Q 1 and ĝ2 = h2 Q 2 so that g = ĝ1 f 1 + ĝ2 f 2 . Moreover, from [START_REF] Mcneal | Convex domains of finite type[END_REF] and [START_REF] Montel | Sur une formule de Darboux et les polynômes d'interpolation[END_REF], and since χ 2 has support in U 2 , it follows for ζ ∈ P κ|ρ(z)| (z)

|ĝ 2 (ζ)| ≤ (c (2) ∞ (g) + c(g)) 1 Q 2 (ζ) sup P 4κ|ρ(z)| (z) |Q 2 | + c(g) 1 Q 1 (ζ) sup P 4κ|ρ(z)| (z) |Q 1 |. (20)
Therefore, in order to prove that g2 is bounded, we will have to prove that Q l (ξ) Q l (ζ) is bounded for ζ ∈ P κ|ρ(z)| (z) and ξ ∈ P 4κ|ρ(z)| (z). This is the aim of the following Fact 3.

Q l (ξ) Q l (ζ) 1. ( 21
)
The proof of fact 3 is analogous to the proof of Lemma 5.1. Without any restriction we assume l = 2.

First case: If |ζ * 0,1 | > 4κ|ρ(z)|, then we have the parametrization of X 2 and it suffices to prove for i / ∈ I 2 that

ξ * 2 -α * 2,i (ξ * 1 ) ζ * 2 -α * 2,i (ζ * 1 )
1.

If

|α 2,i (ξ * 1 )| ≥ |ρ(z)| 1 2 , since from Proposition 3.2 ∂α 2,i ∂ζ * 1 is bounded, |α 2,i (ζ * 1 )| ≥ 1 2 |ρ(z)| 1 2 and |α 2,i (ζ * 1 )| ≥ 1 2 |α 2,i (ξ * 1 )|, so ξ * 2 -α * 2,i (ξ * 1 ) ζ * 2 -α * 2,i (ζ * 1 )
1 is satisfied.

If |α 2 , i(ξ * 1 )| ≤ |ρ(z)| 1 2 , then |ξ * 2 -α * 2,i (ξ * 1 )| |ρ(z)| 1 2
and since by definition of

I 2 , |α 2,i (ζ * 1 )| ≥ 5 2 κ|ρ(z)| 1 2 for all ζ * 1 ∈ ∆ 0 (2κ|ρ(z)|), we have |ζ * 2 -α 2,i (ζ * 1 )| κ|ρ(z)| 1
2 for all ζ ∈ P 2κ|ρ(z)| (z) and so the inequality

ξ * 2 -α * 2,i (ξ * 1 ) ζ * 2 -α * 2,i (ζ * 1 )
1 holds true.

Second case:

If |ζ * 0,1 | < 4κ|ρ(z)| and |ζ * 0,2 | < (4κ|ρ(z)|) 1 2 , then δ(ξ, ζ 0 ) δ(ξ, z) + δ(z, ζ 0 ) |ρ (
∂ α+β ∂ζ * 1 α ∂ζ * 2 β ĥ2 Q 2 |ρ(z)| -α-β 2 c
(2) ∞ (g). Applying the same process with [START_REF] Montel | Sur une formule de Darboux et les polynômes d'interpolation[END_REF] to eP 1 , we get Therefore, ĝ2 satisfies (b) of Theorem 1.2 and of course, ĝ1 also does.

6.3. The L q -case. The assumption, under which a function g holomorphic on D can be written as g = g 1 f 1 +g 2 f 2 with g 1 and g 2 being holomorphic on D and belonging to L q (D), uses a κ-covering P κ|ρ(z j )| (z j ) j∈N in addition to the divided differences.

By transversality of X 1 and bD, and of X 2 and bD, for all j there exists w j in the complex tangent plane to bD ρ(z j ) such that π j , the orthogonal projection on the hyperplane X 2 given by Proposition 2.2 and for l = 1 and l = 2, we still denote by I l,j the set I l,j = {i, ∃z * 1 ∈ ∆ 0 (2κ|ρ(z j )|) such that |α )) and Q l,j = f l P l,j . We define ĥ(j) 1 and ĥ(j) 2 as ĥ1 and ĥ2 in the proof of Theorem 6.4. Instead of defining e (j)

1 and e (j) 2 by integrals over the set {|ξ| = (4κ|ρ(z j )|) 1 2 } as we defined e 1 and e 2 in the proof of Theorem 6.4, here we integrate over {( 72 κ|ρ(z j )|) We therefore have for all j and all z ∈ P κ|ρ(z j )| (z j ):

g(z) = h(j) 1 (z)P 1,j (z) + h(j) 2 (z)P 2,j (z) + P 1,j (z)P 2,j (z)e (j) (z). We split P κ|ρ(z j )| (z j ) in two parts as in Theorem 6. 1 (z) ĥ(j) 1 (z) + χ (j) 1 (z)P 2,j (z)e (j) (z) ,

g(j) 2 (z) = 1 Q (j)
2 (z) ĥ(j) 2 (z) + χ (j) 2 (z)P 1,j (z)e (j) (z) .

Therefore g = g(j)

1 f 1 + g(j) 2 f 2 on P κ|ρ(z j )| (z j ) and in order to apply Theorem 6.1, the assumptions (b) and (c) are left to be shown. As in the proof of fact 1, it follows from Lemma 6.2 and (21) that 1 Q 2,j (z) ĥ(j) 2 (z) uniformly with respect to z ∈ P 2κ|ρ(z j )| (z j ) and j ∈ N and therefore j∈N P 2κ|ρ(z j )| (z j ) 1 Q 2,j (z) ĥ(j) 2 (z) q dV (z) c

(l) q,κ,(z j ) (g). ( 22)

In particular 1 Q 2,j ĥ(j)

2 is an holomorphic function with L q -norm on P 2κ|ρ(z j )| (z j ) lower than (c

(2) q,κ,(z j ) (g)) 1 q . Thus Cauchy's inequalities imply, for all α, β ∈ N and all z ∈ P κ|ρ(z j )| (z j ), that

∂ α+β ∂z * 1 α ∂z * 2 β 1 Q 2,j ĥ(j) 2 (z) (c (l)
q,κ,(z j ) (g))

1
q |ρ(z j )| -3 q -α-β 2 . (23) Since g max(|f 1 |,|f 2 |) belongs to L q (D), g itself belongs to L q (D) and so P 2κ|ρ(z j )| (z j ) |e (j) (z)| q dV (z) |ρ(z j )| -q i 1,j +i 2,j 2 P 4κ|ρ(z j )| (z j )

|g(z)| q dV (z).

In particular, for all α and β and all z ∈ P κ|ρ(z j )| (z j ), we have

∂ α+β e (j) ∂z * 1 α ∂z * 2 β (z) |ρ(z j )| -3 q - i 1,j +i 2,j 2 -α-β 2 . ( 24
)
The inequalities (23) and (24) imply that the hypothesis (c) of Theorem 6.1 is satisfied by g(j)

2 for some large N , the same is also true for g(j)

1 . Now, for z belonging to U (j) 2 , we get from [START_REF] Varopoulos | BMO functions and the ∂-equation[END_REF]: P 2 ) j∈N satisfy the assumption (b) of Theorem 6.1 that we can therefore apply.

Proposition 2 . 4 .

 24 The following inequality holds uniformly for all ζ and z in D |ρ(ζ) + h(ζ, z), ζ -z | δ(ζ, z) + |ρ(ζ)| + |ρ(z)| Proof: We write z as z = ζ + λη ζ + µv ζ where η ζ is the unit outer normal and where v ζ belongs to T C ζ bD ρ(ζ) . With this notation, δ(ζ, z) |λ| + |µ| 2 , Re λ Re h(ζ, z), ζ -z and Im λ Im h(ζ, z), ζ -z . Since ρ is convex, there exists c positive and small such that for all z and ζ in D ρ

2 β

 2 in order to estimate P (j,k) l f l b m and the derivatives of χ (j,k) l

Proof:

  It suffices to construct, for all z near bD, two smooth functions ĝ1 and ĝ2 on P κ|ρ(z)| (z) which satisfy (a) and (b) of Theorem 1.2. Let ζ 0 be a point in bD. If f 1 (ζ 0 ) = 0 then f 1 does not vanish on a neighborhood U 0 of ζ 0 . Then we can define ĝ1 = g f 1 , ĝ2 = 0 which obviously satisfy (a) and (b) for all z ∈ D close to ζ 0 . We proceed analogously if f 2 (ζ 0 ) = 0.

  (ζ * 0,1 , ζ * 0,2 ) the coordinates of ζ 0 in the Koranyi coordinates at z. If |ζ * 0,1 | < 4κ|ρ(z)|, then for l = 1 and l = 2 we set I l = ∅, i l = 0, P l (ζ) = 1 and

1 2

 1 Proof: For l = 1 and l = 2, for all i ∈ I l and for all ζ * 1 ∈ ∆ 0 (4κ|ρ(z)|) we have, from Proposition 3.2, |α l,i (ζ * 1 )| ≤ (3κ|ρ(z)|) provided κ is small enough. Hence |P l (ζ)| |ρ(z)| i l 2 for all ζ ∈ P 4κ|ρ(z)| (z), and with assumption (i), we get for all ζ ∈ P 4κ|ρ(z)| (z)

Fact 3 :

 3 For l = 1 and l = 2, ζ ∈ P 2κ|ρ(z)| (z) and ξ ∈ P 4κ|ρ(z)| (z), we have uniformly with respect to z, ζ and ξ:

2 2 2 2 1 2 2 β 1 2

 22121 z)| and as in the proof of Lemma 5.1, |Q 2 (ξ)| = |f 2 (ξ)| |ρ(z)| p . From proposition 3.1,P 4κ|ρ(z)| (z) ∩ X 2 = ∅ so |f 2 (ζ)| |ρ(ζ)|p and again we are done in this case.Third case:If |ζ * 0,1 | < 4κ|ρ(z)| and |ζ * 0,2 | ≥ (4κ|ρ(z)|) ,then as in the third case of the proof of Lemma 5.1, f 2 (ξ) and f 2 (ζ) are comparable to |ζ * 0,2 | p 2 . Again we are done in this case and Fact 3 is proved. From (20) and (21), we get that ĝ2 is uniformly bounded. However, assumption (b) of Theorem 1.2 is a little stronger and we need that the derivatives of ĝ2 do not explode faster than |ρ(z)| α+ β 2 is P κ|ρ(z)| (z) for all α, β, α and β. Actually, inequality (17) and Cauchy's inequalities implies that, for all ζ ∈ P κ|ρ(z)| (z), |Q 2 (z + ζ * 1 η z + ξv z )|. With Lemma 5.1 and (21), we get

∂ α+β eP 1 ∂ζ * 1 α ∂ζ * 2 βi 1 -i 2 2 supP

 1212 (ζ) |ρ(z)| -α-β 2 c(g) |ρ(z)| 4κ|ρ(z)| (z) |Q 1 | + sup P 4κ|ρ(z)| (z) )| -α-ζ-β+β 2 c(g).

1 2

 1 }, i l,j = #I l,j , P l,j (ζ) = i∈I l,j (ζ * 2 -α (j) l,i (ζ * 1

  j (z * 1 , ξ)P 2,j (z * 1 , ξ)(z * 2 -ξ)dV (ξ).

2 P 1 ,j (ζ) > 1 3 f 2 2 P 2 3 f 2 i 2 ,j 2 P 2 .

 21322232222 κ|ρ(z j )| (z j ), f 1 (ζ)|ρ(z j )| i 1,j (ζ)|ρ(z j )| i 2,j κ|ρ(z j )| (z j ), 2 (ζ)|ρ(z j )|We still denote by χ a smooth function onC 2 \ {0} such that χ(z 1 , z 2 ) = 1 if |z 1 | > 2 3 |z 2 | and χ(z 1 , z 2 ) = 0 if |z 1 | <1 3 |z 2 |; and we set χ

z j +z * 1

 1 ηz j ,vz j [α 1,1 (z * 1 ), . . . , α 1,k (z * 1 )]

U 2 P 2 β

 22 ∩P κ|ρ(z j )| (z j ) 4κ|ρ(z j )| (z j ) |g(ζ * 1 , ξ)| max(|f 1 (ζ * 1 , ξ)|, |f 2 (ζ * 1 , ξ)|) q dV (ξ).Since (P κ|ρ(z j )| (z j )) j∈N is a κ-covering, this yields:j∈N U 2 ∩P κ|ρ(z j )| (z j ) all α, β ∈ N, (z) |ρ(z j )| -α-β 2 , (22) and (25) imply that (g (j)

orthogonal to w j passing through z j , is a covering of X 1 and X 2 . We denote by w * 1 , . . . , w * n an orthonormal basis of C n such that w * 1 = η z j and w * n = w j and we set P ε (z j ) = {z = z j + z * 1 w * 1 + . . .

where dV n-1 is the Lebesgue measure in C n-1 and g (l) = g f l , l = 1 or l = 2. Now we prove the following necessary conditions Theorem 6.5. Let g 1 and g 2 belonging to L q (D), q ∈ [1, +∞[, be two holomorphic functions on D and set g =

(ii) For l = 1 or l = 2 and any κ-covering P κ|ρ(z j )| (z j ) j , we have c (l) q,κ,(z j ) j (g) g l q L q (D) , Proof: The point (i) is trivial and we only prove (ii). As in the proof of Theorem 6.3, for all j ∈ N, all z ∈ P κ|ρ(z j )| (z j ) and all r ∈

After integration for r ∈ [(7/2κ|ρ(z j )|)

], Jensen's inequality yields

Now we integrate the former inequality for z ∈ P κ|ρ(z j )| (z j ) and get

|g l (z)| q dV n (z).

Since P κ|ρ(z j )| (z j ) j∈N is a κ-covering, we deduce from this inequality that c (l) q,κ,(z j ) j∈N (g) g l q L q (D) . Theorem 6.6. Let g be a holomorphic function on D belonging to the ideal generated by f 1 and f 2 , such that c (l) q,κ,(z j ) j (g) is finite and such that

Then there exist two holomorphic functions g 1 and g 2 which belong to L q (D) and such that g = g 1 f 1 + g 2 f 2 .

Proof: We aim to apply Theorem 6.1. For all j in N, in order to construct on P κ|ρ(z j )| (z j ) two functions g(j)

1 and g(j)

2 which satisfy the assumption of Theorem 6.1, we proceed as in the proof of Theorem 6.4. The main difficulty occurs, as in the proof of Theorem 6.4, when we are near a point ζ 0 which belongs to X 1 ∩ X 2 ∩ bD. We denote by (ζ * 0,1 , ζ * 0,2 ) the coordinates of ζ 0 in the Koranyi coordinates at z j . If |ζ * 0,1 | < 4κ|ρ(z j 0 )|, we set i 1,j = i 2,j = 0, I 1,j = I 2,j = ∅, P 1,j = P 2,j = 1, Q 1,j = f 1 and Q 2,j = f 2 . Otherwise, we use the parametrization α


Pierre Castelli

Stéphane Junca

J A Dieudonné

Oscillating waves and optimal smoothing effect for one-dimensional nonlinear scalar conservation laws

Keywords: 35L65, 35B65; Secondary: 35B10, 35B40, 35C20 conservation laws, nonlinear flux, geometric optics, oscillations, Sobolev spaces, smoothing effect, fractional BV spaces

Lions, Perthame, Tadmor conjectured in 1994 an optimal smoothing effect for entropy solutions of nonlinear scalar conservations laws ([19]). In this short paper we will restrict our attention to the simpler one-dimensional case. First, supercritical geometric optics lead to sequences of C ∞ solutions uniformly bounded in the Sobolev space conjectured. Second we give continuous solutions which belong exactly to the suitable Sobolev space. In order to do so we give two new definitions of nonlinear flux and we introduce fractional BV spaces.

Introduction and nonlinear flux definitions

We focus on oscillating smooth solutions for one-dimensional scalar conservations laws:

∂u ∂t + ∂f (u) ∂x = 0, u(0, x) = u 0 (x), t > 0, x ∈ R. (1)
The aim of this paper is to build solutions related to the maximal regularity or the uniform Sobolev bounds conjectured in [START_REF] Lions | A kinetic formulation of multidimensional scalar conservation laws and related equations[END_REF] for entropy solutions. In the one-dimensional case, piecewise smooth solutions with the maximal regularity are obtained in [START_REF] De Lellis | On the optimality of velocity averaging lemmas[END_REF] for power-law fluxes. We seek supercritical geometric optics expansions and some special oscillating solutions. Our constructions are valid for all C ∞ flux and show that one cannot expect a better smoothing effect.

The more complex multidimensional case is dealt with in [START_REF] Junca | High frequency waves and the maximal smoothing effect for nonlinear scalar conservation laws[END_REF][START_REF] Castelli | Oscillating solutions and bounds for the maximal smoothing effect for multidimensional scalar conservation laws[END_REF]. For recent other approaches we refer the reader to [START_REF] Dafermos | Regularity and large time behavior of solutions of a conservation law without convexity[END_REF][START_REF] Cheverry | Regularizing effects for multidimensional scalar conservation laws[END_REF][START_REF] De Lellis | Structure of entropy solutions for multidimensional scalar conservation laws[END_REF][START_REF] Crippa | Regularizing effect of nonlinearity in multidimensional scalar conservation laws[END_REF][START_REF] Jabin | Some regularizing methods for transport equations and the regularity of solutions to scalar conservation laws[END_REF][START_REF] Golse | Optimal regularizing effect for scalar conservation laws[END_REF]. Recall that the first famous BV smoothing effect for uniformly convex flux was given by the Oleinik one-sided Lipschitz condition in the 1950s (see for instance the books [START_REF] Dafermos | Hyperbolic Conservation Laws in Continuum Physics[END_REF][START_REF] Lax | Hyperbolic partial differential equations[END_REF]). For solutions with bounded entropy production, the smoothing effect is weaker than for entropy solutions ( [START_REF] De Lellis | On the optimality of velocity averaging lemmas[END_REF][START_REF] Golse | Optimal regularizing effect for scalar conservation laws[END_REF]).

Let us give various definitions of nonlinear flux from [START_REF] Lions | A kinetic formulation of multidimensional scalar conservation laws and related equations[END_REF][START_REF] Berthelin | Averaging lemmas with a force term in the transport equation[END_REF][START_REF] Junca | High frequency waves and the maximal smoothing effect for nonlinear scalar conservation laws[END_REF][START_REF] Bourdarias | BV s spaces and applications to scalar conservation laws[END_REF]. Throughout the paper, K denotes a compact real interval.

Definition 1 [Lions-Perthame-Tadmor nonlinear flux, [START_REF] Lions | A kinetic formulation of multidimensional scalar conservation laws and related equations[END_REF]] f ∈ C 1 (K, R) is said to be a nonlinear flux on K with degeneracy α if there exists a constant C > 0 such that for all δ > 0, sup

τ 2 +ξ 2 =1 (measure{v ∈ K, |τ + ξ f ′ (v)| < δ}) ≤ Cδ α . (2)
In [START_REF] Lions | A kinetic formulation of multidimensional scalar conservation laws and related equations[END_REF], the authors proved a smoothing effect for entropy solutions in some Sobolev space. They obtained uniform Sobolev bounds with respect to L ∞ bounds of initial data. Moreover, they conjectured a better smoothing effect :

u 0 ∈ L ∞ (R) ⇒ u(t, .) ∈ W s,1 loc (R x
), for all s < α and for all t > 0 (3)

where the parameter α is defined in Definition 1. They proved a weaker smoothing effect which was improved in [START_REF] Tadmor | Velocity averaging, kinetic formulations, and regularizing effects in quasi-linear PDEs[END_REF]. The conjecture [START_REF] Castelli | Lois de conservations scalaires: étude de solutions particulières en dimension 1 d'espace, effet régularisant[END_REF] is still an open problem.

In [START_REF] Junca | High frequency waves and the maximal smoothing effect for nonlinear scalar conservation laws[END_REF] was given another definition related to the derivatives of the flux. It generalizes a notion of nonlinear flux arising in geometric optics ( [START_REF] Chen | Validity of Nonlinear Geometric Optics for Entropy Solutions of Multidimensional Scalar Conservation Laws[END_REF]). The next one-dimensional definition of smooth nonlinear flux is simpler than in the multidimensional case ( [START_REF] Berthelin | Averaging lemmas with a force term in the transport equation[END_REF][START_REF] Junca | High frequency waves and the maximal smoothing effect for nonlinear scalar conservation laws[END_REF]).

Definition 2 [Smooth nonlinear flux, [START_REF] Junca | High frequency waves and the maximal smoothing effect for nonlinear scalar conservation laws[END_REF]

] f ∈ C ∞ (K, R) is said to be a nonlinear flux on K with degeneracy d if d = max u∈K min k ≥ 1, d 1+k f du 1+k (u) = 0 < +∞. ( 4
)
For the Burgers equation or for uniformly convex flux, the degeneracy is d = 1. That is the minimal possible value. For the cubic flux f (u) = u 3 on K = [-1, 1], the degeneracy is d = 2. The cubic flux is "less" nonlinear than the quadratic flux. Notice that, with this definition, a linear flux is not nonlinear: d = +∞ with the natural convention min(∅) = +∞.

This definition is equivalent to Definition 1 for C ∞ flux with α = 1 d ( [START_REF] Berthelin | Averaging lemmas with a force term in the transport equation[END_REF][START_REF] Junca | High frequency waves and the maximal smoothing effect for nonlinear scalar conservation laws[END_REF]). Therefore the Lions-Perthame-Tadmor parameter α is for smooth flux the inverse of an integer.

The conjectured smoothing effect (3) is proved for the first time in fractional BV spaces for the class of nonlinear (degenerate) convex fluxes ( [START_REF] Bourdarias | BV s spaces and applications to scalar conservation laws[END_REF]).

Definition 3 [Nonlinear degenerate convex flux, [START_REF] Castelli | Lois de conservations scalaires: étude de solutions particulières en dimension 1 d'espace, effet régularisant[END_REF][START_REF] Bourdarias | BV s spaces and applications to scalar conservation laws[END_REF]] Let f belong to C 1 (I, R) where I is an interval of R. We say that the degeneracy of f on I is at least p if the continuous derivative a(u) = f ′ (u) satisfies:

0 < inf I×I |a(u) -a(v)| |u -v| p (5)
The lowest real number p, if there exists, is called the degeneracy of f on I. If there is no p such that (5) is satisfied, we set p = +∞. Let f ∈ C 2 (I). We say that a real number

y ∈ I is a degeneracy point of f on I if f ′′ (y) = 0 (i.e. y is a critical point of a). For instance, if f is the power-law flux on [-1, 1]: f (u) = |u| 1+α where α > 0, then the degeneracy is p = max(1, α), ([3, 2]).
Remark 1 Definition 3 implies the convexity (or the concavity) of the flux f . Indeed, by definition there exists

C > 0 such that |f ′ (u) -f ′ (v)| ≥ C|u -v| p . Hence the difference f ′ (u) -f ′ (v) never vanishes for u = v.
Since the flux is continuous, it has got a constant sign for u > v, which implies the monotonicity of f ′ and then the convexity (or the concavity) of the flux.

Remark 2 Definition 3 is less general than Definition 1. Nevertheless, if f satisfies [START_REF] Chen | Validity of Nonlinear Geometric Optics for Entropy Solutions of Multidimensional Scalar Conservation Laws[END_REF] then it also satisfies ( 2) with α = 1 p , and also [START_REF] Castelli | Oscillating solutions and bounds for the maximal smoothing effect for multidimensional scalar conservation laws[END_REF] with

d = p when f is smooth.
The paper is organized as follows. The sequence given in Section 2 is exactly uniformly bounded in the Sobolev space conjectured in [START_REF] Lions | A kinetic formulation of multidimensional scalar conservation laws and related equations[END_REF]. Furthermore, this sequence is unbounded in all smoother Sobolev spaces. In Section 3, we build solutions with the suitable regularity (3).

Supercritical geometric optics 4

We give a sequence of high frequency waves with small amplitude exactly uniformly bounded in the Sobolev space conjectured in [START_REF] Lions | A kinetic formulation of multidimensional scalar conservation laws and related equations[END_REF]. The construction uses a WKB expansion ( [START_REF] Chen | Validity of Nonlinear Geometric Optics for Entropy Solutions of Multidimensional Scalar Conservation Laws[END_REF][START_REF] Rauch | Hyperbolic Partial Differential Equations and Geometric Optics[END_REF]).

Theorem 4 Let f ∈ C ∞ (K, R) be a nonlinear flux with degeneracy d defined by [START_REF] Castelli | Oscillating solutions and bounds for the maximal smoothing effect for multidimensional scalar conservation laws[END_REF]. There exists a constant state u ∈ K such that for any smooth periodic function U 0 satisfying for all 0 < ε ≤ 1, for all x ∈ R, u ε 0 (x) = u + εU 0 x ε d ∈ K, the following properties hold: 1. there exists a positive time T such that the entropy solution u ε of equation ( 1) with

u 0 = u ε 0 is smooth on [0, T ]×R for all 0 < ε ≤ 1 , 2. the sequence (u ε ) is uniformly bounded in W s,1 loc ([0, T ] × R) for s = α = 1 d and unbounded for s > α when U ′ 0 = 0 a.e.
The key point is to construct a sequence of very high frequency waves near the state u where the maximum in ( 4) is reached. Next we compute the optimal Sobolev bounds uniformly with respect to ε on the WKB expansion:

u ε (t, x) = u + ε U t, ϕ(t, x) ε d + ε r ε (t, x).
To estimate the remainder in Sobolev norms, we build a smooth sequence of solutions. It is quite surprising to have such smooth sequence on uniform time strip [0, T ]. Indeed, it is a sequence of solutions with no entropy production, without shock. But for any higher frequency, the life span T ε of u ε as a continuous solution goes towards 0 and oscillations are canceled ( [START_REF] Junca | High frequency waves and the maximal smoothing effect for nonlinear scalar conservation laws[END_REF]). Thus the construction is optimal.

Remark 3 The uniform life span of the smooth sequence

(u ε ) is at least T ∼ 1 sup θ d U 0 d θ ,
as one can see in [START_REF] Junca | High frequency waves and the maximal smoothing effect for nonlinear scalar conservation laws[END_REF]. So we can build such smooth sequence for any large time T and any non constant initial periodic profile U 0 small enough in C 1 . But we cannot take T = +∞ since shocks always occur when U 0 is not constant. [START_REF] De Lellis | On the optimality of velocity averaging lemmas[END_REF].

In this case, u = 0 and the sequence is simply u ε (t, x) = εU t, x ε p , the exact entropy solution of (1) and U(t, θ) is the entropy solution of

∂ t U + ∂ θ |U| 1+p = 0, U(0, θ) = U 0 (θ).
Proof: We give a sketch of the proof (see [START_REF] Junca | High frequency waves and the maximal smoothing effect for nonlinear scalar conservation laws[END_REF] for more details).

• Existence of u: the map u → min{k ≥ 1, f (1+k) (u) = 0} is upper semi-continuous, so it achieves its maximum on the compact K.

• WKB expansion ( [START_REF] Diperna | The validity of nonlinear geometric optics for weak solutions of conservation laws[END_REF][START_REF] Junca | A two-scale convergence result for a nonlinear conservation law in one space variable[END_REF][START_REF] Chen | Validity of Nonlinear Geometric Optics for Entropy Solutions of Multidimensional Scalar Conservation Laws[END_REF][START_REF] Junca | High frequency waves and the maximal smoothing effect for nonlinear scalar conservation laws[END_REF]): we plug the ansatz

u ε (t, x) = u + εU ε t, ϕ(t, x) ε d
into [START_REF] Berthelin | Averaging lemmas with a force term in the transport equation[END_REF]. Notice that the exact profile U ε depends on ε.

Set λ = f ′ (u) and b = f (1+d) (u) (1 + d)! = 0. After simplification,
the Taylor expansion of the flux f (u

+ εU ε ) = f (u) + ε λ U ε + ε 1+d b U 1+d ε -ε 2+d R ε (U ε )
gives an equation for the exact profile U ε and the phase ϕ:

∂U ε ∂t +b ∂U 1+d ε ∂θ = ε ∂R ε (U ε ) ∂θ , U ε (0, θ) = U 0 (θ), ϕ(t, x) = x-λt. (6)
The profile, which does not depend on ε, is

∂U ∂t + b ∂U 1+d ∂θ = 0, U(0, θ) = U 0 (θ). (7)
• Existence of smooth solutions for a time T > 0 independent of ε: it is a consequence of the method of characteristics. Indeed, the characteristics of equation ( 6) are a small perturbation of characteristics of equation [START_REF] Cheverry | Regularizing effects for multidimensional scalar conservation laws[END_REF].

• Approximation in C 1 ([0, T ]×R): it comes again from the method of characteristics since εR ε → 0. Notice that the expansion is valid in L 1 loc after shock waves ( [START_REF] Chen | Validity of Nonlinear Geometric Optics for Entropy Solutions of Multidimensional Scalar Conservation Laws[END_REF]). But it is not enough to estimate the Sobolev norms.

• Sobolev estimates: roughly speaking, the order of growth of the

s fractional derivative d s dx s U 0 x ε d is ε -sd .
For the profile U, this estimate is propagated along the characteristics on [0, T ]. We have the same estimate for U ε since U ε is near U in C 1 . Then we get the Sobolev bounds for u ε .

Oscillating solutions

In this section we give exact continuous solutions with the Sobolev regularity conjectured in [START_REF] Lions | A kinetic formulation of multidimensional scalar conservation laws and related equations[END_REF]. Indeed, we choose a suitable initial data such that the regularity is not spoiled by the nonlinearity of the flux for a positive time T . Furthermore, the conjectured smoothing effect is proved for the first time in fractional BV spaces ( [START_REF] Bourdarias | BV s spaces and applications to scalar conservation laws[END_REF]) for the degenerate convex class of nonlinear flux given by Definition 3. The next theorem shows the optimality of this smoothing effect. The optimality was also given in [START_REF] De Lellis | On the optimality of velocity averaging lemmas[END_REF] in Besov spaces framework. Let us introduce the BV s spaces.

Definition 5 (Fractional BV spaces)

Let I be a non empty interval of R. A partition σ of the interval I is a finite ordered subset:

σ = {x 0 , x 1 , • • • , x n } ⊂ I, x 0 < x 1 < • • • < x n .
We denote by S(I) the set of all partitions of I. Let s belong to ]0, 1] and p = 1 s ≥ 1. The s-total variation of a real function u on I is

T V s u{I} = sup σ∈S(I) n k=1 |u(x k ) -u(x k-1 )| p .
BV s (I) is the space of real functions u such that T V s u{I} < +∞.

BV s spaces are introduced in [START_REF] Bourdarias | BV s spaces and applications to scalar conservation laws[END_REF] for applications to conservation laws. These spaces measure the regularity of regulated functions: BV = BV 1 ⊂ BV s ⊂ L ∞ . Indeed, BV s (K) is very close to the Sobolev space W s,1/s (K) ( [START_REF] Bourdarias | BV s spaces and applications to scalar conservation laws[END_REF]):

• BV s (K) ⊂ W s-η,1/s (K) for all 0 < η < s . • BV s (K) = W s,1/s (K)
We now give continuous functions which have the BV s regularity.

Proposition 1 (A continuous BV s function [START_REF] Castelli | Lois de conservations scalaires: étude de solutions particulières en dimension 1 d'espace, effet régularisant[END_REF]) Let 0 < s < 1, 0 < η < 1-s and let g = g s,η be the real function defined on [0, 1] by g(0) = 0 and for all x ∈]0, 1] :

g(x) = x b cos π x c , where b = s + s 2 η and c = s η .
The function g belongs to BV s ([0, 1])∩C 0 ([0, 1]) but not to BV s+η ([0, 1]).

Notice that such example do not provide a function which belongs to BV s but not to η>0 BV s+η . Proof: The extrema of g are achieved

on x k = k -1/c . Let p = 1 s > 1 , q ≤ p and V q = +∞ k=1 |g(x k+1 ) -g(x k )| q .
Since qb/c = q(s + η), the asymptotic behavior |g(x k+1 )g(x k )| q ∼ 2 q k -qb/c when k → +∞ yields V q = +∞ when q = 1/(s + η) and V p < +∞. First this implies g / ∈ BV s+η . Second, for such oscillating function with diminishing amplitudes, we choose the optimal infinite partition to compute the s-total variation (see Proposition 2.3. p. 6 in [START_REF] Bourdarias | BV s spaces and applications to scalar conservation laws[END_REF]). Then g belongs to BV s .

We are now able to find oscillating initial data with the critical Sobolev exponent propagated by the nonlinear conservation law [START_REF] Berthelin | Averaging lemmas with a force term in the transport equation[END_REF].

Theorem 6 Assume f ∈ C ∞ (K, R) be
u(t, •) ∈ BV s (R, R) and u(t, •) / ∈ BV s+η (R, R).
The idea follows the K-S Cheng construction ( [START_REF] Cheng | The space BV is not enough for hyperbolic conservation laws[END_REF]) with the function g given in Proposition 1.

Proof: Let u ∈ K a point where the maximum of degeneracy of f is achieved. We also suppose that u ∈

• K (the proof of Theorem 6 is quite similar if u ∈ ∂K).

We define the initial condition u 0 by :

           u 0 (x) = u if x < 0 u 0 (x) = u + δg(x) if 0 ≤ x ≤ 1 u 0 (x) = u -δ if 1 < x ,
where δ > 0 is chosen such that for all x ∈ [0, 1], u + δg(x) ∈ K. Notice that for all x ∈ [0, 1], -1 ≤ g(x) ≤ 1 and g(1) = -1.

Then, following the method of characteristics, we define the function u(t, x) by : ]). Let be t > 0 and for all y, θ t (y) = y + ta(u + δg(y)).

           u(t, x) = 0 if x < 0 u(t, x) = u + δg(y) if x = y + ta(u + δg(y)), 0 ≤ y ≤ 1 u(t, x) = u -δ if 1 + ta(u -δ) < x . u 0 ∈ BV s ([0 , 1]) and u 0 / ∈ BV s+η ([0 , 1
Considering the change of variable y = xa(u)t, we can assume without loss of generality that f ′ (u) = a(u) = 0. Since f ∈ C ∞ (K, R), we derive from a Taylor expansion that

a(u) = 1 d! a (d) (u)(u -u) d + u u (u -s) d a (1+d) (s)ds . Defining I n (y) = 1 d! 1 0 (1 -r) d a (1+d) (u + r δ g(y))dr, J n (y) = 1 d! 1 0 r(1 -r) d a (2+d) (u + r δ g(y))dr,
we get then : We can take δ > 0 small enough such that T δ > T . Thus for all t ∈]0, T ], θ t is an homeomorphism between [0, 1] and [0, 1 + ta(uδ)]. Then u(t, x) is a continuous solution of equation ( 1) on [0, T ] × R. Furthermore, since u 0 ∈ BV s (I) and u 0 / ∈ BV s+η (I), where I = [0, 1], we deduce that for all t ∈]0, T ], u(t, •) ∈ BV s (J) and u(t, •) / ∈ BV s+η (J), where J = θ t (I) = [0, 1 + ta(uδ)]. Finally, as u(t, •) is constant outside J, we have proved that u(t, •) ∈ BV s (R) and u(t, •) / ∈ BV s+η (R).

θ t (y) = y + tδ d g(y) d 1 d! a (d) (u) + δg(y)I n (y) .
Remark 5 As in Remark 4, Theorem 6 is restricted for critical exponent s such that 1 s ∈ N. To obtain all exponent s ∈]0, 1], following [START_REF] De Lellis | On the optimality of velocity averaging lemmas[END_REF], we can consider a power-law flux with p = 1 s : f (u) = |u| 1+p . Our construction is quite similar as in the proof of Theorem 6 with u = 0 and δ > 0 small enough.

Remark 4

 4 For C ∞ flux, the parameter α in Definition 1 is always the inverse of an integer. To get supercritical geometric optics expansions for all α ∈]0, 1] and not only α ∈ 1 n , n ∈ N * , we shall consider power-law flux f (u) = |u| 1+p , where p = 1 α ∈ [1, +∞[, as in

  nonlinear in the sense of Definition 2. We denote by d its degeneracy and s = 1 d . For any η > 0 and any time T > 0 there exists a solution u ∈ C 0 ([0, T ] × R, R) such that for all t ∈ [0 , T ]

b cos π y c d- 1 b y b- 1 y c d- 1 b |y| c cos π y c + πc sin π y c .

 111c Note that g, I n , J n are bounded on [0, 1]. For y = 0, since b d = 1 + c, we have |g(y)| d y = O (y c ) at 0. Thus θ t is differentiable at 0 and dθ t dy (0) = 1. For y = 0, we havedθ t dy (y) = 1 + tδ d h n (y),whereh n (y) = g(y) d-1 g ′ (y) 1 (d -1)! a (d) (u) + (d + 1)δg(y)I n (y) + δ 2 g(y) 2 J n (y) .For y = 0, since b d = 1 + c, we haveg(y) d-1 g ′ (y) = y cos π y c + πcy b-c-1 sin π y c , g(y) d-1 g ′ (y) ≤ cos π Thus g(y) d-1 g ′ (y) is bounded on [0, 1].As h n is bounded on [0, 1], there exists T δ > 0 such that for all y ∈ [0, 1] and for all t ∈]0, T ], dθ t dy (y) > 0. Notice that lim δ→0 T δ = +∞.