Philippe Philippe Charpentier
email: philippe.charpentier@math.u-bordeaux.fr

Y Dupain

ON WEIGHTED L p -SOBOLEV ESTIMATES FOR SOLUTIONS OF THE ∂ -EQUATION ON LINEALLY CONVEX DOMAINS OF FINITE TYPE AND APPLICATION

Keywords: 2010 Mathematics Subject Classification. 32T25, 32T27 lineally convex, finite type, ∂ -equation, Bergman projection. 1

We obtain some weighted L p -Sobolev estimates with gain on p and the weight for solutions of the ∂ -equation in lineally convex domains of finite type in C n and apply them to obtain weighted L p -Sobolev estimates for weighted Bergman projections of convex domains of finite type for quite general weights equivalent to a power of the distance to the boundary.

INTRODUCTION

The motivation of this paper is to extend the weighted L p -Sobolev estimates for weighted Bergman projections obtained in [START_REF]Estimates for weighted Bergman projections on pseudo-convex domains of finite type in C n[END_REF] (1) of Theorem 1.1] for convex domains of finite type in C n to more general weights.

For this we use the method of Section 4 of [START_REF] Charpentier | Weighted and boundary l p estimates for solutions of the ∂ -equation on lineally convex domains of finite type and applications[END_REF] which needs weighted L p -Sobolev estimates for solutions of the ∂ -equation with suitable gains, the weight being a power of the distance to the boundary. So in this paper we try to obtain such estimates.

Estimates for derivatives of solutions of the ∂ -equation in convex domains of finite type has already a long history. The first result was obtained by W. Alexandre for C k estimates in [START_REF]C k -estimates for the ∂b -equation on convex domains of finite type[END_REF][START_REF]C k estimates for ∂ on convex domain of finite type[END_REF]. For L p -Sobolev estimates, with gain on the Sobolev index, partial results were obtained by Ahn H. and Cho H. in [START_REF] Ahn | Optimal sobolev estimates for ∂ on convex domains of finite type[END_REF] and complete results announced by Yao L. in [START_REF] Yao | Sobolev and Hölder estimates for homotopy operators of ∂ -equations on convex domains of finite multitype[END_REF].

Here we obtain partial weighted L p -Sobolev estimates, with both gain on p and the power of the weight, for solutions of the ∂ -equation on the more general case of lineally convex domains of finite type. We use the operator constructed and used in [START_REF] Charpentier | Estimates for Solutions of the ∂ -Equation and Application to the Characterization of the Zero Varieties of the Functions of the Nevanlinna Class for Lineally Convex Domains of Finite Type[END_REF][START_REF] Charpentier | Weighted and boundary l p estimates for solutions of the ∂ -equation on lineally convex domains of finite type and applications[END_REF] and we apply them to weighted Bergman projections.

The paper is organized as follows. In Section 2 we fix our notations and state the theorems about the estimates on solutions of the ∂ -equation. In Section 3 we give the main properties of the geometry of lineally convex domains of finite type, in Section 4 we recall the construction of the operator solving the ∂ -equation, and, in Section 5 we give the proofs of the Theorems stated in Section 2. Finally, in Section 6 we give the estimates for weighted Bergman projections we can obtain with our method.

NOTATIONS AND MAIN RESULTS

Throughout this paper Ω = {ρ < 0} is a smoothly bounded lineally convex domain of finite type ≤ m in C n which means that (c.f. [START_REF] Charpentier | Estimates for Solutions of the ∂ -Equation and Application to the Characterization of the Zero Varieties of the Functions of the Nevanlinna Class for Lineally Convex Domains of Finite Type[END_REF]) for all point p in the boundary ∂ Ω of Ω, there exists a neighborhood W of p such that, for all point z ∈ ∂ Ω ∩W ,

z + T 1,0 z ∩ (Ω ∩W ) = / 0,
where T 1,0 z is the holomorphic tangent space to ∂ Ω at the point z and that there exists a smooth defining function ρ of Ω such that, for δ 0 sufficiently small, the domains Ω t = {ρ(z) < t}, -δ 0 ≤ t ≤ δ 0 , satisfy the same condition. Theorem 2.1. Let Ω = {ρ < 0} be a smoothly bounded lineally convex domain of finite type ≤ m. Let p ∈ [1, +∞[. Let γ and γ ′ be two real numbers such that γ ′ > -1 and γp /m ≤ γ ′ ≤ γ. Then there exists a linear operator T , depending on ρ and γ, such that, for any ∂ -closed (0, r)-form (1 ≤ r ≤ n -1) whose coefficients are in L p Ω, δ γ Ω , T f is a solution of the equation ∂ (T f ) = f such that:

(1) For 1 ≤ p < m(γ

+ n) + 2 -(m -2)(r -1), T f L q δ γ ′ Ω f L p (δ γ Ω ) ,
where p, γ, q, and γ ′ satisfy the following condition:

if p = 1 and γ ≥ 0, 1 q > 1 p - 1-m p (γ-γ ′ )
m(γ ′ +n)+2-(m-2)(r-1) and 1 q = 1 p -

1-m p (γ-γ ′ ) m(γ ′ +n)+2-(m-2)(r-1)
otherwise.

(2) For p > m(γ

+ n) + 2 -(m -2)(r -1), T f Λ α f L p (δ γ Ω ) , with α = 1 m 1 -m(γ+n)+2-(m-2)(r-1) p .
Remark. For γ = γ ′ this statement is identical to [CD18, Theorem 2.1] but (1) is stronger when γ = γ ′ . For p = m(γ + n) + 2 -(m -2)(r -1) it could be proved that T f ∈ BMO(Ω) but we will not do it here.

Theorem 2.2. Let Ω = {ρ < 0} be a smoothly bounded lineally convex domain of finite type ≤ m. Let p ∈ [1, +∞[. Let γ and γ ′ be two real numbers such that γ ′ > -1 and γp /m ≤ γ ′ ≤ γ. Then there exists a linear operator T , depending on ρ and γ, such that, for any ∂ -closed (0, r)-form whose coefficients are in

L p 1 Ω, δ γ Ω , T f is a solution of the equation ∂ (T f ) = f such that: T f L q 1,r-1 δ γ ′ Ω f L p 1,r (δ γ Ω ) ,
where p, γ, q, and γ ′ satisfy the following condition:

1 q = 1 p -α with: (1) If γ ≥ p -1: α < 1-m p (γ-γ ′ ) m(γ ′ +n)+2-(m-2)(r-1) ; (2) If γ < p -1: γ ′ ≥ γ -γ+1 p and α < min 1-m p (γ-γ ′ ) m(γ ′ +n)+2-(m-2)(r-1) , γ+1 p -(γ-γ ′ ) 2n+γ ′ . Theorem 2.3.
Let Ω be a smoothly bounded lineally convex domain of finite type ≤ m. Let p ∈ [1, +∞[. Let d ≥ 2 be an integer. Let γ and γ ′ be two real numbers such that γ ′ > -1 and γp /m ≤ γ ′ ≤ γ. Then there exists a linear operator T , depending on ρ and γ, such that, for any

∂ -closed (0, r)-form (1 ≤ r ≤ n -1) f whose coefficients are in L p d Ω, δ γ Ω and p ∈ [1, +∞[, T f is a solution of the equation ∂ (T f ) = f such that: T f L q d,r-1 δ γ ′ Ω f L p d,r (δ γ Ω )
, if one of the three following conditions is satisfied:

(

1) f ∈ L p d,0,r Ω, δ γ Ω , (2) f ∈ L p d,r Ω, δ γ Ω and γ ≥ d p -1, (3) f ∈ L p d,r Ω, δ
γ Ω and d(p -1) ≤ γ < d p -1, with p, γ, q, and γ ′ satisfy:

• when condition (1) or (2) is verified, 1 q > 1 p - 1-m p (γ-γ ′ ) m(γ ′ +n)+2-(m-2)(r-1) , • when condition (3) is verified, 1 q > 1 p -min 1-m p (γ-γ ′ ) m(γ ′ +n)+2-(m-2)(r-1) , γ+1 p -(γ-γ ′ ) 2n+γ ′ .

GEOMETRY OF LINEALLY CONVEX DOMAINS OF FINITE TYPE

To simplify the reading of this paper we recall here the main properties of the geometry of lineally convex domains of finite type taken from [START_REF] Conrad | Anisotrope optimale Pseudometriken für lineal konvex Gebeite von endlichem Typ (mit Anwendungen)[END_REF], [START_REF] Charpentier | Estimates for Solutions of the ∂ -Equation and Application to the Characterization of the Zero Varieties of the Functions of the Nevanlinna Class for Lineally Convex Domains of Finite Type[END_REF] and [START_REF] Charpentier | Weighted and boundary l p estimates for solutions of the ∂ -equation on lineally convex domains of finite type and applications[END_REF].

For ζ close to ∂ Ω and ε ≤ ε 0 , ε 0 small, define, for all unitary vector v,

(3.1) τ (ζ , v, ε) = sup {c such that ρ (ζ + λ v) -ρ(ζ ) < ε, ∀λ ∈ C, |λ | < c} .
Note that the lineal convexity hypothesis implies that the function

(ζ , v, ε) → τ(ζ , v, ε) is smooth. In particular, ζ → τ(ζ , v, δ Ω (ζ )) is a smooth function. The pseudo-balls B ε (ζ )
(for ζ close to the boundary of Ω) of the homogeneous space associated to the anisotropic geometry of Ω are

(3.2) B ε (ζ ) = {ξ = ζ + λ u with |u| = 1 and |λ | < c 0 τ(ζ , u, ε)}
where c 0 is chosen sufficiently small depending only on the defining function ρ of Ω.

Let ζ and ε be fixed. Then, an orthonormal basis

(v 1 , v 2 , . . . , v n ) is called (ζ , ε)-extremal (or ε-extremal, or simply extremal) if v 1 is the complex normal (to ρ) at ζ ,
and, for i > 1, v i belongs to the orthogonal space of the vector space generated by (v 1 , . . . , v i-1 ) and minimizes τ (ζ , v, ε) in the unit sphere of that space. In association to an extremal basis, we denote

(3.3) τ(ζ , v i , ε) = τ i (ζ , ε). so i → τ i (ζ , ε) is increasing. Moreover the finite type condition implies that (3.4) τ 1 (ζ , ε) = ε and ε 1/2 τ i (ζ , ε) ε 1/m , i ≥ 1.
If v is any unit vector and λ ≥ 1,

λ 1/m τ(ζ , v, ε) ≤ 1 C τ(ζ , v, λ ε) ≤ Cλ τ(ζ , v, ε),
where m is the type of Ω.

Then we define the polydiscs AP ε (ζ ) by

(3.5) AP ε (ζ ) = AP(ζ , ε) = z = ζ + n ∑ k=1 λ k v k such that |λ k | ≤ c 0 Aτ k (ζ , ε) . P ε (ζ ) = P(ζ , ε) being the corresponding polydisc with A = 1. We choose c 0 < 1 so that z ∈ P ε (ζ ) implies (3.6) |δ Ω (z) -δ Ω (ζ )| ≤ ε/2.
Remark. Note that there is neither unicity of the extremal basis (v 1 , v 2 , . . . , v n ) nor of associated polydisk P ε (ζ ). However the polydisks associated to two different (ζ , ε)-extremal basis are equivalent. Thus in all the paper P ε (ζ ) will denote a polydisk associated to any (ζ , ε)-extremal basis and τ i (ζ , ε) the radius of P ε (ζ ).

Moreover there exists C > 0 such that for ζ close to ∂ Ω and ε > 0 small (see [START_REF] Conrad | Anisotrope optimale Pseudometriken für lineal konvex Gebeite von endlichem Typ (mit Anwendungen)[END_REF] and [START_REF] Charpentier | Estimates for Solutions of the ∂ -Equation and Application to the Characterization of the Zero Varieties of the Functions of the Nevanlinna Class for Lineally Convex Domains of Finite Type[END_REF]):

If v is a unit vector then:

(1)

z = ζ + λ v ∈ P ε (ζ ) implies |λ | ≤ Cτ(ζ , v, ε), (2) z = ζ + λ v with |λ | ≤ τ(ζ , v, ε) implies z ∈ CP ε (ζ ).
Lemma 3.1. For z close to ∂ Ω, ε small and ζ ∈ P ε (z) or z ∈ P ε (ζ ), we have, for all

1 ≤ i ≤ n: (1) τ i (z, ε) = τ (z, v i (z, ε) , ε) ≃ τ (ζ , v i (z, ε) , ε) where (v i (z, ε)) i is a (z, ε)-extremal basis; (2) τ i (ζ , ε) ≃ τ i (z, ε); (3) In the coordinate system (z i ) associated to the (z, ε)-extremal basis, ∂ ρ ∂ z i (ζ ) ε τ i
where τ i is either τ i (z, ε) or τ i (ζ , ε).

Remark.

In (1) above τ (ζ , v i (z, ε) , ε) is not τ i (ζ , ε
) because the extremal basis at z and ζ are different but (2) implies that these quantities are equivalent.

We also define

d(ζ , z) = inf {ε such that z ∈ P ε (ζ )} .
The fundamental result here is that d is a pseudo-distance. Moreover the pseudo-balls B ε and the polydiscs P ε are equivalent in the sense that there exists a constant K > 0 depending only on Ω and ρ such that

(3.7) 1 K P ε (ζ ) ⊂ B ε (ζ ) ⊂ KP ε (ζ ) so d(ζ , z) ≃ inf {ε such that z ∈ B ε (ζ )} and (3.8) d(ζ , z) ≃ d(z, ζ ). Moreover (3.9) δ (z) + d(z, ζ ) ≃ δ (ζ ) + d(z, ζ ).
Lemma 3.2. For z ∈ Ω, close to ∂ Ω, δ small, k = 0, . . . , n -1 and -1 < µ < 1, (3.10)

P δ (z) dλ (ζ ) |z -ζ | 2(n-k)-1+µ µ τ k+1 (z, δ ) 1-µ k ∏ j=1 τ 2 j (z, δ ).
For 0 < µ < 2, k = 0, . . . , n -1, T being the real tangent space to ρ at the point z (3.11)

P δ (z)∩T dσ (ζ ) |z -ζ | 2(n-k-1)+µ µ    τ 1 (z, δ )τ k+1 (z, δ ) 2-µ ∏ k j=2 τ j (z, δ ) 2 if k ≥ 2 τ 1 (z, δ )τ 2 (z, δ ) 2-µ if k = 1 τ 1 (z, δ ) 1-µ if k = 0 and 0 < µ < 1. For -1 < β ≤ 0, k = 1, . . . , n -1, 0 ≤ µ < 1 and δ ≥ δ Ω (ζ )
(3.12)

P δ (ζ ) δ Ω (z) β dλ (z) |z -ζ | 2(n-k)-1+µ β ,µ δ β τ k+1 (ζ , δ ) 1-µ k ∏ j=1 τ 2 j (ζ , δ ) .
Proof. We use the coordinate ξ = (ξ i ) 1≤i≤n system associated to a (z, δ )-extremal basis and we denote ξ ′ = (ξ k+1 , . . . , ξ n ) and, if k < n -1, ξ ′′ = (ξ k+2 , . . . , ξ n ).

• Proof of (3.10): Let

E 0 = ζ ∈ P δ (z) such that z ′ -ζ ′ ≤ 2τ k+1 (z, ζ )
and

E j = ζ ∈ P δ (z) such that |z ′′ -ζ ′′ | ∈ 2 j-1 τ k+1 (z, δ ), 2 j τ k+1 (z, δ ) , if k < n -1 / 0, if k = n -1.
(note that (see (3.6)) P δ (z) = ∪ j≥0 E j ) Then

E 0 dλ (ζ ) |z -ζ | 2(n-k)-1+µ µ τ k+1 (z, δ ) 1-µ k ∏ j=1 τ j (z, δ ) 2 and, if k < n -1, E j dλ (ζ ) |z -ζ | 2(n-k)-1+µ µ 2 j τ k+1 (z, δ ) -1-µ k+1 ∏ j=1 τ j (z, δ ) 2 ,
which proves (3.10). • Proof of (3.11): analog to the proof of (3.10).

• Proof of (3.12): Let T be the real tangent space to ρ at ζ and ν the inward real normal to ρ at ζ . For z ∈ P δ (ζ ), let us write z = Z(z) + t(z)ν, Z(z) ∈ T , and let t 0 such that Z(z)

+ t 0 ν = W (z) ∈ ∂ Ω so z = W (z) +tν with t = t(z) -t 0 . Then δ Ω (z) ≃ t and |z -ζ | ≃ |Z(z) -ζ | + |t(z)
| and, by hypothesis on δ and (3.6), δ Ω (z) ≤ 2δ , thus

P δ (ζ ) δ Ω (z) β dλ (z) |z -ζ | 2(n-k)-1+µ P δ (ζ )∩T 2δ 0 t β dt |W -tν -ζ | 2(n-k)-1+µ dσ P δ (ζ )∩T δ β +1 |Z -ζ | 2(n-k-1)+1+µ) dσ (Z) δ β +1 δ τ k+1 (ζ , δ ) 2-(1+µ) ∏ k j=2 τ j (ζ , δ ) 2 if k ≥ 1 δ β τ k+1 (ζ , δ ) 1-µ if k = 0 by (3.11) = δ β τ k+1 (ζ , δ ) 1-µ ∏ k j=1 τ j (ζ , δ ) if k ≥ 1, δ β τ k+1 (ζ , δ ) 1-µ if k = 0,
which ends the proof.

DEFINITION AND PROPERTIES OF A SOLUTION OF THE ∂ -EQUATION

To solve the ∂ -equation we use the method introduced in [START_REF] Charpentier | Estimates for Solutions of the ∂ -Equation and Application to the Characterization of the Zero Varieties of the Functions of the Nevanlinna Class for Lineally Convex Domains of Finite Type[END_REF]. Let us briefly recall it:

If

f is a smooth (0, r)-form ∂ -closed on Ω, then f (z) = (-1) r+1 ∂ z Ω f (ζ ) ∧ K r N (z, ζ ) - Ω f (ζ ) ∧ P r N (z, ζ ),
where K r N (resp. P r N ) is the component of a kernel K N (N ≥ 1), recalled in formula (4.4) below, of bi-degree (0, r -1) in z and (n, nr) in ζ (resp. the component of P N of be-degree (0, r) in z and (n, nr) in ζ ) constructed with the method of [AB82], using the Diederich-Fornaess support function constructed in [START_REF] Diederich | Lineally convex domains of finite type: holomorphic support functions[END_REF] (see also Theorem 2.2 of [START_REF] Charpentier | Estimates for Solutions of the ∂ -Equation and Application to the Characterization of the Zero Varieties of the Functions of the Nevanlinna Class for Lineally Convex Domains of Finite Type[END_REF]) and the function

G(ξ ) = 1 ξ N with a sufficiently large number N (instead of G(ξ ) = 1 ξ in formula (2.7) of [CDM14a]).
Then, the form Ω f (ζ ) ∧ P r N (z, ζ ) is ∂ -closed and the operator T solving the ∂ -equation in theorems 2.1, 2.2 and 2.3 is defined by

(4.1) T f (z) = (-1) r+1 Ω f (ζ ) ∧ K r N (z, ζ ) -∂ * N Ω f (ζ ) ∧ P r N (z, ζ ) ,
Lemma 4.1. Let r ≥ 1, -1 < γ, d ∈ N. Let f be a ∂ -closed (0, r)-form with coefficients in L 1 (Ω, δ γ Ω ) and let g = Ω f (ζ ) ∧ P r N (z, ζ ). Then for N ≥ N 0 = N 0 (d) ∂ * N (g) is a solution of the equation ∂ u = g satisfying ∂ * N (g) C d (Ω) ≤ C k,N f L 1 (Ω,δ γ ) .
Finally the proofs of our theorems are reduced to the proofs of estimates for the operator T K defined by

(4.2) T K ( f ) = Ω f (ζ ) ∧ K r N (z, ζ ).
Let us now recall (from [START_REF] Charpentier | Estimates for Solutions of the ∂ -Equation and Application to the Characterization of the Zero Varieties of the Functions of the Nevanlinna Class for Lineally Convex Domains of Finite Type[END_REF]) the explicit formula for the kernel K N : Let S 0 (z, ζ ) be the holomorphic support function of Diederich-Fornaess [START_REF] Diederich | Lineally convex domains of finite type: holomorphic support functions[END_REF] (see also Theorem 2.2 of [START_REF] Charpentier | Estimates for Solutions of the ∂ -Equation and Application to the Characterization of the Zero Varieties of the Functions of the Nevanlinna Class for Lineally Convex Domains of Finite Type[END_REF]). Define two

C ∞ functions χ 1 (z, ζ ) = χ (|z -ζ |) and χ 2 (z) = χ (δ ∂ Ω (z)) (δ ∂ Ω denoting the distance to the boundary of Ω) where χ and χ are C ∞ functions, 0 ≤ χ, χ ≤ 1, such that χ ≡ 1 on [0, R /2] and χ ≡ 0 on [R, +∞[ and χ ≡ 1 on [0, η 1/2]
and χ ≡ 0 on [η 0 , +∞[, where R and η 1 are small enough. Then we define

χ(z, ζ ) = χ 1 (z, ζ )χ 2 (ζ ) and S(z, ζ ) = χ(z, ζ )S 0 (z, ζ ) -(1 -χ(z, ζ ))|z -ζ | 2 = n ∑ i=1 Q i (z, ζ ) (z i -ζ i ) .
Define the two forms s and Q used in [START_REF] Andersson | Henkin-Ramirez formulas with weight factors[END_REF] in the construction of the Kopelman formula by

s(z, ζ ) = n ∑ i=1 ζ i -z i d (ζ i -z i ) and Q(z, ζ ) = 1 K 0 ρ(ζ ) n ∑ i=1 Q i (z, ζ )d (ζ i -z i ) ,
where K 0 is a large constant chosen so that [CD18, formula (3.12) p. 204] Lemma 4.2. For z 0 close to ∂ Ω, ε small and z, ζ ∈ P ε (z 0 ), in the coordinate system (ζ i ) associated to a (z 0 , ε)-extremal basis, we have:

ℜe ρ(ζ ) + 1 K 0 S(z, ζ ) < ρ(ζ ) 2 . The Q i (z, ζ ) are C ∞ in Ω× Ω
(1) |Q i (z, ζ )| + |Q i (ζ , z)| ε τ i , (2) ∂ Q i (z,ζ ) ∂ ζ j + ∂ Q i (z,ζ ) ∂ ζ j + ∂ Q i (z,ζ ) ∂ z j ε τ i τ j ,
where τ i are either τ i (z, ε), τ i (ζ , ε) or τ i (z 0 , ε).

Moreover we have ([CD18, Formula (3.12) and Lemma 3.5])

(4.3) ρ(ζ ) + 1 K 0 S(z, ζ ) |ρ(ζ )| + |ρ(z)| + d(z, ζ ). Recall that if ζ ∈ ∂ Ω, ℜeS(z, ζ ) < 0 for z ∈ Ω, and, that s satisfies |z -ζ | 2 = | s, z -ζ | ≤ C |z -ζ |, z, ζ ∈ Ω. Then the kernel K N is (formula (2.7) of [CDM14a]) (4.4) K N (z, ζ ) = C n n-1 ∑ k=0 (n -1)! k! ρ(ζ ) 1 K 0 S(z, ζ ) + ρ(ζ ) N+k s(z, ζ ) |ζ -z| 2(n-k) ∧ (dQ) k ∧ (ds) n-k-1 .
The main articulations of the proofs are as follows:

The proof of Theorem 2.1 is given in Section 5.1 and the proofs of Theorems 2.2 and 2.3 are given in next sections.

To estimate derivatives of T K ( f ) we note that, for z ∈ Ω, the coefficients of ζ → K N (z, ζ ) and their derivatives up to order N -1 are smooth in Ω \ {z} and equal to 0 when ζ ∈ ∂ Ω. Thus we define K N (z, ζ ) the kernel obtained extending the coefficients of k) . Moreover we use an other trick (inspired by [START_REF] Wu | Sobolev and Hölder Estimates for ∂ on Bounded Convex Domains in C 2[END_REF]) on the derivatives of 1 K 0 S(z, ζ ) + ρ(ζ ). Finally the proofs are done using the following Lemma which is easily obtained reading the proof of [CD18, Lemma 3.10] (see also [START_REF] Range | Holomorphic Functions and Integrals Representations in Several Complex Variables[END_REF]Appendix B]): Lemma 4.3. Let Ω be a smoothly bounded domain in C n . Let µ and ν be two positive measures on Ω. Let K be a measurable function on Ω × Ω and let us consider the linear operator T defined by

K N by 0 if (z, ζ ) ∈ Ω × C n \ Ω . Then T K ( f ) = C n f (ζ ) ∧ K r N (z, ζ ) and, making the change of variable ζ -z = ξ we can derivate T K ( f ) (less than N -1 times) without derivating the expression 1 |z-ζ | 2(n-
T f (z) = Ω K(z, ζ ) f (ζ )dµ(ζ ).
Denote δ Ω the distance to the boundary of Ω. Assume that there exists a real number s ≥ 1 such that:

(1) If there exists a positive number κ 0 > 0 and, for all 0 < κ ≤ κ 0 , a positive constant C κ such that,

Ω |K(z, ζ )| s δ Ω (ζ ) -κ dµ(ζ ) ≤ C κ δ Ω (z) -κ , z ∈ Ω and Ω |K(z, ζ )| s δ Ω (z) -κ dν(z) ≤ C κ δ Ω (ζ ) -κ , ζ ∈ Ω, then T is bounded from L p (Ω, µ) to L q (Ω, ν) for all 1 < p < s s-1 with 1 q = 1 p -s-1 s . (2) If there exists a positive constant C such that Ω |K(z, ζ )| s dν(z) ≤ C, ζ ∈ Ω, then T is bounded from L 1 (Ω, µ) to L q (Ω, ν) with q = s. For 0 ≤ k ≤ n -1 and f a (0, r) form (1 ≤ r ≤ n -1), let us denote T k f (z) = (n -1)! k! Ω ρ(ζ ) 1 K 0 S(z, ζ ) + ρ(ζ ) N+k s(z, ζ ) |ζ -z| 2(n-k) ∧ (dQ) k ∧ (ds) n-k-1 ∧ f (ζ ) = Ω K k (z, ζ ) ∧ f (ζ ) so that T K f = ∑ n-1 k=0 C n T k f .

PROOFS OF THEOREMS

As sharp estimates of the coefficients of K N using the finite type hypothesis are made only when ζ and z are close and close to ∂ Ω, we consider a recovering of Ω as follows: let ε 0 be a small positive number, P l , 1 ≤ l ≤ M, points in ∂ Ω, V l = B (P l , ε 0 ), 1 ≤ l ≤ M, the euclidean balls centered at P l and radius ε 0 and V 0 an open set relatively compact in Ω such that {V l , 0 ≤ l ≤ M} is an open covering of Ω. Precisely ε 0 is chosen sufficiently small so that:

• For z and ζ in the balls 2V l = B (P l , 2ε 0 ), 1 ≤ l ≤ M, all the properties and es- timates associated to the finite type hypothesis are valid (see (3.1) to (3.12) and Lemma 4.2) including Lemmas 5.1 to 5.4;

• For z and ζ in 2V l ∩ Ω, 1 ≤ l ≤ M, Q(z, ζ ) is holomorphic in the z variable.
Then we choose smooth cut-off functions χ l ∈ C ∞ 0 (V l ), ∑ M l=0 χ l ≡ 1 on Ω and we write

T k f (z) = M ∑ l=0 Ω K k (z, ζ )χ l (ζ ) ∧ f (ζ ) = M ∑ l=0 T l k f (z) so T K f (z) = M ∑ l=0 n-1 ∑ k=0 C n T l k f (z).
By standard regularization procedure, to prove Theorem 2.1, Theorem 2.2 and Theorem 2.3 under condition (2) or (3) (resp. Theorem 2.3 under condition (1)) we can assume that the coefficients of f are C ∞ -smooth in Ω (resp. in D(Ω)). So in all the proofs below the coefficients of the form f are in C ∞ Ω .

Moreover, for l ≥ 1, if z ∈ 2V l , using the holomorphic property of Q, an elementary calculus shows that, in the above expression of

T K f (z), T l k f (z) appears only for k ≤ n -r and the kernel K k (z, ζ )χ l (ζ ) is equal to (5.1) K k (z, ζ )χ l (ζ ) = χ l (ζ )C n,k ρ(ζ ) 1 K 0 S(z, ζ ) + ρ(ζ ) N+k s(z, ζ ) |ζ -z| 2(n-k) ∧ ∂ ζ Q(z, ζ ) k ∧ ∂ ζ s(z, ζ ) n-k-r ∧ ∂ z s(z, ζ ) r-1 with s(z, ζ ) = ∑ i ζ i -z i dζ i and Q(z, ζ ) = 1 K 0 ρ(ζ ) ∑ i Q i (z, ζ )dζ i .
Lemma 5.1. Let D be a derivative of order j in z and ζ . Denote

F(z, ζ ) = ρ(ζ )∂ ζ as Q b s (z, ζ ) + Q b s (z, ζ )∂ ζ as ρ(ζ ).
Then for z and ζ close and close to the boundary of

Ω (1) |F(z, ζ )| α 2 τ as τ bs ; (2) |DF(z, ζ )| 1 |ρ(ζ )| α 2 τ as τ bs if j = 1; (3) |DF(z, ζ )| 1 1 |ρ(ζ )| 2
α 2 τ as τ bs for j ≥ 2;

where α = δ Ω (z) + d(z, ζ ) ≃ δ Ω (ζ ) + d(z, ζ ) and τ i = τ i (z, α) ≃ τ i (ζ , α).
Proof. Comes from (3) of Lemma 3.1 and Lemma 4.2. 5.1. Proof of Theorem 2.1. The result stated here is slightly stronger than the one given in [START_REF] Charpentier | Weighted and boundary l p estimates for solutions of the ∂ -equation on lineally convex domains of finite type and applications[END_REF] and we give a detailed proof. We assume N > 2 |γ| + 1 + N 0 .

Lemma 5.2. If z and ζ close to the boundary and close together

|K k (z, ζ )| ρ(ζ ) α N-1 1 |z -ζ | 2(n-k)-1 k ∏ i=1 τ -2 i where τ i = τ i (z, α) or τ i = τ i (ζ , α) with α = δ (z) + d(z, ζ ) ≃ δ (ζ ) + d(z, ζ ), otherwise |K k (z, ζ )| 1 |z -ζ | 2(n-k)-1 .
This Lemma follows easily formula (5.1) and Lemma 5.1.

Proof of (1) of Theorem 2.1. Let us consider the operators T l k , 0 ≤ l ≤ M. We use Lemma 4.3, with s = 1 + µ and 0 ≤ µ < 1/(2n -1), writing

T l k f (z) = Ω K l k (z, ζ ) f (ζ )δ (ζ ) γ-γ ′ p dµ(ζ ) with K l k (z, ζ ) = χ l (ζ )K k (z, ζ )δ (ζ ) -γ+γ ′ p -γ ′ and dµ(ζ ) = δ (ζ ) γ ′ dλ (ζ ):
Thus to prove that T k maps L p (δ γ dλ ) into L q δ γ ′ dλ (that is T l k defined by T l k g(z)

= Ω K l k (z, ζ )g(ζ )dµ(ζ ) maps L p (dµ) into L q (dν) with dν(z) = δ (z) γ ′ dλ (z))
we have to estimate the two following integrals:

I 1 (z) = Ω K l k (z, ζ ) 1+µ δ (ζ ) γ ′ -ε dλ (ζ ), 0 ≤ l ≤ M and I 2 (ζ ) = Ω K l k (z, ζ ) 1+µ δ (z) γ ′ -ε dλ (z), 0 ≤ l ≤ M, for 0 ≤ ε < min {γ ′ + 1, 1}. If l = 0 the estimate of I 2 (z) is trivial because • if |z -ζ | ≤ dist (V 0 , ∂ Ω) /2, δ Ω (z) ≃ δ Ω (ζ ) and δ Ω (ζ ) -(1+µ) -γ+γ ′ p -γ ′ δ Ω (z) γ ′ -ε is bounded, • if |z -ζ | > dist (V 0 , ∂ Ω) /2, |K k (z, ζ )| 1+µ δ Ω (ζ ) -(1+µ) -γ+γ ′ p -γ ′ is bounded and γ ′ -ε > -1, so I 2 (ζ ) ε,µ 1 δ Ω (ζ ) -ε .
If l ≥ 1 and z / ∈ 2V l then:

|K k (z, ζ )| 1+µ δ Ω (ζ ) -(1+µ) -γ+γ ′ p -γ ′
is bounded, trivially if -γ+γ ′ p -γ ′ ≤ 0 and other- wise by Lemma 5.2, and we obtain the same result because γ ′ -ε > -1.

Similar result is easily obtained for I 1 (z).

So we only have to estimate these integrals when l ≥ 1 and z

∈ 2V l . For -1 < γ ′ ≤ γ, γ -γ ′ < p/m, γ < m + n + 1 let µ 0 (k) =        1-m p (γ-γ ′ ) 2(n-k)-1+m(k+1+γ ′ + γ-γ ′ p ) if k = 0 1-γ-γ' p 2n-1+γ ′ + γ-γ ′ p if k = 0 , Proposition 5.1. For l ≥ 1, z ∈ 2V l and µ ≤ µ 0 (k), I 1 (z) ε δ (z) -ε if 0 < ε < γ ′ + 1. Proof. Denote P o (z) = P(z, δ (z)) and P i (z) = P z, 2 i δ (z) \ P z, 2 i-1 δ (z) for i ≥ 1. Then, using Lemma 5.2 or a direct calculus (N > 2 |γ|), if ζ ∈ P i (z), by (3.6) K l k (z, ζ ) 1+µ 0 δ (ζ ) γ ′ -ε |K k (z, ζ )| 1+µ 0 2 i δ (z) (1+µ 0 ) γ ′ -γ p -µ 0 γ ′ -ε .
(1) Case k = 0 By Lemma 3.2, (3.10), properties (3.4)

P i (z) |K k (z, ζ )| 1+µ 0 dλ (ζ ) τ k+1 z, 2 i δ (z) 1-µ 0 (2(n-k)-1) k ∏ j=1 τ -2 j z, 2 i δ (z) µ 0 2 i δ (z) 1 m (1-µ 0 (2(n-k)-1))-µ 0 (k+1) .
Then

P i (z) K l k (z, ζ ) 1+µ 0 δ (ζ ) γ ′ -ε dλ (ζ ) 2 i δ (z) 1 m (1-µ 0 (2(n-k)-1))-µ 0 (k+1)+(1+µ 0 ) γ ′ -γ p -µ 0 γ ′ -ε 2 i δ (z) -ε-µ 0 2(n-k)-1 m +k+1+ γ-γ ′ p +γ ′ + 1 m + γ ′ -γ p 2 i δ (z) -ε .
(2) Case k = 0 Similarly

P i (z) K l 0 (z, ζ ) 1+µ 0 δ (ζ ) γ ′ -ε dλ (ζ ) 2 i δ (z) -ε-µ 0 γ ′ -(2n-1)µ 0 +(1+µ 0 ) γ ′ -γ p 2 i δ (z) -ε .
Thus I 1 (z) ε δ (z) -ε which ends the proof of the Proposition.

Proposition 5.2. For l ≥ 1, z ∈ 2V l I ′ 2 (ζ ) = Ω∩2V l K l k (z, ζ ) 1+µ δ (z) γ ′ -ε dλ (z) ε δ (ζ ) -ε
with 0 < ε < min {γ ′ + 1, 1} and µ ≤ µ 0 (k) or ε = 0 with µ ≤ µ 0 (k) if γ < 0 and µ < µ 0 (k) otherwise.

Proof. Assume k = 0. By Lemma 5.2 or trivially if -γ+γ ′ p -γ ′ ≤ 0, for z ∈ P i (ζ )

K l k (z, ζ ) 1+µ δ (z) γ ′ -ε |K k (z, ζ )| 1+µ 2 i δ Ω (ζ ) -γ+γ ′ p -γ ′ δ (z) γ ′ -ε
and, by (3.12), γ ′ -ε > -1 implies

I ′ 2 (ζ ) 2 i δ Ω (ζ ) -ε-µ 2(n-k)-1 m +k+1+ γ-γ ′ p +γ ′ + 1 m + γ ′ -γ p .
The case k = 0 is easily treated as in the previous proposition which ends the proof.

The proof of (1) of Theorem 2.1 is complete.

Proof of (2) of Theorem 2.1. We give two different methods for the study of T 0 f and T k f , k ≥ 1.

• Case k ≥ 1. As in [START_REF] Charpentier | Weighted and boundary l p estimates for solutions of the ∂ -equation on lineally convex domains of finite type and applications[END_REF] we use the Hardy-Littlewood lemma, so, we have to prove the following inequality

|∇ z T k f | δ Ω (z) α-1 , if p > m(γ + n) + 2 -(m -2)(r -1) with α = 1 m 1 -m(γ+n)+2-(m-2)(r-1) p
. By Hölder inequality it follows from (5.2)

Ω |∇ z K k (z, ζ )χ l (ζ )| p p-1 δ Ω (ζ ) -γ p-1 δ Ω (z) p(α-1) p-1 . If l = 0 |∇ z K k (z, ζ )χ l (ζ )| p p-1 δ Ω (ζ ) -γ p-1 1 |z -ζ | (2n-1) p p-1 1 |z -ζ | 2n 2n-2 2n-1 and (5.2) is trivial. If l ≥ 1 and z / ∈ 2V l |∇ z K k (z, ζ )χ l (ζ )
| is bounded (by Lemma 5.2) and (5.2) is clear because γ p-1 < 1. In the last case, the Q i being holomorphic in z, it remains only the terms corresponding to k ≤ nr and by Lemma 5.2, Lemma 5.1, the fact that for s = s ′ a s = a s ′ and b s = b s ′ and the increasing property of the τ i , if z ∈ 2V l we have

∇ z K k (z, ζ )χ l (ζ )δ Ω (ζ ) -γ/p 1 |z -ζ | 2(n-k)-1 1 δ (z) + d(z, ζ ) 1+ γ p k ∏ i=1 τ i (z, δ (z) + d(z, ζ )) -2 + 1 |z -ζ | 2(n-k) 1 δ (z) + d(z, ζ ) γ p k ∏ i=1 τ i (z, δ (z) + d(z, ζ )) -2 = W 1 + W 2 .
Let P 0 (z) = P(z, δ (z)) and P j (z) = P z, 2 j δ (z) \ P z, 2 j-1 δ (z) . Then, if ζ ∈ P j (z), by (3.6),

|W 1 | p p-1 1 |z -ζ | 2(n-k)-1 p p-1 2 j δ (z) -p p-1 1+ γ p k ∏ i=1 τ i (z, δ (z) + d(z, ζ )) -2 p p-1
and by Lemma 3.2

P j (z) 1 |z -ζ | 2(n-k)-1 p p-1 dλ (ζ ) τ k+1 (z, 2 i δ Ω (z)) 1-(2(n-k)-1) 1 p-1 k ∏ i=1 τ i (z, 2 j δ Ω (z)) 2 . Then P j (z) |W 1 | p p-1 τ k+1 (z, 2 j δ Ω (z)) 1-(2(n-k)-1) p-1 2 j δ (z) 1- (2(n-k)-1) p-1 k ∏ i=1 τ i (z, 2 j δ Ω (z)) -2 p-1 2 j δ (z) 1 m 1- (2(n-k)-1) p-1 - p+γ+k+1 p-1 by (3.4) 2 j δ (z) p m(p-1) 1-m- m(γ+n-r+1)+2r p (because k ≤ n -r) = 2 j δ (z) p(α-1) p-1 . Similarly, if ζ ∈ P j (z), |W 2 | p p-1 1 |z -ζ | 2(n-k) p p-1 2 j δ (z) -γ p-1 k ∏ i=1 τ i z, 2 j δ (z) -2p p-1
and as Lemma 3.2 implies

P j (z) 1 |z -ζ | 2p(n-k) p-1 τ k z, 2 j δ (z) 2- 2(n-k) p-1 k-1 ∏ i=1 τ i z, 2 j δ (z) 2 we get P j (z) |W 2 | p p-1 2 j δ (z) -1 p-1 (2n-2+γ) if k > 1 2 j δ (z) -1 p-1 (2n+γ) if k = 1 2 j δ (z) p(α ′ -1) p-1 with α ′ = 1 -2n+γ p > α.
• Case k = 0. In that case as we cannot derive T 0 f we make a direct computation for kernels of the form

K 0 (z, ζ ) = ζ t -z t |z -ζ | 2n ρ(ζ ) 1 K 0 S(z, ζ ) + ρ(ζ ) N χ l (ζ ).
We have to estimate the difference

|u 0 (z 0 ) -u 0 (z 1 )| ≤ Ω |K 0 (z 0 , ζ ) -K 0 (z 1 , ζ )| | f (ζ )| dλ (ζ ) Ω | f (ζ )| p δ Ω (ζ ) γ dλ (ζ ) 1/p Ω |K 0 (z 0 , ζ ) -K 0 (z 1 , ζ )| p p-1 δ Ω (ζ ) -γ p-1 dλ (ζ ) p-1 p .
Let us write, z 0 and z 1 being fixed,

K 0 (z 0 , ζ ) -K 0 (z 1 , ζ ) = ζ t -z 0,t |z 0 -ζ | 2n - ζ t -z 1,t |z 1 -ζ | 2n ρ(ζ ) 1 K 0 S(z 0 , ζ ) + ρ(ζ ) N χ l (ζ ) +   ρ(ζ ) 1 K 0 S(z 0 , ζ ) + ρ(ζ ) N - ρ(ζ ) 1 K 0 S(z 1 , ζ ) + ρ(ζ ) N   ζ t -z 1,t |z 1 -ζ | 2n χ l (ζ ) = K 1 0 (ζ ) + K 2 0 (ζ ).
Let us denote T 0 the real tangent space to ρ and η 0 the inward real normal to ρ at the point z 0 . Let Z ∈ T 0 and W ∈ ∂ Ω such that ζ = W + tη 0 and W = Z + t 1 η 0 . Let us first estimate

I (z 0 , z 1 ) = Ω K 1 0 (ζ ) p p-1 δ Ω (ζ ) -γ p-1 dλ (ζ ). Let P 0 (z 0 ) = B (z 0 , 3 |z 0 -z 1 |), P j (z 0 ) = B z 0 , 3 j+1 |z 0 -z 1 | \ B z 0 , 3 j |z 0 -z 1 |
and Q j (z 0 ) = P j (z 0 ) ∩ T 0 where B(z, r) denotes the euclidean ball of center z and radius r in C n .

Let us first consider I

0 (z 0 , z 1 ) = P 0 (z 0 ) K 1 0 (ζ ) p p-1 δ Ω (ζ ) -γ p-1 dλ (ζ ): I 0 (z 0 , z 1 ) P 0 (z 0 ) δ Ω (ζ ) -γ p-1 |z 0 -ζ | (2n-1) p p-1 dλ (ζ ) + P 0 (z 0 ) δ Ω (ζ ) -γ p-1 |z 1 -ζ | (2n-1) p p-1 dλ (ζ ) = I 1 0 (z 0 , z 1 ) + I 2 0 (z 0 , z 1 ) .
To calculate the integral I 1 0 (z 0 , z 1 ) we use the coordinate system (Z,t) ∈ T × Rη 0 :

δ (ζ ) ≃ t, |z 0 -ζ | ≃ |Z -z 0 | + |t + t 1 | (note that γ p-1 < 1) I 1 0 (z 0 , z 1 ) |Z-z 0 |≤3|z 0 -z 1 | |Z -z 0 | -1 p-1 (γ+(2n-1)p)+1 |z 0 -z 1 | 1 p-1 (p-(γ+2n)) .
Now it is easy to see that the same estimate is valid for I 2 0 (z 0 , z 1 ). Let us now look at

I j (z 0 , z 1 ) = P j (z 0 ) K 1 0 (ζ ) p p-1 δ Ω (ζ ) -γ p-1 dλ (ζ ), j ≥ 1: using that ζ t -z 0,t |z 0 -ζ | 2n-1 - ζ t -z 1,t |z 1 -ζ | 2n-1 |z 0 -z 1 | 3 j |z 0 -z 1 | -2n
we get

I j (z 0 , z 1 ) |z 0 -z 1 | p p-1 3 j |z 0 -z 1 | -2np p-1 +2n-1-γ p-1 +1 3 -j(2n+γ) p-1 |z 0 -z 1 | p-(2n+γ) p-1 Thus ∑ j I j (z 0 , z 1 ) |z 0 -z 1 | p-(2n+γ) p-1
. Finally we have

Ω K 1 0 (ζ ) p p-1 δ Ω (ζ ) -γ p-1 dλ (ζ ) p-1 p |z 0 -z 1 | 1-2n+γ p .
To estimate

Ω K 2 0 (ζ ) p p-1 δ Ω (ζ ) -γ p-1 dλ (ζ )
we use the pseudo-balls

P 0 (z 1 ) = P (z 1 , |z 0 -z 1 |) and P k (z 1 ) = P z 1 , 2 k |z 0 -z 1 | \ P z 1 , 2 k-1 |z 0 -z 1 | . For P 0 (z 1 ) K 2 0 (ζ ) p p-1 δ Ω (ζ ) -γ p-1 dλ (ζ )
we argue as for I 1 0 (z 0 , z 1 ) using Lemma 3.2. This ends the proof because α < 1 -2n+γ p .

5.2. Proofs of Theorems 2.2 and 2.3. Denotes f l = χ l f , so

T K f (z) = M ∑ l=0 n-1 ∑ k=0 Ω K k (z, ζ ) ∧ f l (ζ ).
As explained in Section 4 we extend the kernels

K k to (z, ζ ) ∈ Ω × C n with K k (z, ζ ) = K k (z, ζ ) if (z, ζ ) ∈ Ω × Ω, 0 if (z, ζ ) ∈ Ω × (C n \ Ω),
and we also extend the coefficients of the (0, r)-form f in C ∞ 0 (C n ). Thus, for 0 ≤ k ≤ nr, making the change of variable ξ = z -ζ we get

(5.3) T l k f (z) = - C n K k (z, z -ξ ) ∧ f l (z -ξ ).
Note that, f being assumed smooth, the estimate (4.3) implies that T l k f (z) is smooth in Ω and it's derivatives are given derivating under the sign of integration and does not depend on the order of derivation.

As ∂ ∑ n-1 k=0 T k f = f -P N f , by Lemma 4.1, all derivatives of ∑ n-r k=0 T k f involving an anti-holomorphic derivative are derivatives of a coefficient of f up to a function smooth in Ω, so we need to estimate derivatives of the T k f involving only holomorphic derivatives (This is not necessarily but useful to simplify the notation in the proofs that follow).

To fix notations, for any integer

J let D J = J j=1 ∂ ∂ z i j = ∂ J ∂ z i 1 ...∂ z i J
, i j ∈ {1, . . . , n}, a derivative of order J in the z variable and denote

∆ J = J j=1 ∂ ∂ z i j + ∂ ∂ ζ i j the derivative obtained replacing ∂ ∂ z i j by ∂ ∂ z i j + ∂ ∂ ζ i j
, 1 ≤ j ≤ J, in D J . Then calculating D J T l k f with (5.3) and making the change of variable z -ξ = ζ we obtain that D J T l k f is a sum of integrals of the form

(5.4) I l J,k = Ω z t -ζ t |z -ζ | 2(n-k) ∆ J   ρ(ζ ) N-k 1 1 K 0 S(z, ζ ) + ρ(ζ ) N+k k ∏ s=1 ρ(ζ )∂ ζ as Q b s (z, ζ ) + ∂ ρ/∂ ζ a s Q b s (z, ζ ) f .
where f is a coefficient of χ l f . 5.2.1. Proof of Theorem 2.2. We assume N > 2m(n + 1) + 5n + 2 |γ| + N 0 (1) + 1. By (5.4) we have to estimate the operators associated to the integrals

I l 1,k (z) = Ω z t -ζ t |z -ζ | 2(n-k) ∆   ρ(ζ ) N-k 1 1 K 0 S(z, ζ ) + ρ(ζ ) N+k k ∏ s=1 ρ(ζ )∂ ζ as Q b s (z, ζ ) + ∂ ρ/∂ ζ a s Q b s (z, ζ ) f (ζ ) dλ (ζ )
where

∆ = ∂ ∂ z j + ∂ ∂ ζ j .
If ∆ acts on f the integral corresponds to an operator treated in Section 5.1, so we assume that ∆ does not act on f . By a straightforward calculus we are lead to consider the integral operators of the following type:

(5.5)

I = Ω * z t -ζ t |z -ζ | 2(n-k)   ρ(ζ ) N ′ -k 1 1 K 0 S(z, ζ ) + ρ(ζ ) N ′ +k+1 k ∏ s=1 ρ(ζ )∂ ζ as Q b s (z, ζ ) + ∂ ρ/∂ ζ a s Q b s (z, ζ ) f (ζ )dλ (ζ )
where N ′ = N or N -1 and * is a smooth function or

(5.6) I = Ω z t -ζ t |z -ζ | 2(n-k)   ρ(ζ ) N-k 1 1 K 0 S(z, ζ ) + ρ(ζ ) N+k ∆ k ∏ s=1 ρ(ζ )∂ ζ as Q b s (z, ζ ) + ∂ ρ/∂ ζ a s Q b s (z, ζ ) f (ζ )dλ (ζ ).
If l = 0 the kernels in I are, by Lemma 5.2 of the form * |z-ζ | 2(n-k)-1 , and, if l ≥ 1 and z / ∈ 2V l the kernels are smooth. Thus the integrals corresponding are easy to estimate so we consider only the cases l ≥ 1, z ∈ 2V l and k ≤ nr to study I because of the holomorphic property of the Q i .

The following Lemma is elementary:

Lemma 5.3. Let p ∈ [1, +∞[ and γ > -1. There exists a constant C > 0 such that, for any function g

∈ C 1 (Ω), g L p (δ ν Ω dλ ) ≤ C ν g L p 1 (δ γ Ω dλ
) where δ Ω denotes the distance to the boundary of Ω and

ν > -1 if γ < p -1, γ -p if γ ≥ p -1.
Proof of (1) of Theorem 2.2. By Lemma 5.1 the factors of f in (5.5) and (5.6) are bounded by

1 |z -ζ | 2(n-k)-1 ρ(ζ ) 1 K 0 S(z, ζ ) + ρ(ζ ) M 1 |ρ(ζ )| k ∏ i=1 τ i (z, δ Ω (z) + d(z, ζ )) -2 . By Lemma 5.3 δ Ω (ζ ) f p ∈ L p δ γ+ε Ω
, for all ε > 0, and we can apply the last result of the proof of Theorem 2.1 to conclude.

Proof of (2) of Theorem 2.2. If γ < p -1 we use a transformation similar to one used by Deyun Wu in [START_REF] Wu | Sobolev and Hölder Estimates for ∂ on Bounded Convex Domains in C 2[END_REF] for the integrals with k ≥ 1: Lemma 5.4. Let η z be the complex normal at the point z and let η

* ζ = ∂ ∂ η z -∂ ∂ η z acting on the variable ζ . Then for ζ ∈ V l and z ∈ 2V l : (1) η * ζ ρ(ζ ) |z -ζ | d(z, ζ ) 1/m 1 K 0 S(z, ζ ) + ρ(ζ ) 1/m (2) η * ζ S(z, ζ ) 1 (3) 1 1 K 0 S(z,ζ )+ρ(ζ ) M = * η * α ζ 1 1 K 0 S(z,ζ )+ρ(ζ ) M-α
where * is smooth in Ω.

Proof. ρ(ζ ) and S(z, ζ ) are C ∞ functions in the variable ζ on Ω and η * ζ ρ(z) = 0, and, by the definition of the support function S 0 , η * ζ S(z, z) 1 so (2) is valid if the balls V l are sufficiently small and, for (1) we use (4.3).

The two integrals (5.5) and (5.6) are treated similarly so we only treat the first one. Applying the preceding Lemma the integral (5.5) becomes

I = Ω * z l -ζ l |z -ζ | 2(n-k) ρ(ζ ) N ′ -k k ∏ s=1 ρ(ζ )∂ ζ as Q b s (z, ζ ) + ∂ ρ/∂ ζ a s Q b s (z, ζ ) η * ζ   1 1 K 0 S(z, ζ ) + ρ(ζ ) N ′ +k   f (ζ )dλ (ζ ).
Now we integrate by parts the derivative η * ζ . As the integrant is identically zero on the boundary of Ω ∩V l we get

I = Ω K 1 1 (z, ζ ) f (ζ )dλ (ζ ) + Ω K 2 1 (z, ζ )η * ζ f (ζ )dλ (ζ ) = I 1 1 + I 2 1
and iterating this procedure for I 1 1 (which is possible because 2(nk) -1 ≤ 2n -3)

I 1 1 = Ω K 1 2 (z, ζ ) f (ζ )dλ (ζ ) + Ω K 2 2 (z, ζ )η * ζ f (ζ )dλ (ζ ) = I 1 2 + I 2 2 .
If possible (integrability of a derivative of z l -ζ l |z-ζ | 2(n-k) ) we iterate this procedure to I 1 2 and so on. Finally we get a family of integrals

I 1 α = Ω K 1 α (z, ζ ) f dλ (ζ )
for α = (α 1 , α 2 , α 3 )

K 1 α (z, ζ ) = * η * ζ α 1 z l -ζ l |z -ζ | 2(n-k) η * ζ α 2 ρ(ζ ) N ′ -k η * ζ α 3 k ∏ s=1 ρ(ζ )∂ ζ as Q b s (z, ζ ) + ∂ ρ/∂ ζ a s Q b s (z, ζ ) 1 1 K 0 S(z, ζ ) + ρ(ζ ) N ′ +k+1-|α|
, the procedure being stopped when α 1 = 2k or α 2 or α 3 "great enough" (i. e. α 2 = 2m(n + 1), α 3 = 3n -1). By Lemma 5.1

η * ζ β ρ(ζ )∂ ζ as Q b s (z, ζ ) + ∂ ρ/∂ ζ a s Q b s (z, ζ )      ε 2 τ as (ζ ,ε)τ bs (ζ ,ε) if β = 0 ε τ as (ζ ,ε)τ bs (ζ ,ε) if β = 1 1 if β ≥ 2
where ε = δ Ω (ζ ) + d(z, ζ ) which implies, using that the {a s } and {b s } are distinct and the increasing property of the τ i ,

η * ζ α 3 k ∏ s=1 ρ(ζ )∂ ζ as Q b s (z, ζ ) + ∂ ρ/∂ ζ a s Q b s (z, ζ ) ε 2k-α ′ 3 k ∏ i=1 τ i (ζ , ε) -2 where α ′ 3 = α 3 if α 3 ≤ 2k 2k if α 3 ≥ 2k .
By (1) of Lemma 5.4

η * ζ α 2 ρ(ζ ) N ′ -k |ρ(ζ )| N ′ -k-α 2 A (α 2 )
where

A (α 2 ) η * ζ ρ(ζ ) α 2 + ρ(ζ ) α 2 -1 2 1 K 0 S(z, ζ ) + ρ(ζ ) α 2 2m
.

Finally

K 1 α (z, ζ ) 1 |z -ζ | 2(n-k)-1+α 1 ρ(ζ ) 1 K 0 S(z, ζ ) + ρ(ζ ) N ′ -k-α 2 1 K 0 S(z, ζ ) + ρ(ζ ) α 1 -1+α 3 -α ′ 3 + α 2 2m k ∏ i=1 τ i (ζ , ε) -2 .
Note first that

K 1 α (z, ζ ) 1 |z -ζ | 2(n-k)-1+α 1 |ρ(ζ )| α 1 -1 ρ(ζ ) 1 K 0 S(z, ζ ) + ρ(ζ ) M k ∏ i=1 τ i (ζ , ε) -2 for M = N ′ -k -α 2 -α 1 + 1 > 2 |γ| + 1.
On the other hand if α 2 or α 3 is large then, by Lemma 5.2,

K 1 α (z, ζ ) 1 |z -ζ | 2n-1 ρ(ζ ) 1 K 0 S(z, ζ ) + ρ(ζ ) M .
If α 1 = 2k then, using (3.4),

K 1 α (z, ζ ) 1 |z -ζ | 2n-1 1 K 0 S(z, ζ ) + ρ(ζ ) 2k-1 ρ(ζ ) 1 K 0 S(z, ζ ) + ρ(ζ ) M k ∏ i=1 τ i (ζ , ε) -2 1 |z -ζ | 2n-1 1 1 K 0 S(z, ζ ) + ρ(ζ ) ρ(ζ ) 1 K 0 S(z, ζ ) + ρ(ζ ) M =: W 1 (z, ζ ). Now K 2 α (z, ζ ) = * K 1 α ′ (z, ζ ) 1 K 0 S(z, ζ ) + ρ(ζ ) with |α ′ | = |α| -1, so K 2 α (z, ζ ) 1 |z -ζ | 2(n-k)-1+α 1 |ρ(ζ )| α 1 ρ(ζ ) 1 K 0 S(z, ζ ) + ρ(ζ ) M k ∏ i=1 τ i (ζ , ε) -2 .
If α 1 is even (α 1 = 2 (k -k 1 )), using (3.4),

K 2 α (z, ζ ) 1 |z -ζ | 2(n-k 1 )-1 |ρ(ζ )| 2(k-k 1 ) ρ(ζ ) 1 K 0 S(z, ζ ) + ρ(ζ ) M k 1 ∏ i=1 τ i (ζ , ε) -2 k ∏ i=k 1 +1 τ i (ζ , ε) -2 1 |z -ζ | 2(n-k 1 )-1 ρ(ζ ) 1 K 0 S(z, ζ ) + ρ(ζ ) M k 1 ∏ i=1 τ i (ζ , ε) -2 =: W 2 (z, ζ ). If α 1 is odd then K 2 α (z, ζ ) K 2 α ′ (z, ζ ) + K 2 α ′′ (z, ζ ) with α ′ = (α 1 -1, α 2 , α 3
) and α ′′ = (α 1 + 1, α 2 , α 3 ) so there exists k ′ 1 and k ′′ 1 such that

K 2 α (z, ζ ) 1 |z -ζ | 2(n-k ′ 1 )-1 k ′ 1 ∏ i=1 τ i (ζ , ε) -2 + 1 |z -ζ | 2(n-k ′′ 1 )-1 ρ(ζ ) 1 K 0 S(z, ζ ) + ρ(ζ ) M k ′′ 1 ∏ i=1 τ i (ζ , ε) -2 .
Finally I is a sum of integrals applied to f with kernels controlled by W 1 (z, ζ ) and applied to η * ζ f with kernels controlled by W 2 (z, ζ ). Theorem 2.1 has been proved applying Lemma 4.3. If the conditions of Lemma 4.3, for parameter s = 1 + µ, are verified then the operator T maps L p (δ γ ) into L q δ γ ′ for 1 q = 1 p -µ 1+µ . For all p ≥ 1 under hypothesis of Theorem 2.1 we have proved the result if

µ ∈ 0, 1 -m p (γ -γ ′ ) m(γ ′ + n) + 2 -(m -2)(r -1) = [0, µ o [
(note that µ 0 < 1 2n-1 ). The integrals applied to η * ζ f have kernels bounded by those appearing in the proof of Theorem 2.1 and are well controlled when µ < µ 0 .

For the integrals involving only f we have to estimate the operator T given by T f

(z) = Ω K(z, ζ ) f (ζ )dλ (ζ ) with (5.7) K(z, ζ ) = 1 |z -ζ | 2n-1 ρ(ζ ) ρ(ζ ) + 1 K 0 S(z, ζ ) M 1 ρ(ζ ) + 1 K 0 S(z, ζ ) χ l (ζ ). Writing T f (z) = Ω K(z, ζ )δ (ζ ) -α-γ ′ (δ α (ζ ) f (ζ )) dµ(ζ ) with dµ(ζ ) = δ γ ′ (ζ )dλ (ζ ) and α = γ-γ ′ -β p , by Lemma 5.3 δ α f L p (δ γ ′ ) ≤ C α f L p 1 (δ γ ) with (5.8) γ -γ ′ < β < γ + 1.
and it suffices to prove that the operator

T β g(z) = Ω K(z, ζ )δ (ζ ) -α-γ ′ |g(ζ )| dµ(ζ ) maps
L p δ γ ′ into L q δ γ ′ under the conditions on β given in (5.8).

• Case p > 1. By Lemma 4.3 this result will be obtained proving the two following inequalities (5.9)

I 1 (z) = Ω |K(z, ζ )| 1+µ δ (ζ ) (-α-γ ′ )(1+µ)+γ ′ -ε dλ (ζ ) ε δ (z) -ε

and

(5.10)

I 2 (ζ ) = Ω |K(z, ζ )| 1+µ δ (ζ ) (-α-γ ′ )(1+µ) δ (z) γ ′ -ε dλ (z) ε δ (ζ ) -ε , with µ < min β -(γ-γ ′ ) 2np-(β -(γ-γ ′ ))+pγ ′ , µ 0 =: µ 1 (β ). Let us verify first I 1 (z). If l = 0, |K(z, ζ )| 1+µ δ (ζ ) (-α-γ ′ )(1+µ)+γ ′ -ε 1 |z -ζ | (2n-1)(1+µ)
and the result is trivial as µ < 1/2n -1. If l > 0: if z ∈ Ω \ 2V l the inequality is trivial for similar reasons, so, we show it if z ∈ 2V l . By (5.7) and (4.3)

|K(z, ζ )| 1+µ δ (ζ ) (-α-γ ′ )(1+µ)+γ ′ -ε ≤ 1 |z -ζ | (2n-1)(1+µ) |ρ(ζ )| M(1+µ) ρ(ζ ) + 1 K 0 S(z, ζ ) M(1+µ)+1+µ δ (ζ ) (-α-γ ′ )(1+µ)+γ ′ -ε 1 |z -ζ | (2n-1)(1+µ) 1 δ (ζ ) + d(z, ζ ) -γ ′ +(α+γ ′ )(1+µ)+1+µ+ε , thus, if ζ ∈ P k (recall that P k (z) = P z, 2 k δ (z) \ P z, 2 k-1 δ (z) if k ≥ 1 and P 0 (z) = P (z, δ (z)
) and that the hypothesis on µ imply 0 ≤ (2n -1)µ < 1)

(5.11)

P k (z) |K(z, ζ )| 1+µ δ (ζ ) (-α-γ ′ )(1+µ)+γ ′ -ε ≤ 2 k δ (z) γ ′ -(α+γ ′ )(1+µ)-1-µ-ε P k (z) 1 |z -ζ | (2n-1)(1+µ) ≤ 2 k δ (z) 1-(2n-1)µ+γ ′ -(α+γ ′ )(1+µ)-1-µ-ε
, by (3.10).

Then I 1 (z) µ δ Ω (z) -ε if 0 ≤ µ < µ 1 .

For I 2 (ζ ), using that δ (z) 2 k δ (ζ ) if z ∈ P k (ζ ) and (3.12) the conclusion follows as before if µ

< µ 1 . If µ < γ ′ +1 2np-(γ ′ +1)+pγ ′ there exists β , γ ′ < β < γ + 1, such that µ < µ 1 (β )
achieving the proof for p > 1, the case k = 0 being trivial. • Case p = 1. Formula (5.11) and Lemma 4.3 imply the result. 5.2.2. Proof of Theorem 2.3. Now we assume N > 2 |γ| + d + 2m(n + 1) + 5n + N 0 (d) + 1. To prove the theorem it suffices to prove that there exists a constant C depending only on Ω and N such that for all (0, r)-form f with coefficients in C ∞ Ω with condition (1) and in D(Ω) with condition (2)

T K ( f ) L q k,r-1 δ γ ′ Ω ≤ C f L p k,r (δ γ Ω ) ,
with T K ( f ) given by (4.2) and p, γ, q and γ ′ as in the theorem.

First we have to calculate the derivatives of the T k f . As usual we assume l > 0 and z ∈ 2V l .

Using (5.4) for J ≥ 1, we have For the last point we use the above Lemma (with d ′ = d -1) and Theorem 2.2.

∆ J   ρ(ζ ) 1 K 0 S(z, ζ ) + ρ(ζ ) N+k k ∏ s=1   ∂ ζ as Q b s (z, ζ ) ρ(ζ ) + ∂ ρ/∂ ζ a s Q b s (z, ζ ) ρ 2 (ζ )   f   is a sum of expressions of the form ∆ J 1   ρ(ζ ) 1 K 0 S(z, ζ ) + ρ(ζ ) N+k   ∆ J 2   k ∏ s=1   ∂ ζ as Q b s (z, ζ ) ρ(ζ ) + ∂ ρ/∂ ζ a s Q b s (z, ζ ) ρ 2 (ζ )     ∆ J 3 f where ∆ J t = J t j=1 ∂ ∂ z i j + ∂ ∂ ζ i j with J 1 + J 2 + J 3 = J. Proposition 5.3. Let D z J = J j=1 ∂ ∂ z i j = ∂ J ∂ z i j ...∂ z i J , i j ∈ {1, . . . , n} be a derivative of order J, l > 0. Then D z J T K f is controlled by a sum of integrals Ω∩2V l K i,l (z, ζ )D ζ i f (ζ ), 0 ≤ i ≤ J, where D ζ i f is a derivative of f of order J -i and K i,l (z, ζ )   r-1 ∑ k=0 1 |z -ζ | 2(n-k)-1 ρ(ζ ) ρ(ζ ) + 1 K 0 S(z, ζ ) M k ∏ j=1 τ j (z, α) -2   1 ρ(ζ ) + 1 K 0 S(z, ζ ) i χ l (ζ ), with α = δ Ω (z) + d(z,

APPLICATION TO WEIGHTED BERGMAN PROJECTIONS

In view of theorems 2.2 and 2.3 we have: Proposition 6.1. Let d ∈ N * , p 0 ≥ 1, γ 0 γ 1 , (d -1)p 0 -1 < γ 0 ≤ γ 1 . There exists an operator T solving the ∂ -equation and ε > 0 such that for p ∈ [1, p 0 ] and γ ∈ [γ 0 , γ 1 ]:

(1) T maps continuously L p d δ γ Ω in L p+ε Proposition 6.3. Let D be a smoothly bounded convex domain of finite type. Let g the gauge of D, ρ 0 = g 4 e 1-1/g -1 and ω 0 = (-ρ 0 ) r with r a non negative rational number. Let P ω the weighted Bergman projection of D associated to the Hilbert space L 2 (D, ω 0 dλ ), dλ denoting the Lebesgue measure. Then for s ∈ N, p ∈ ]1, +∞[ and -1 < β < p (r + 1) -1, P ω 0 maps the Sobolev space L p s D, δ β ∂ Ω continuously into itself.

The following Proposition extends partially this result: Using Proposition 6.1 and the two previous Propositions, the proof is completely similar to the proof done in Section 4 of [START_REF] Charpentier | Weighted and boundary l p estimates for solutions of the ∂ -equation on lineally convex domains of finite type and applications[END_REF] based on the following comparison formula ϕP ω (u) = P ω 0 (ϕu) + Id -P ω 0 • A P ω (u) ∧ ∂ ϕ , where ω = ϕω 0 and A is an operator solving the ∂ -equation.

For d a nonΩ

  negative integer and p ∈ [1, +∞[, δ Ω being the distance to the boundary of Ω, we denote by L p d Ω, δ γ Ω , -1 < γ , the space of functions f such that, for any derivative D of order α, 0 ≤ α ≤ d, D f belongs to L p Ω, of the set of C ∞ functions with compact support in Ω. Moreover we denote by L p d,r Ω, δ γ Ω or simply L p d,r (δ γ ) the space of forms of degree r whose coefficients belong to r (δ γ ) the space of forms of degree r whose coefficients belong to L

  where ∂ * N is the canonical solution of the ∂ -equation derived from the theory of the ∂ -Neumann problem on pseudoconvex domains of finite type. By [CD18, Lemma 3.2] and Sobolev lemma we get (see [CD18, Lemma 3.3]):

  and satisfy the following estimates ([CD18, Lemma 3.6]):

  ζ ) and M large (N being large depending on J) and δ Ω denotes the distance to the boundary. Proof. Comes from Lemma 5.1 and Lemma 5.2. Lemma 5.5. Let p ∈ [1, +∞[, γ ∈ R, d ′ ∈ N and g ∈ C ∞ (Ω). Then under one of the two following condition(1) γ ≥ d ′ p -1 (2) g has compact support in Ω (i.e. g ∈ D(Ω))for ν > γd ′ p there exists a constant C ν such that g L p (δ ν Ω dλ ) ≤ C ν g L p d ′ (δ γ Ω dλ ) .Proof. Assume condition (1) satisfied. Then, by induction on i = 1, . . . , d ′ , Lemma 5.3 givesg L p d ′ -i (δ ν Ω ) ≤ C ν g L p d ′ (δ γ Ω ) for ν > γip.Assume now condition (2) satisfied. Remark first that the following Claim is an elementary consequence of Hölder inequality:Claim. Let p ∈ [1, +∞[ and η < p -1. Then for ν > ηp and any function h ∈ D(Ω), h L p (δ ν Ω dλ ) ≤ C ν h L p 1 (δ η Ω dλ ) .Then using the Claim and Lemma 5.3 the Lemma 5.5 is easily obtained by induction on i = 1, . . . ,d ′ : for ν > γip, g L p d ′ -i (δ ν Ω ) ≤ C ν g L p d ′ (δ γ Ω ) .Let us now finish the proof of Theorem 2.3. For the two first points of the theorem we use this Lemma (with d ′ = d) and Proposition 5.3 (recall ρ(ζ ) + 1 K 0 S(z, ζ ) δ Ω (ζ )).

  We use the operator T defined in (4.1) withN > 2 |γ 1 | + 2m(n + 2)+ 5n + d + 1.The following Proposition is a special case of [CDM15, Theorem 2.1]: Proposition 6.2. Let D be a smoothly bounded convex domain of finite type in C n . Let ρ be a smooth defining function of D. Let r ∈ R + , be a non negative real number, η ∈ C ∞ (D) be strictly positive and ω = η |ρ| r . Then, for any integer d, the weighted Bergman projection P Ω ω of the Hilbert space L 2 (Ω, ωdλ ) maps continuously the weighted Sobolev space L 2 d (Ω, ωdλ ) into itself. In [CDM14b, (1) of Theorem 1.1] the following result was proved:

Proposition 6. 4 .

 4 Let d ∈ N * . Let D be a smoothly bounded convex domain of finite type in C n . Let χ be any C ∞ non negative function in D which is equivalent to the distance δ D to the boundary of D and let η be a strictly positive C ∞ function on D. Let P ω be the (weighted) Bergman projection of the Hilbert space L 2 (D, ωdλ ) where ω = η χ r with r a non negative rational number. Then: (1) For p ∈ [2, +∞[ and -1 < β ≤ r, P ω maps the Sobolev space L p 1 D, δ β D continuously into itself. (2) If d ≥ 2, for p ∈ [2, +∞[ if r > d p -1 and d(p -1) -1 < β ≤ r, P ω maps continuously the Sobolev space L p d D, δ β D into itself.


Borros Arneth

The Movement of Binary Star Systems and Stars in Galaxies and a Novel Explanation: Complex Gravitation

Here, we discuss the literature regarding the roles of stellar mass, dark energy, and gravity in the rotation of galaxies and binary star systems. Furthermore, a novel explanation for the movement of stars in galaxies as well as for the movement of stars in binary star systems, called complex gravitation, is introduced here. In addition, a slightly modified law of gravitation is described.

Review: The Movement of Binary Star Systems and Stars in Galaxies I. Summary A recent study by Chae [START_REF] Chae | Breakdown of the Newton-Einstein standard gravity at low acceleration in internal dynamics of wide binary stars[END_REF] investigated a gravitational anomaly in the orbital dynamics of wide binary stars under conditions of weak gravitational acceleration. The orbital motions of these binary stars are analyzed using data from the Gaia DR3 database, revealing deviations from the expected Newtonian range at accelerations below 10^(-9) m/s². These deviations indicate that gravity behaves differently under specific acceleration conditions. The study also compares these observations with predictions made in aquadratic Lagrangian (AQUAL) theory, noting a significant 10σ deviation from the Newtonian expectation at gN = 10^(-10.15) m/s², where the observed Newton-predicted acceleration ratio is 1.43 with a deviation of 0.06. These findings suggest that standard gravity does not fully explain the dynamics of wide binary star motion, especially at low accelerations, necessitating the proposal and empirical evaluation of novel theories to account for these observations.

II. Systematic Review

McGaugh [START_REF] Mcgaugh | Predictions and outcomes for the dynamics of rotating galaxies[END_REF] reviews previous predictions regarding the motion dynamics of rotating galaxies, with a focus on modified Newtonian dynamics (MOND) data corroborated in numerous previous studies. From his observations, three main categories emerge: proven MOND predictions, galaxies mimicking MOND behaviors, and novel, unexplored aspects of binary star system motion. These data also highlight the presence of dark matter. Feng [START_REF] Feng | Rotating disk galaxies without dark matter based on scientific reasoning[END_REF] and Vitral [START_REF] Vitral | Galaxy dancers: A dynamical study of globular star clusters[END_REF] argue that the absence of visible mass providing gravitational support for the orbital velocity of matter in rotating disk galaxies is evidence of nonbaryonic dark matter. Newton's laws are used to determine the mass distribution in rotating galaxies, as they form closed systems subject to the universal laws of physics.

According to Arcones and Thielemann [START_REF] Arcones | Origin of the elements[END_REF], all elements in the universe originate from cosmic events such as the Big Bang and stellar explosions. These authors assess early papers on nucleosynthesis and related nuclear physics processes, which, depending on conditions such as temperature, density, pressure, and neutron/proton ratios, produce elements such as lead, lithium, and other rare earth metals. Arcones and Thielemann [START_REF] Arcones | Origin of the elements[END_REF] aim to enhance our understanding of astrophysical models by improving existing stellar system representations, analyzing the composition of ejecta from stellar winds, exploring supernova core collapse mechanisms, and examining the energy transition into hypernova and bursts of gamma radiation. According to this study, nova explosions result from combinations of stellar systems, X-rays, type 1a supernovae, and mergers between black holes and neutron stars, leading to the ejection of materials rich in rare metals, often in isotopic form, which become integral to the formation of future stars.

As low-mass and intermediate-mass stars, including those in binary systems, transition to the asymptotic giant branch, they become complex cosmic objects whose mechanisms depend on various kinematic processes [START_REF] Cherchneff | A chemical study of the inner winds of asymptotic giant branch stars[END_REF][START_REF] Cernicharo | Molecular shells in IRC+10216: tracing the mass loss history[END_REF][START_REF] Engels | Phase-lag distances of OH Masing AGB stars[END_REF]. New chemical elements are formed, leading to reactions that contribute to the creation of other novel components [START_REF] Engels | Phase-lag distances of OH Masing AGB stars[END_REF]. Dust grains form in the upper atmospheres of stellar objects, resulting in accelerated winds that scatter photons through gas collisions, aided by the bodies' angular momenta. Despite the abundance of chemical elements and other forms of mass in the universe, dark matter and dark energy also constitute a significant portion of the universe. Elbeze [START_REF] Elbeze | On the existence of another source of heat production for the earth and planets, and its connection with gravitomagnetism[END_REF] reports that observations of the rotation curve of galaxies on a flat plane can only be explained by the existence of dark energy in the cosmos. Applying Newton's laws and the theory of gravitation in such cases, with the Hubble constant as a limit, reveals the presence of an unaccounted-for spacetime variable, E0, which may represent dark energy. This view builds upon Elbeze's [START_REF] Elbeze | Hubble's constant and flat rotation curves of stars: Are dark matter and energy needed?[END_REF] earlier work and is supported by Riess [START_REF] Riess | A 2.4% determination of the local value of the Hubble constant[END_REF].

In the July issue of The Astrophysical Journal, an article was published on the mathematical description of binary star systems by Kyu-Hyun Chae [START_REF] Chae | Breakdown of the Newton-Einstein standard gravity at low acceleration in internal dynamics of wide binary stars[END_REF].

A significant number of binary star systems have been newly discovered with the Gaia telescope.

Importantly, data on the movements of these binary star systems are also now available.

Chae notes that the movements of binary star systems and the rotations of stars in galaxies cannot be described using the classic Kepler approach.

Compared to what is indicated by Kepler's approach, binary star systems have a mean acceleration that is too high by a factor of 1.4 on average. The article postulates that one should instead choose the Milgrom approach.

In the Milgrom approach, also called the MOND approach, Newton's acceleration law is slightly modified. Building on this concept, it should be noted that the MOND approach is not the only way to solve the problem by slightly modifying a physical law. Here, it is suggested that not only Newton's law of acceleration but also Newton's law of gravitation should be slightly modified.

Specifically, Newton's law of gravitation and Coulomb's law of electricity can be unified into a common law that uniformly describes the attractive properties of both gravity and electricity in terms of a complex quantity. This view is not a complication but rather a simplification of the problem. Electricity is stronger than gravity by a factor of b=10^10. It follows that in such a unified description, nothing changes when considering small masses. The newly created mass-charge interactions only become relevant when considering large masses on the order of 1E10 kg.

A simple calculation shows that the deviation of the movement of binary star systems described in the cited article can be easily explained to within an order of magnitude.

With the appropriate choice of units, the following law applies, which we call "complex gravitation":

Mass1*mass2+ mass1*charge2 + charge1*mass2 + charge1*charge2 = 1.4 times the force 1+1*x+x*1 +x^2=1.4
This means that for charges of 0.225 Coulomb and 1.77 Coulomb, there is already a deviation in force and acceleration by a factor of 1.4.

For a star that is heavier than 1E10 kg, a corresponding deviation will occur only with a correspondingly higher charge. It is therefore quite easy to explain the deviation in acceleration in binary star systems described in the cited article if one starts from a unified complex law of gravity and electricity. Moreover, it is not necessary to assume exceptionally high charges. Even small charges are sufficient. It is only important that these charges exist between the two stars and are not completely compensated.

III. Conclusion

Chae's [START_REF] Chae | Breakdown of the Newton-Einstein standard gravity at low acceleration in internal dynamics of wide binary stars[END_REF] study illustrates the inability of standard gravity to account for the motion dynamics of wide binary stars, especially at low accelerations. The present paper also includes a systematic literature review to highlight the roles of stellar mass, gravity, and dark energy in the rotation of binary star systems and galaxies. The key conclusion of this analysis is that novel theories are needed to sufficiently account for these observations. Furthermore, it is proposed that complex gravitation can serve as an alternative explanation for the differences between Newton's laws and the observations of binary star movements. In this regard, complex gravitation is similar to the MOND theory.

Reference number Experimental results

Part A: Movement of binary stars [START_REF] Cernicharo | Molecular shells in IRC+10216: tracing the mass loss history[END_REF] Emission maps from the IRC+10216 star, examined under the IRAM 30 m telescope, show the presence of dense spherical structures. The data collected in this study show that the central star, known as CW Leo, emits a lot of carbon (II) oxide gas, with the higher emissions occurring at the nucleus. This phenomenon, according to the authors, is caused by photodissociation. All astronomical models created and used in this study are in line with Newton. AGB describes an evolutionary stellar period for low-to medium-mass stars approaching the ends of their lives. They turn into red giants with luminosity several thousand times greater than the Sun's. Most AGB stars with dense outer layers are in line with Newton but some display altered characteristics.

Part B: Movement of stars in galaxies [START_REF] Chae | Breakdown of the Newton-Einstein standard gravity at low acceleration in internal dynamics of wide binary stars[END_REF] A gravitational anomaly has been discovered where gravitational acceleration (gN) is significantly weak with data less than 10 -9 m/s 2 . This data was obtained from an analysis of wide binary star dynamics in the Gaia DR3 database. This study's findings are in line with MOND.

[4] This paper involved an analysis of the core collapse of a star and resultant tidal interactions and mass segregation. It modeled globular clusters using simulations built from data collected by the Hubble Space Telescope and the Gaia astrometric mission. The results point to the existence of a dark matter mini-halo.

[5] The authors of this study investigated the effect of angular momentum on various galaxies using kinematic data from early-type variants. The findings showed that the rotation of the outer region of these galaxies (halo spin) does not account for the observable differences in the properties of spiral galaxies.

However, the data supported scenarios in which these galaxies would preserve a portion of their initial energies (10-60%) during the expansion process. These findings are in line with Newton.

[6] The authors of this study discovered a pattern showing that elliptical and spiral galaxies follow the paths predetermined by their angular momenta and mass.

This process was described to be similar to the Tully Fisher relation. The authors found that elliptical galaxies are shaped based on their mass, size, and rotation properties. This phenomenon is known as the mass-size-rotation conspiracy.

These findings challenge earlier theories based on the idea that elliptical galaxies accumulate their angular momenta during mergers. These findings are in line with Newton.

[7]

According to this study, the turbulent motions observed in most galaxies are consistent with the Toomre instability parameter (Q = 1). This value associates a galaxy's motion with the critical gravitational disk instability threshold.

Therefore, instability is a significant factor in the occurrence of turbulence within galactic formations. These findings are in line with Newton.

[9] Stars on the asymptotic giant branch experience a mass loss due to gravityrelated factors, especially in the wind-forming regions. The findings of this study are in line with Newton.

[10] Inner winds are a characteristic of the motion dynamics of the asymptotic giant branch. There are four parent molecules in the four objects, indicating that these chemical species form at random stages of the evolution of asymptotic giant branch galaxies. This study focuses on the chemical aspect of the universe, but all physics-related computations, assertions, and findings are in line with Newton.

Part C: Dark matter, dark energy, and their effects on galactic rotation [START_REF] Mcgaugh | Predictions and outcomes for the dynamics of rotating galaxies[END_REF] MOND data confirms the motion dynamics of rotating galaxies. Post hoc explanations for observations in terms of dark matter are possible in this case;

hence, the data is reliable. These findings can be used to predict mass loss in galaxies as well. The findings of this study are in line with MOND.

[3] Solving a force balance equation based on Newton's laws from measured rotation curves is the most reliable method for determining the distribution of mass in rotating disk galaxies. The findings of this study are in line with Newton.

[14] According to the authors, Hubble's constant is not affected by dark energy and dark matter. Additionally, the most cited evidence for nonbaryonic dark matter is the apparent lack of visible mass offering gravitational support to the observed orbital velocity of matter in rotating disk galaxies. The findings of this study are in line with Newton.

Part D: The role of universal constants, chemical formation, and electromagnetic factors in star system dynamics [START_REF] Arcones | Origin of the elements[END_REF] All elements in the universe are formed from the Big Bang, stellar explosions, and other cosmic events. Most heavy atoms were formed from the nucleosynthesis of supernovae. This study is in line with Newton. [START_REF] Elbeze | On the existence of another source of heat production for the earth and planets, and its connection with gravitomagnetism[END_REF] According to this study, gravitomagnetism causes planets to generate heat in their cores. Additionally, Earth is currently in a phase of thermal balance. This study is in line with Newton.

[15] Observations from the Hubble Space Telescope confirm the Hubble constant to be 2.4%. Near-infrared observations of Cepheid variables were leveraged to achieve the improvement from an initial value of 3.3%. This study's findings are in line with Newton. Part A: Movement of binary stars [START_REF] Cernicharo | Molecular shells in IRC+10216: tracing the mass loss history[END_REF] Changes in the density of shells around stars and galaxies reveal a gradual loss of mass, which affects their rotational dynamics over time. As these formations evolve, the expansion of carbon (II) oxide envelopes fills the voids left behind by the lost mass. This research offers great insights into the formation and mobility of galaxies and stars despite not precisely explaining the dynamics involved.

I. Key Findings

This paper is in line with Newton.

[12] The paper presents the phase-lag approach and how it can be used to estimate Part B: Movement of stars in galaxies [START_REF] Chae | Breakdown of the Newton-Einstein standard gravity at low acceleration in internal dynamics of wide binary stars[END_REF] Wide binary start systems with weak gravitational acceleration often have predictable orbital dynamics following Newtonian physics. However, based on the observations of this study, the correlation between acceleration and other properties consistently diverges from expectation. For instance, the findings showed a deviation of 0.007 from the expected results, translating to a statistical significance of 10 sigma. Despite these slight deviations, this study was in line with MOND.

[4] The thesis improved current data modeling standards in the construction of galactic density profiles, motion measurements, and Jeans kinematic modeling. It also highlighted new constraints to consider when computing globular cluster formations, such as the dark matter mini-halo. This study is in line with Newton.

[5] The paper established the role of angular momentum in the physical mechanisms of the evolution of j * and m * galaxies, involving outflows, stripping, collapse bias, and merging processes. The vector sum of these processes, according to the results, matches the observations made. Based on these results, this study is in line with Newton.

[6] In early-type galaxies, there are observable links between the angular momentum of stars. However, spiral galaxies of the same mass have higher angular momenta than their elliptical counterparts, often as much as 4:1 and 3:1. The results of this study are in line with Newton.

[7] Baryon fraction, angular momentum, and pressure create turbulence. However, the resultant kinematics need to satisfy the Toomre instability parameter of 1 to be considered a major driver of galactic turbulence. These findings are in line with Newton.

[9] Stars within the asymptotic giant branch experience mass loss due to factors related to gravity. In the case of galaxies, their rotational mechanics and dynamics lead to gradual mass shedding, especially in the wind-forming regions.

These findings are in line with Newton.

[10] Observations of key molecules in stars in the asymptotic giant branch, at both low and high excitation levels, were carried out using APEX and JCMT telescopes at millimeter and submillimeter wavelengths. These observations showed that the inner winds of stars in the asymptotic giant branch are based on nonequilibrium chemistry, showing these regions have a chemical imbalance.

These results are in line with Newton.

Part C: Dark matter, dark energy, and their effects on galactic rotation [START_REF] Mcgaugh | Predictions and outcomes for the dynamics of rotating galaxies[END_REF] MOND data confirms the motion dynamics of rotating galaxies. Post hoc explanations for observations in terms of dark matter are possible in this case;

hence, the data is reliable. These findings can be used to predict mass loss in galaxies as well. These findings are in line with MOND.

[3] The most dependable and accurate method for estimating the distribution of mass within rotating disc galaxies is the solution of a force balance equation derived from Newton's laws of motion applied to galactic rotation curves. The approach proposed by these authors, therefore, is in line with Newton.

[14] Hubble's constant is not affected by dark energy and dark matter. Additionally, the most cited evidence for nonbaryonic dark matter is the apparent lack of visible mass offering gravitational support to the observed orbital velocity of matter in rotating disk galaxies. The findings of this paper are in line with Newton.

Part D: The role of universal constants, chemical formation, and electromagnetic factors in star system dynamics [START_REF] Arcones | Origin of the elements[END_REF] All elements in the universe are formed from the Big Bang, stellar explosions, and other cosmic events. Most heavy atoms were formed from the nucleosynthesis of supernovae. All computations and findings in this paper are in line with Newton.

[13] Rocky planets such as Earth depend on gravitomagnetics to generate heat in their cores. Based on current data, the Earth is in a phase of thermal balance. These findings are in line with Newton.

[15] Observations from the Hubble Space Telescope confirm the Hubble constant to be 2.4%. Near-infrared observations of Cepheid variables were leveraged to achieve the improvement from an initial value of 3.3%. The computations and observations that were used to arrive at these figures are in line with Newton.

[ 12 ]

 12 The authors observed 20 OH/IR stars within the Milky Way Galaxy with the Nancay radio telescope. The results show these stars have variable emissions, and this data was used to compute the phase lag of each star based on the outflow of materials from their surrounding shells. While ongoing observation and data collection continue to refine current knowledge about these systems, existing data have high uncertainties exceeding 20%. The approach proposed in this study (phase lag) offers a way to determine distances between Earth and stars in the Asymptotic Giant Branch (AGB) characterized by dense outer layers.

  distances from Earth to stars in the Asymptotic Giant Branch. It focuses on stars with optically thick circumstellar shells. The phase lag of stars satisfying this condition (20 OH/IR) in the galactic disk has been determined based on a monitoring program with the Nancay radio telescope. The findings offer great insight into galactic dynamics, but their accuracy cannot be established since the study is ongoing. Most AGB stars with dense outer layers conform to Newton but some do not.

Table 1 Table 1 :

 11 Experimental results

Table 2 :

 2 Key findings

	Reference	Key findings
	number


Thomas Ferrandiz
email: thomas.ferrandiz@isae.fr

Fabrice Frances
email: fabrice.frances@isae.fr

Christian Fraboul
email: christian.fraboul@enseeiht.fr

Using Network Calculus to compute end-to-end delays in SpaceWire networks

The SpaceWire network standard is promoted by the ESA and is scheduled to be used as the sole on-board network for future satellites. This network uses a wormhole routing mechanism that can lead to packet blocking in routers and consequently to variable end-to-end delays. As the network will be shared by real-time and non realtime traffic, network designers require a tool to check that temporal constraints are verified for all the critical messages.

Network Calculus can be used for evaluating worstcase end-to-end delays. However, we first have to model SpaceWire components through the definition of service curves. In this paper, we propose a new Network Calculus element that we call the Wormhole Section. This element allows us to better model a wormhole network than the usual multiplexer and demultiplexer elements used in the context of usual Store-and-Forward networks.

I. INTRODUCTION

SpaceWire [START_REF] Parkes | Spacewire: A spacecraft onboard network for real-time communications[END_REF], [START_REF] Ecss | Spacewire -links, nodes, routers and networks[END_REF] is an on-board network for satellites designed by the European Space Agency and the University of Dundee. It uses high-speed serial, pointto-point links and simple routers to interconnect satellite equipment using arbitrary topologies. In the future, the ESA plans to use SpaceWire as the sole on-board network in their satellites. The idea is to use the same network for both the payload and command/control traffic. This will simplify the network architecture and reduce the costs of the satellites.

At the moment, SpaceWire provides enough bandwidth (up to 200 Mbps) to carry both types of traffic at the same time. However, in order to reduce memory size (radiation-hardened memory is very expensive), SpaceWire uses wormhole routing to transmit the data packets across the network. The downside of this technique is that it can lead to blocked packets and thus huge variations in the end-to-end delays of those packets. Furthermore, SpaceWire does not provide built-in realtime mechanisms that guarantee the timely delivery of urgent packets.

Thus, network designers need a tool to check that temporal constraints are verified for urgent packets before SpaceWire can be used to carry command/control traffic.

Using simulations is not possible because covering every scenario would be very long and costly. A better solution is to design an analytical model to compute an upperbound on the worst-case end-to-end delay of a flow.

We proposed a first model in [START_REF] Ferrandiz | A method of computation for worst-case delay analysis on spacewire networks[END_REF] that allowed us to compute such an upper-bound for a SpaceWire network. This model works well in most cases but has some limitations. As it did not require a specific model of input traffic, it was pessimistic when the network was not fully loaded.

As a consequence, we chose to create a new model based on Network Calculus theory [START_REF] Leboudec | Network calculus a theory of deterministic queuing systems for the internet[END_REF]. It is a theory designed to study deterministic queueing systems which we have already successfully used in [START_REF] Frances | Using network calculus to optimize the afdx network[END_REF] to study the AFDX network. One strong point of Network Calculus is that it allows us to model the input traffic precisely by using traffic envelopes.

However, Network Calculus has been mostly used to study Internet components and not wormhole routers. As a consequence, the usual multiplexer and demultiplexer elements are not adequate to model a wormhole network. Thus, in Section II, we propose a new network element we call a "Wormhole Section". We can then divide the path of a flow into a series of Wormhole Sections to deduce an end-to-end service curve.

Furthermore, existing models are either not suited to a wormhole network or very pessimistic. In Section III, we give a new model to compute the delay caused by a flow when it leaves a Wormhole Section.

Finally, we conclude and discuss future work in Section IV.

II. THE WORMHOLE SECTION ELEMENT

A complete overview of Network Calculus would be beyond the scope of this paper. Readers not familiar with this theory can refer to [START_REF] Boudec | A short tutorial on network calculus. i. fundamental bounds in communication networks[END_REF] for a short introduction.

Here, we just recall this fundamental theorem. Theorem 1: Concatenation of two systems Assume that a flow traverses two systems S 1 and S 2 in sequence. Assume that S 1 and S 2 offer the service curves β 1 and β 2 respectively. Then the concatenation of the two systems offers the service curve β 1 ⊗ β 2 to the flow.

β 1 ⊗ β 2 (t) = inf 0≤s≤t {β 1 (t -s) + β 2 (s)} is the Min-Plus convolution of β 1 and β 2 .
This allows us to combine the network elements successively traversed by a flow to obtain an end-to-end service curve offered to the flow by the network as a whole.

A. Assumptions

We consider a network composed of SpaceWire routers and terminals. Each terminal has only one SpaceWire interface but can be the source and/or destination of any number of flows. Each flow f is modeled by a source, a destination, a path through the network and an arrival curve α f . Usually, the arrival curve is a staircase function which gives a more precise model of the input traffic than an affine function.

Since SpaceWire routers use static routing, we consider only a static network. We also assume that the network is stable, i.e. that no link is required to transfer more data than its capacity. In Network Calculus terms, this can be written as follows (see [START_REF] Cruz | A calculus for network delay, part i: Network elements in isolation[END_REF]). For each link j, we note I j the number of flows traversing that link and α j i the arrival curve of flow f i at link j. The stability condition is now:

∀j, ∀i ∈ {1, . . . , I j }, lim t→+∞ [β j - Ij i=1 α j i ](t) = +∞ (1)

B. A new network element

To use Network Calculus theory, we first need to determine a service curve for each element traversed by a flow. We can then compute an End-To-End (ETE) service curve using Theorem 1. However, in a wormhole network, the service offered by a router is not independent from the service offered by downstream routers. Because of the flow control, a router can output data at an average rate far inferior to its nominal capacity.

For this reason, we do not propose a classic multiplexer/demultiplexer model of each router. Instead, we adopt a macroscopic view of the network which tries to optimize the ETE service curve of each flow while accounting for the interdependency between routers.

When a flow encounters interferences with other flows, the impact of each conflict is twofold. First, there is a delay when the flow is multiplexed with the interfering flow. Then, when the interfering flow is demultiplexed, the flow control mechanism will propagate its own delays backward to the studied flow.

To properly model this, we propose a new network element that we call a "Wormhole Section". The basic idea is to divide the path followed by a flow into a serie of sections. Each section is composed of a set of successive output ports shared by the same flows. Thus, a Section corresponds to the arrival or the departure of an interfering flow from the path of the studied flow.

Each wormhole section offers a service curve that depends on the arrival curves of the interfering flows. Once we have computed the service curve of every section in the path of a flow, we can deduce the endto-end service curve by using Theorem 1.

Of course, once an interfering flow has left the path of the studied flow, it will go through other wormhole sections before reaching its destination. The delays caused in those sections are those that will be carried over to the studied flow. As an example, consider the network in Figure 1. Each flow f i goes from its source S i to its destination D i . When a destination is shared by two flows i and j it is denoted D i,j .

Let us take a closer look at f 1 . f 1 has to go through three wormhole sections (S 1 , S 2 , S 4 ) to reach D 1,4 with sections S 1 and S 4 shared with other flows. S 1 and S 4 are thus the two sources of direct delays for f 1 . In addition, since after leaving S 1 , f 2 traverses section S 3 which it shares with f 3 , S 3 will be another source of delay for f 1 but only indirectly.

As can be seen in this example, the wormhole section network element makes it easier to analyse the conflicts in a wormhole network. In the next section, we present a detailed model of this new network element.

C. Section sharing

We will present the model when there is only one interfering flow. We note f 1 the studied flow. f 2 is the interfering flow. f i has α Sin i as an arrival curve at the entrance of section S. Let us assume that section S comprises M routers.

For j = 1, . . . , M , β j is the service curve offered by the output port of router R j to the two flows. Because of the flow control mechanism, the amount of data which is transmitted by the port may actually be inferior to β j . Thus, β j should be seen as an intermediate parameter used to determine the service guaranteed to a given flow by this section of the network. Once we have analysed the conflicts in each output port traversed by a flow, we can use Theorem 1 to compute a service curve for the section, then for the complete path. This end-to-end curve is now valid because it takes into account the influence of all the ports used by a flow, including the indirect delays. As a consequence, it represents the real end-to-end output of the network.

Since all the routers in the section are shared between the two flows and no other interference occurs, we can view these routers as a single router with service curve β S = M j=1 β j . SpaceWire routers use a simplified Round-Robin access scheme that guarantees that each input port gets a fair acces to each output port. However, each packet can use the output port for an unlimited duration. The usual round-robin model attributes some weight to each flow and shares the bandwidth in proportion to those weights. Since the SpaceWire standard does not define such weights, we have no way of using this model and have to rely on the more pessimistic "blind multiplexing" model [START_REF] Leboudec | Network calculus a theory of deterministic queuing systems for the internet[END_REF], Theorem 6.2.1.

Thus, the service curve offered by the section to f 1 is

β S 1 = ( M j=1 β j -α Sin 2 ) ↑ . (2)
(β) ↑ is the positive and non-decreasing upper closure of β defined as (β) ↑ = max(sup 0≤s≤t β(s), 0).

β S 1 is a worst-case service curve that implies that all the interfering flows have a higher priority than f 1 and can instantly preempt it. Of course, in reality the packets are not interrupted but this model ensures that we have a worst-case service curve for any possible scheduling of the packets.

III. SECTION DEMULTIPLEXING

A. Limits of existing models

In a classic Store-And-Forward network model, the packets are instantaneously demultiplexed. Furthermore, once a packet has left the router, the delays it can endure are not propagated backward on the network. Thus, the demultiplexer has no impact on the delay (see [START_REF] Cruz | A calculus for network delay, part i: Network elements in isolation[END_REF] for example).

However, this is not true for a wormhole network. In fact, after two flows f 1 and f 2 have been demultiplexed, f 2 can still have an impact on f 1 . This is because the flow control mechanism will carry over the delays caused to f 2 on the end of its path backward to f 1 . A possible way to model this phenomenon is given in [START_REF] Qian | Analysis of worst-case delay bounds for best-effort communication in wormhole networks on chip[END_REF]. In this paper, the authors consider the situation described on Figure 2. In this network, after they have been demultiplexed both flows f 1 and f 2 have to go through a flow controller. Flow controller τ i models the impact of the flow control on the downstream output link on flow f i with a service curve β τi . In turn, β τi depends on the service curve of the downstream router.

The authors consider that, as far as the aggregate flow f {1,2} is concerned, τ 1 and τ 2 are parallel traffic regulators. As a consequence, the service offered to the aggregate flow f {1,2} is β {1,2} = min(β τ1 , β τ2 ). This aggregate service is then shared between the two flows to derive the service offered to each individual flow.

This service is valid but is is very pessimistic. Let us consider the example in Figure 2. The arrival curves are affine: α i (t) = r i .t+b i and the service curves are linear:

β i (t) = C i .t
We use the following values:

f 1 f 2 r i (Mbps) 50 10 b i (bits) 1000 200 C i (Mbps) 100 20
On the one hand, the service offered to

f {1,2} is β {1,2} (t) = min(C 1 , C 2 ).t = C 2 .t. On the other hand, the arrival curve of f {1,2} is α {1,2} (t) = α 1 (t)+α 2 (t) = r 1,2 .t + b 1,2 with r 1,2 = r 1 + r 2 = 60 Mbps and b 1,2 = b 1 + b 2 = 1200 bits.
The horizontal distance h(α {1,2} , β {1,2} ) between the two curves is clearly infinite and we have to conclude that the network cannot carry both flows. This is very pessimistic because it is easy to see that this network can handle both flows.

Thus we need a new, more precise network model of wormhole demultiplexer.

B. A new model of demultiplexing

Since both flows go in different directions in the network, we cannot determine an exact service for the aggregated flow. Each flow receives its own service but can still cause delays to the other flow thanks to the flow control mechanism.

The delay actually caused to an individual packet of f 1 is hard to determine because it depends on which exact conflicts occur farther on the path of f 2 . However, we can determine an upper-bound on this delay.

In fact, the maximum delay caused by f 2 , which we will denote d 2 , is at most the maximum delivery delay from the demultiplexer to the destination of f 2 . Thus,

d 2 = h(α Sout 2 , β dest 2 ) (3)
where α Sout 2 is the arrival curve of f 2 at the end of S and β dest 2 the service curve offered to f 2 between the end of S and its destination.

Here, since we have assumed a blind multiplexing with f 2 as the higher priority flow,

α Sout 2 = α Sin 2 M j=1 β j (4)
where (α β)(t) = sup s≥0 {α(t + s) -β(s)} is the Min-Plus deconvolution of α and β.

With this model, in the example in Figure 2 the delay caused by f 2 to f 1 is only h(α 2 , β 2 ) = 200 10 7 s = 10µs. As can be seen, this model is far less pessimistic than the model from [START_REF] Qian | Analysis of worst-case delay bounds for best-effort communication in wormhole networks on chip[END_REF].

C. Complete service curve offered by a Wormhole Section

We can combine the results from Subsection II-C and III-B to obtain the complete service curve offered by section S. When two flows share Section S, the service curve offered to f 1 is

β S 1 = ( M j=1 β j -α Sin 2 ) ↑ ⊗ δ h(α S out 2 ,β dest 2 ) (5)
with the delay function δ T (t) = 0 if t ≤ T and δ T (t) = +∞ if t ≥ T for any T ≥ 0. This formula can be easily generalized when there are more than one interfering flow.

D. Computing an end-to-end service curve

Once we can compute a service curve for each Wormhole Section, we have to combine them to determine an end-to-end service curve. When interfering flows arrive and depart in various routers along the path of a flow, we cannot directly use the model presented here.

We would have to use a Section for each router crossed by the flow which would give us a suboptimal result. Rather, we try to optimize the ETE service curve by combining a set of flows sharing several consecutive routers as an aggregate flow. We can then determine Wormhole Sections for this aggregate flow and deduce the service curves offered to it. Those service curves will then be shared between the individual flows composing the aggregate flow.

To automate this, we plan to use the three interference patterns defined in [START_REF] Qian | Analysis of worst-case delay bounds for best-effort communication in wormhole networks on chip[END_REF] to determine the order in which the residual services are computed.

IV. CONCLUSION

In this paper, we have defined a new Network Calculus element that can be used to model a SpaceWire network more accurately than the usual multiplexer and demultiplexer elements. We call this new element a Wormhole Section since it represents a part of the path followed by a flow. The Wormhole Section should allow us to obtain tighter bounds by considering successive routers as only one element. Furthermore, we also described a new way to compute the delay caused by a flow leaving a wormhole section. We showed on a simple example that our method is a lot less pessimistic than existing demultiplexer models for a wormhole network.

We plan to implement a software tool based on the Wormhole Section element and the interference patterns defined in [START_REF] Qian | Analysis of worst-case delay bounds for best-effort communication in wormhole networks on chip[END_REF] to study some industrial configurations provided by Thales Alenia Space.

Figure 1 .

 1 Figure 1. A network divided into wormhole sections

Figure 2 .

 2 Figure 2. Demultiplexing of two flows in a wormhole network

ACKNOWLEDGMENT

This work was supported by a PhD grant from the CNES and Thales Alenia Space.