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

Y Dupain

ZEROS SETS OF H p FUNCTIONS IN LINEALLY CONVEX DOMAINS OF FINITE TYPE IN C n

Keywords: 2010 Mathematics Subject Classification. 32T25, 32T27 lineally convex, finite type, d-equation, ∂ -equation, zero sets, Hardy classes

In this note we extend N. Th. Varopoulos result

INTRODUCTION

The study of the zero-sets of holomorphic functions in a given class of a smoothly bounded domain in C n is a very classical problem which has been intensively studied. When n = 1 those sets are characterized, by the Blaschke condition, for the Nevanlinna and Hardy classes, but when n ≥ 2, the situation is more complicated.

Such characterizations are only known for the Nevanlinna under additional hypothesis on the domain: G. M. Henkin ([Hen75]) and H. Skoda [START_REF] Skoda | Valeurs au bord pour les solutions de l'opérateur ∂ , et caractérisation des zéros des fonctions de la classe de Nevanlinna[END_REF] independently obtained the case of strictly pseudo-convex domains, D. C. Chang A. Nagel and E. Stein ( [START_REF] Chang | Estimates for the ∂ -Neumann problem for pseudoconvex domains in C 2 of finite type[END_REF]) proved the same result for pseudo-convex domains of finite type in C 2 , and much later the case of convex domains of finite type ([BCD98, Cum01, DM01]), and, recently, the case of lineally convex domains of finite type were obtained ( [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]).

In [START_REF] Th | Zeros of h p functions in several complex variables[END_REF] (see also [START_REF] Andersson | On varopoulos's theorem about zero sets of H p -functions[END_REF]) N. Th. Varopoulos proved that, in a strictly pseudo-convex domain, a divisor satisfying a special Carleson condition is always defined by a function in some Hardy space H p (Ω). Tentatives to generalize this result, for example to convex domains of finite type, were done in [START_REF] Bruna | Zero sets of H p functions in convex domains of strict finite type in C n[END_REF] and [START_REF] Nguyen | Un théorème de la couronne H p et zéros des fonctions de H p dans les convexes de type fini[END_REF], but some gaps in the proofs leave the problem open until a recent paper of W. Alexandre ([Ale17]). In this last paper, the author make a strong use of the estimates of the Bergman metric obtained by Mc. Neal.

In this note, we show that Varopoulos result extends to lineally convex domains of finite type with a classical method using only the anisotropic geometry, described in [START_REF] Conrad | Anisotrope optimale Pseudometriken für lineal konvex Gebeite von endlichem Typ (mit Anwendungen)[END_REF], of those domains.

MAIN RESULTS

Let us first recall the definition of a lineally convex domain: Definition 2.1. A domain Ω in C n , with smooth boundary is said to be lineally convex at a point p ∈ ∂ Ω if there exists a neighborhood W p of p such that, for all point z ∈ ∂ Ω ∩W p ,

Ä z + T 10 z ä ∩ (Ω ∩W p ) = / 0,
where T 10 z is the holomorphic tangent space to ∂ Ω at the point z. In all the paper, we assume that ∂ Ω is of finite type and lineally convex at every point of ∂ Ω. We may assume that there exists a C ∞ defining function ρ for Ω and a number η 0 > 0 such that ∇ρ(z) = 0 at every point of W = {-η 0 ≤ ρ(z) ≤ η 0 } and the level sets {z ∈ W such that ρ(z) = η} , are lineally convex of finite type.

We thus assume that the defining function ρ satisfies this hypothesis.

In the next section, for z ∈ Ω in a fixed small neighborhood V of ∂ Ω, we recall the definition of the two fundamental quantities τ(z, u, δ ), for 0 < δ ≤ δ 0 , δ 0 > 0 depending only on Ω, and u a non zero complex vector and k(z, u) = δ Ω (z) τ(z,u,δ Ω (z)) where δ Ω (z) denotes the dis- tance of z to the boundary of Ω. The lineal convexity hypothesis implies that (z, u) → k(z, u) is a continuous function in V × C n * and, for 0 < δ 1 < δ 0 and K < +∞, there exists constants c > 0 and C < +∞ such that for z ∈ V ∩ {δ Ω (ζ ) ≥ δ 1 } and 1 K ≤ |u| ≤ K, c ≤ k(z, u) ≤ C. So, if u(z) is a continuous vector field in Ω, 1 K ≤ |u| ≤ K, k(z, u(z)) can be extended to Ω in a continuous function satisfying c ≤ k(z, u) ≤ C in Ω ∩ {δ 0 ≥ δ Ω (ζ ) ≥ δ 1 }.

To state our main result we have to recall the notion of Carleson measure in our context (see [CD]): a bounded measure µ in Ω is called a Carleson measure if

µ W 1 (Ω) := sup z∈∂ Ω, 0<ε<ε 0 |µ| (P ε (z) ∩ Ω) σ (P ε (z) ∩ ∂ Ω) + |µ| (Ω) < +∞,
ε 0 = αδ 0 , for α small enough, where P ε (z) is the extremal polydisk defined in the next section and σ the surface measure on ∂ Ω. W 1 (Ω) will denote the space of Carleson measures on Ω. Then, following ideas initiated in [START_REF] Bruna | Zeros varieties for the Nevanlinna class in convex domains of finite type in C n[END_REF] and adapted by W. Alexandre ([Ale17, Definition 1.2]) to non smooth forms we consider the following terminology (see section 4 for more details): A current ϑ = n i, j=1 ϑ i j dz i ∧ dz j of degree (1, 1) and order zero in

Ω is called a Car- leson current (in Ω) if ϑ W 1 (Ω) := sup u 1 ,u 2 δ Ω |ϑ (u 1 , u 2 )| k (•, u 1 ) k (•, u 2 ) W 1 (Ω) + (δ Ω |ϑ |) (Ω) < +∞,
where supremum is taken over all smooth vector fields u

1 = u i 1 i and u 2 = u i 2 i never vanishing in Ω, |ϑ (u 1 , u 2 )| is the absolute value of the measure ϑ (u 1 , u 2 ) = i, j ϑ i j u i 1 u j 2 and k (•, u k ) the continuous function z → k (z, u k ) defined before.
Similarly, a current ω of degree 1 and order zero in Ω is called a Carleson current (in

Ω) if ω W 1 (Ω) := sup u |ω(u)| k (•, u) W 1 (Ω) + |ω| (Ω) < +∞.
Remark. In the above definitions, the expressions are independent of the modulus of the vector fields. Thus they can always be chosen of modulus one.

Main Theorem.

Let Ω be a smoothly bounded lineally convex domain of finite type in C n . Let X be a divisor in Ω and ϑ X the associated (1, 1)-current of integration. Then, if ϑ X is a Carleson current and if the cohomology class of X in H 2 (Ω, Z) is zero, there exist p > 0 and f in the Hardy space H p (Ω) such that X is the zero set of f .

The general scheme of the proof is now standard (see [Var80, AC90, BG99, Ngu01, Ale17]): following Lelong's theory, we have to find a plurisubharmonic function u such that i∂ ∂ u = ϑ X satisfying a BMO estimate on ∂ Ω, and, as in [START_REF] Th | Zeros of h p functions in several complex variables[END_REF] (and [START_REF] Nguyen | Un théorème de la couronne H p et zéros des fonctions de H p dans les convexes de type fini[END_REF]), the conclusion will follow the John-Nirenberg theorem ( [START_REF] John | On functions of bounded mean oscillation[END_REF]). The two main steps are the resolution of the equation idw = ϑ with a Carleson estimate and the resolution of the ∂ b - equation with a BMO estimate on the boundary of Ω:

Theorem 2.1. Let ϑ be a closed Carleson current of degree (1, 1) and order 0 in Ω such that his canonical cohomology class in H 2 (Ω : C) is zero. Then there exists a Carleson current of degree 1 and order 0 ω satisfying dω = ϑ . Furthermore, if ϑ is real, ω can be chosen real.

Remark. This theorem could have been stated in the more general context of geometrically separated domains Ω introduced in [START_REF]Extremal bases, geometrically separated domains and applications[END_REF]. But, as we cannot prove the Main Theorem in that case, and because the technical details of the proof would be much more complicated, we restrict us to the case of lineally convex domains of finite type.

The second step is based on the proof of [CD, Theorem 2.4]: Theorem 2.2. There exists a constant C > 0 such that, for all ∂ -closed Carleson current of degree (0, 1) and order 0 ω on Ω, there exists a solution of the equation

∂ b u = ω such that u BMO(∂ Ω) ≤ C ω W 1 (Ω) .
Note that in [CD] this last result is stated, for smooth forms, with ω(ζ ) dλ W 1 instead of ω W 1 (Ω) , but we will see in section 4 that, for smooth forms, these two quantities are equivalent.

Theorem 2.1 is proved in section 5: after a regularization procedure we will essentially follow the general scheme developed in [START_REF] Andersson | On varopoulos's theorem about zero sets of H p -functions[END_REF] (see also [START_REF] Skoda | Valeurs au bord pour les solutions de l'opérateur ∂ , et caractérisation des zéros des fonctions de la classe de Nevanlinna[END_REF] and [START_REF] Th | Zeros of h p functions in several complex variables[END_REF]), the technical part being a strong modification of the calculus made in [START_REF] Bruna | Zero sets of H p functions in convex domains of strict finite type in C n[END_REF].

Theorem 2.2 is proved in section 6: once again, after a convenient regularization we use the methods developed in [CD] and in [START_REF] Skoda | Valeurs au bord pour les solutions de l'opérateur ∂ , et caractérisation des zéros des fonctions de la classe de Nevanlinna[END_REF].

GEOMETRY OF LINEALLY CONVEX DOMAINS OF FINITE TYPE

The anisotropic geometry of lineally convex domains of finite type is described in [START_REF] Conrad | Anisotrope optimale Pseudometriken für lineal konvex Gebeite von endlichem Typ (mit Anwendungen)[END_REF]. Let us just recall the basic estimates (from [CD]) we will use in the next section.

For ζ close to ∂ Ω and ε ≤ ε 0 , ε 0 small, define, for all non zero 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 where c 0 is chosen sufficiently small depending only on the defining function ρ of Ω and we define

(ζ , ε) → τ(ζ , v, ε) is smooth. In particular, ζ → τ(ζ , v, δ Ω (ζ ))
d(ζ , z) = inf {ε such that z ∈ B ε (ζ )} .
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).
Then we defined polydiscs AP ε (ζ ) by

(3.4) AP ε (ζ ) = z = ζ + n k=1 λ k v k such that |λ k | ≤ c 0 Aτ k (ζ , ε) . P ε (ζ )
being the corresponding polydisc with A = 1 and we also define

d 1 (ζ , z) = inf {ε such that z ∈ P ε (ζ )} .
Remark. Note that there are neither uniqueness of the extremal basis (v 1 , v 2 , . . . , v n ) nor of associated polydisk P ε (ζ ). However the functions τ i and the polydisks associated to two different (ζ , ε)-extremal basis are equivalent. Thus in all the paper P ε (ζ ) = P(ζ , ε) will denote a polydisk associated to any (ζ , ε)-extremal basis and τ i (ζ , ε) the radius of P ε (ζ ).

The fundamental result here is that d and d 1 are equivalent pseudo-distance which means that there exists a constant K and, ∀α > 0, constants c(α) and C(α) such that

(3.5) for ζ ∈ P ε (z), P ε (z) ⊂ P Kε (ζ ), and
(3.6) c(α)P ε (ζ ) ⊂ P αε (ζ ) ⊂ C(α)P ε (ζ ) and P c(α)ε (ζ ) ⊂ αP ε (ζ ) ⊂ P C(α)ε (ζ ).
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 such that (3.7) 1

K P ε (ζ ) ⊂ B ε (ζ ) ⊂ KP ε (ζ ), so d(ζ , z) ≃ d 1 (ζ , z).
Let us recall for ζ close to ∂ Ω and ε > 0 small, other basic properties of this geometry (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]): Lemma 3.1.

(1) Let w = (w 1 , . . . , w n ) be an orthonormal system of coordinates centered at ζ . Then

∂ |α+β | ρ(ζ ) ∂ w α ∂ wβ ε i τ (ζ , w i , ε) α i +β i , |α + β | ≥ 1. (2) If (v 1 , . . . , v n ) is a (ζ , ε)-extremal basis and γ = n 1 a j v j = 0, then 1 τ(ζ , γ, ε) ≃ n j=1 a j τ j (ζ , ε) . (3) If v is a unit vector then: (a) z = ζ + λ v ∈ P ε (ζ ) implies |λ | τ(ζ , v, ε), (b) z = ζ + λ v with |λ | ≤ τ(ζ , v, ε) implies z ∈ CP ε (ζ ). (4) If ν is the unit complex normal vector, then τ(ζ , v, ε) = ε and if v is any unit vector and λ ≥ 1, (3.8) λ 1/m τ(ζ , v, ε) τ(ζ , v, λ ε) λ τ(ζ , v, ε),
where m is the type of Ω.

Lemma 3.2 (Lemma 3.4 of [CD]). 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 the (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 3.1. Clearly, for δ Ω (z) ≤ δ 1 and all non zero vector v, we can extend smoothly the functions τ (z, v, ε) to all ε and we can also define vectors e i (z, ε) and polydisks P(z, ε), so that the above properties remain true with constants depending on A for ε and λ ε ∈ ]0, A],

δ Ω (z) and δ Ω (ζ ) ≤ δ 1 .

Of course the new (e i (z, ε)) i are not extremal basis in the original sense but we will call them again extremal basis.

SOME PROPERTIES OF CARLESON CURRENTS

In the previous section, we defined the terminology of Carleson current of degree (1, 1) or 1. We extend it to general currents T of degree 2 with the same definition:

Let T = i< j T 0 i, j dz i ∧ dz j + T 1 i, j dz i ∧ dz j + i< j T 2 i, j dz i ∧ dz j then

T W 1 (Ω) = sup u 1 ,u 2 δ Ω |T (u 1 , u 2 )| k (•, u 1 ) k (•, u 2 ) W 1 (Ω) + (δ Ω |T |) (Ω) < +∞,
where supremum is taken over all smooth vector fields u 1 = u i 1 i and u 2 = u i 2 i never vanishing in Ω, and |T (u 1 , u 2 )| is the absolute value of the measure

T (u 1 , u 2 ) = i< j T 0 i, j u i 1 u j 2 + T 1 i, j u i 1 u j 2 + i< j T 2 i, j u i 1 u j 2 .
Moreover, let V be an open set in Ω and T a current of degree 1 or 2 and order zero in V . We say that T is a Carleson current in Ω if the current χ V T , where χ V is the characteristic function of V , is a Carleson current in Ω and we denote

T W 1 (Ω) := χ V T W 1 (Ω) . Note that, if V is relatively compact in Ω, a current T in V is a Carleson current (in Ω) if the coefficients of T are bounded measures.
In the two next sections we need to regularize Carleson currents to be able to write explicit formulas solving the d or the ∂ equation. This is done classically using convolutions (see M. Andersson and H. Carlsson, [AC90, page 472] in the case of strictly pseudo-convex domains), and, because of the definition of the W 1 (Ω) norm for currents, we give below some details (for currents of degree (1, 1) to simplify notations

) when V is contained in a small neighborhood of a point of ∂ Ω (V ⊂ {δ Ω (z) < β δ 1 }).
For ε > 0 sufficiently small, let

ϕ ε = 1 ε 2n ϕ z ε where ϕ is a C ∞ -smooth non negative function supported in the ball {|z| < 1 /2} of C n such that ϕ = 1. Let T = T I,J dz I ∧ dz J be a Carleson current of order zero in an open set V of Ω. Let V ε = {z ∈ V such that δ ∂V (z) > ε} . Then for z ∈ V ε define T ε = I,J T I,J * ϕ ε dz I ∧ dz J so that T ε is a smooth form in V ε .
Proposition 4.1. With the above notations, if T is a closed Carleson current of degree 2 or 1 in V , then the forms T ε are closed and T ε W 1 (Ω)

T W 1 (Ω) .

Proof. To simplify the notations, we make the proof for T of degree (1, 1), that is T = i, j T i, j dz i ∧ dz j and

T ε = i, j T ε i, j dz i ∧ dz j with T ε i, j (z) = {|z-ζ |<1/2} ϕ ε (ζ -z)dT i, j (ζ ). Let Z ∈ ∂ Ω. If t < cε (c small enough depending only on Ω) then B(Z,t) ∩V ε = / 0.
Let us assume cε < t ≤ ε 0 . We have to estimate

I = χ(z) δ Ω (z) k(z, u)k(z, v) i, j T ε i, j (z)u i (z)v j (z) dλ (z) = sup | f |≤1 f (z)χ(z) δ Ω (z) k(z, u(z))k(z, v(z)) Ñ i, j T ε i, j (z)u i (z)v j (z) é dλ (z)
where χ is the characteristic function of B(Z,t) ∩ V ε and u and v are smooth vector fields never vanishing in Ω. Using the definition of T ε we get

I = sup | f |≤1 i, j Å f (z)χ(z) δ Ω (z)u i (z)v j (z) k(z, u(z))k(z, v(z)) ϕ ε (ζ -z)dλ (z) ã dT i, j (ζ ).
Note now that the function

z → f (z)χ(z) δ Ω (z)u i (z)v j (z) k(z, u(z))k(z, v(z)) ϕ ε (ζ -z) is supported in B(Z,t)∩{δ Ω > ε /2}. Moreover, for z ∈ V ε and |ζ -z| < ε /2, δ Ω (z) ≃ δ Ω (ζ ),
and, ζ ∈ P (z, K 1 δ Ω (z)). Then by (1) of Lemma 3.2 and (4) of Lemma 3.1, k(z, u(z

)) ≃ k(ζ , u(z)) and k(z, v(z)) ≃ k(ζ , v(z)), so I sup | f |≤1 i, j Å f (z)χ(z) δ Ω (ζ )u i (z)v j (z) k(ζ , u(z))k(ζ , v(z)) ϕ ε (ζ -z)dλ (z) ã dT i, j (ζ ).
Making the change of variables ζz = ξ and applying Fubini theorem, we get (

χ(ζ - ξ ) = 0 implies ζ ∈ P(Z, Kt)) I ϕ ε (ξ ) ñ sup | f |≤1 f (ζ -ξ )χ(ζ -ξ ) δ Ω (ζ ) k(ζ , u(ζ -ξ ))k(ζ , v(ζ -ξ )) i, j u i (ζ -ξ )v j (ζ -ξ )dT i, j (ζ )   dλ (ξ ) = ϕ ε (ξ ) ñ B(Z,Kt) χ(ζ -ξ ) δ Ω (ζ ) k(ζ , u(ζ -ξ ))k(ζ , v(ζ -ξ )) d i, j τ ξ (u i ) τ ξ (v j ) T i, j (ζ )   dλ (ξ ). where τ ξ (u i ) (ζ ) = u i (ζ -ξ ), τ ξ (v i ) (ζ ) = v i (ζ -ξ ).
Finally, (3.6) gives, denoting

u ξ (ζ ) = u(ζ -ξ ) and v ξ (ζ ) = v(ζ -ξ ) I sup |ξ |<ε/2 B(Z,Kt)∩{δ Ω >ε/2} δ Ω (ζ ) k(ζ , u ξ (ζ ))k(ζ , v ξ (ζ )) d T (u ξ , v ξ )
which concludes the proof, as the smooth vector fields u ξ and v ξ can be viewed as smooth vector fields in Ω never vanishing, and, for β small enough, V is contained in the union of the tents.

If T is globally defined in Ω, then the forms T ε are defined in {z ∈ Ω such that δ Ω (z) > ε} so there exists a constant C (depending only on ρ) such that they are defined in Ω ε = {ρ < -Cε}, ε small enough, but they are not Carleson currents in Ω ε in general. Then to be able to use this regularization procedure in the last section we have to introduce a notion of s-Carleson current.

Let s > 0 small. We say that a measure

µ in Ω is a s-Carleson measure if µ W 1 s (Ω) := sup z∈∂ Ω, s<ε<ε 0 |µ| (P ε (z) ∩ Ω) σ (P ε (z) ∩ ∂ Ω) + |µ| (Ω) < +∞,
and we say that a 1-current ω of order zero is a s

-Carleson current in Ω if ω W 1 s (Ω) := sup u |ω(u)| k (•, u) W 1 s (Ω) + |ω| (Ω) < +∞.
Then:

Proposition 4.2. There exists a constant C depending only on ρ such that, if T is a closed Carleson current in Ω then, for ε small, the closed forms T ε are ε-Carleson currents in

Ω ε = {ρ < -Cε} and T ε W 1 ε (Ω ε ) T W 1 (Ω) .
Proof. By Proposition 4.1 it suffices to show that

T ε W 1 ε (Ω ε ) T ε W 1 (Ω)
. Let z ∈ ∂ Ω ε and let Z be the projection of z on ∂ Ω. For t > ε, P(z,t) ⊂ P(Z, Kt) and the proposition follows (3.8) and Lemma 3.2.

Finally, to solve the d and ∂ equations with good estimates, we need to compare the notion of Carleson current for smooth currents of degree 2 or 1 T with a convenient punctual norm T (ζ ) dλ W 1 (Ω) : we define

T (ζ ) = sup v i ∈C n , v i =1 |T (v 1 , v 2 ) (ζ )| k (ζ , v 1 ) k (ζ , v 2 ) ,
for forms of degree 2 and

T (ζ ) = sup v =1 |T (v)(ζ )| k (ζ , v) ,
for forms of degree (0, 1). If there were smooth vector fields ( e i ) 1≤i≤n such that, at each point z, ( e i (z)) i is a (z, δ Ω (z))-extremal basis, this comparison would be immediate and, as noted by several authors, many points of the theory of convex (and lineally convex) domains of finite type would be simplified. Unfortunately this is not the case, and, in the case of convex domains of finite type and smooth currents, W. Alexandre overcomes this difficulty using a base of the Bergman metric (and estimates of this metric proved by J. Mc Neal, see [Ale17, Proposition 2.12]). The same result could be proved in our context of lineally convex domains using the results of [START_REF]Extremal bases, geometrically separated domains and applications[END_REF]. However, we do not use this method because it is quite easy to show, in general, that the W 1 (Ω)-norm of a current is controlled by vector fields "almost extremal": Proposition 4.3. Let ψ be a current of order zero of degree 2 or 1 in an open set U ⊂ {δ Ω (z) < β δ 1 } of Ω.

(1) There exist n smooth vector fields u i never vanishing in Ω such that, if ψ is of degree 2,

ψ W 1 (Ω) ≃ i, j δ Ω ψ (u i , u j ) k(•, u i )k(•, u j ) W 1 (Ω)
, the vector fields u i coinciding, outside a set of |ψ|-measure arbitrary small, with extremal basis in the sense of geometrically separated domains, and

ψ W 1 (Ω) ≃ i |ψ(u i )| k(•, u i ) W 1 (Ω)
if it is of degree 1, the constants in the equivalence being independent of ψ.

(2) Moreover, if ψ is smooth. Then: (a) ψ W 1 (Ω) ≃ δ Ω χ U ψ dλ W 1 (Ω) if ψ is of degree 2, and ψ W 1 (Ω) ≃ χ U ψ dλ W 1 (Ω) if not, the constants in the equivalence being indepen- dent of ψ; (b) for s > 0, if ψ is of degree 1, ψ W 1 s (Ω) ≃ χ U ψ dλ W 1 s (Ω)
the constants in the equivalence being independent of ψ.

Remark.

(1) In [Ale17, Proposition 2.12] W. Alexandre proved (2) of the proposition for convex domains of finite type, using universal vector fields (i.e. depending only on Ω but not on ψ) related to the Bergman metric.

(2) The extremal basis in the sense of geometrically separated domains ( [START_REF]Extremal bases, geometrically separated domains and applications[END_REF]) are not stricto-sensus extremal basis in J. Mc. Neal and M. Conrad sense, but they give the same homogeneous space.

(3) The equivalences of (2) of the proposition can be proved directly without using any extremal basis, using simply the continuity of the functions z → δ Ω (z)ψ(u,v)(z)

k(z,u(z))k(z,v(z))
which gives an equivalent ot Lemma 1 below on small euclidean balls. The final construction of the vectors fields (u i ) i is analog (and easyer). (4) Even if the proof of Proposition 5.1 needs only the second part of the proposition, we thought that it is interesting to present the assertion in the general case of non smooth currents.

Proof of Proposition 4.3. We only do the proof for currents ψ of degree 2. The inequality is trivial, so we prove the converse one.

Lemma 1. Let w ∈ U and (e i ) i be a δ Ω (w)-extremal basis at w. Let u and v be two smooth non vanishing vectors fields.

(1) Assume the coefficients of ψ are measures. Then, for every measurable set D ⊂ P (w, δ Ω (w)) ∩U, we have

(4.1) D δ Ω (ξ )d |ψ (u, v)| (ξ ) k(ξ , u(ξ ))k(ξ , v(ξ )) i, j D δ Ω (ξ )d ψ (e i , e j ) (ξ ) k (ξ , e i ) k (ξ , e j ) .
(2) Moreover, if ψ smooth. Then, for ξ ∈ P (w, δ Ω ) ∩U we have

(4.2) |ψ (u, v) (ξ )| k(ξ , u(ξ ))k(ξ , v(ξ )) i, j ψ (e i , e j ) (ξ ) k (ξ , e i ) k (ξ , e j ) .
Proof. Decomposing u and v on the basis (e i ) i , we get

|ψ (u, v) (ξ )| k(ξ , u(ξ ))k(ξ , v(ξ )) ψ (e i , e j ) (ξ ) |u i (ξ )| v j (ξ ) k(ξ , u(ξ ))k(ξ , v(ξ )) ,
if ψ is smooth, and, if not

D δ Ω (ξ )d |ψ (u, v)| (ξ ) k(ξ , u(ξ ))k(ξ , v(ξ )) i, j D δ Ω (ξ ) |u i (ξ )| v j (ξ ) d ψ (e i , e j ) (ξ ) k(ξ , u(ξ ))k(ξ , v(ξ )) .
Now, by (4) of Lemma 3.1 and (2) of Lemma 3.2,

k(ξ , u(ξ )) ≃ δ Ω (ξ ) τ (ξ , u(ξ ), δ Ω (w)) ≃ δ Ω (ξ ) τ (w, u(ξ ), δ Ω (w))
and, by (2) of Lemma 3.1,

1 τ (w, u(ξ ), δ Ω (w)) ≃ max |u i (ξ )| τ i (w, δ Ω (w))
and 1

k(ξ , u(ξ )) min τ i (w, δ Ω (w)) |u i (ξ )| δ Ω (ξ ) -1 .
Then

|u i (ξ )| v j (ξ ) k(ξ , u(ξ ))k(ξ , v(ξ )) τ i (w, δ Ω (w)) τ j (w, δ Ω (w)) δ Ω (ξ ) -2 1 k (ξ , e i ) k (ξ , e j )
because τ i (w, δ Ω (w)) = τ (w, e i , δ Ω (w)) ≃ τ (ξ , e i , δ Ω (ξ )) and the lemma is proved. which proves the result for β small enough.

We now finish the proof of the Proposition 4.3, proving both parts at the same time. Let

P i = P Ä Z i , δ Ω (Z i ) K ä , i ∈ N, be a minimal covering of U ∩ Ω.
For each i fixed and N i to be precised later, let

A j i = ß z ∈ U ∩ Ω such that d e (z, P i ) < δ Ω (Z i ) K j N i ™ , j = 1, . . . , N i ,
and A 0 i = P i . We assume that K is chosen so that, for all j,

A j i ⊂ P (Z i , δ Ω (Z i )). Let B j i = A j i \ A j-1 i . Let I k = i ∈ N such that δ Ω (Z i ) ∈ 2 -k , 2 -k+1 and M k = #I k the cardinal of I k . For each i ∈ I k let us choose N i = N(k) sufficiently large so that there exists s(i) ≥ 1 such that |ψ| Ä B s(i) i ä ≤ 1 2 k+4 1 M k ψ W 1 (Ω) . Let C i = A s(i)-1 i . Note that (C i ) i is an open covering of U ∩ Ω. Let ∆ = i B s(i) i
and let D j the connected components of (U ∩ Ω) \ ∆.

Let J( j) = t such that D j ⊂ C t . Let w j one of the points Z t such that t ∈ I( j). Let (ψ j ) j be a family of smooth functions such that 0 ≤ ψ j ≤ 1, ψ j ≡ 1 on D j , Supp (ψ j ) ⊂

D j ∪ ¶ A s(t) t
such that t ∈ J( j) © , and j ψ j ≡ 1 on U ∩ Ω.

For each j let Ä e j l ä l be a δ Ω (w j )-extremal basis at w j . Note that we can chose e j l so that the component of e j l on the first vector of the canonical basis ( f k ) k of C n is non negative. If we denote v j l = e j l + δ 2 Ω (w j ) f 1 and u l = j ψ j v j l , the vector fields u l , 1 ≤ l ≤ n, are smooth, don't vanish on U, and the proposition follows the lemmas.

PROOF OF THEOREM 2.1

The main point in the proof is the following local version of the theorem: Proposition 5.1. For each point p ∈ Ω there exist two neighborhoods W and V of p in Ω, W ⋐ V (in Ω) such that:

(1) If ϑ is a closed current of order 0 and degree 2 supported in V ∩ Ω such that ϑ is a Carleson current in Ω, there exists w a solution of the equation dw = ϑ in W such that w is a Carleson current in Ω.

(2) If ω is a closed current of order 0 and degree 1 supported in V ∩ Ω such that ω is a Carleson current in Ω, there exists f a solution of the equation

d f = ω in W such that δ 1/m-1 Ω f is a Carleson measure in Ω.
For convenience of the reader, let us briefly indicate how Theorem 2.1 is a simple consequence of the Proposition (this follows [START_REF] Skoda | Valeurs au bord pour les solutions de l'opérateur ∂ , et caractérisation des zéros des fonctions de la classe de Nevanlinna[END_REF], [START_REF] Th | Zeros of h p functions in several complex variables[END_REF], and [START_REF] Andersson | On varopoulos's theorem about zero sets of H p -functions[END_REF]).

We consider the following three sheaves F 0 , F 1 and

F 2 : Let U be an open set in Ω. ϑ ∈ Γ (U, F 2 ) if ϑ is a closed 2-current supported in U ∩ Ω and χϑ is a 2-Carleson current supported in U ∩Ω for all χ ∈ C ∞ 0 (U); w ∈ Γ (U, F 1 ) if w is a 1-current supported in U ∩ Ω, dw ∈ Γ (U, F 2 ) and χw is a 1-Carleson current supported in U ∩ Ω for all χ ∈ C ∞ 0 (U); f ∈ Γ (U, F 0 ) if f is a measure in U ∩ Ω, d f ∈ Γ (U, F 1 ) and χδ 1/m-1 Ω f is a Carleson measure supported in U ∩ Ω for all χ ∈ C ∞ 0 (U).
Then F 0 and F 1 are fine sheaves and, by Proposition 5.1, the sequence

0 → C i → F 0 d → F 1 d → F 2 → 0
is exact, and, by a standard cohomology argument,

Γ Ω, F 2 /dΓ Ω, F 1 ≃ H 2 Ω, C ≃ H 2 (Ω, C)
and Theorem 2.1 is proved.

For p ∈ Ω, choosing V and W to be euclidean balls relatively compact in Ω, the proposition only means that if ϑ (resp ω) is a current whose coefficients are bounded measures in V there exists w (resp. f ) solution of dw = ϑ (resp. d f = ω) in W whose coefficients are bounded measures in W . As this is standard, we don't give any details here.

We now prove Proposition 5.1 for a fixed point p ∈ ∂ Ω.

There exists a strictly positive real number δ 1 such that, for z ∈ Ω satisfying δ Ω (z) ≤ δ 1 the polydisk P(z, ε) are well defined. Then we choose the neighborhoods V and W of p as follows: let r 1 , r 2 and η 1 three positive real numbers, δ 1 > r 1 > 4r 2 > 8η 1 , such that (denoting by B e an euclidean ball):

• B e (p, r 1 ) ∩ Ω ⊂ {z ∈ Ω such that δ Ω (z) < δ 1 }; • there exists a point A(p) ∈ B e (p, r 1 ) ∩ {ξ such that 2r 2 < δ Ω (ξ ) < r 1/2} ∩ Ω; • for ζ ∈ B e (p, r 2 ), P (ζ , η 1 ) ∩ Ω ⊂ B e (p, 2r 2 ); • for ζ ∈ B e (p, 2r 2 ) and all ξ ∈ B e (p, 2r 2 ), if ν(ξ ) = ∇ρ(ξ ) is the normal at ξ , ¨ν(ξ ), # » A(p)ζ ∂ ≥ 1 2 # » A(p)ζ ν(ξ ) . Then we define V = B e (p, r 1 ) ∩ Ω and W = B e (p, r 2 ) ∩ {ζ ∈ Ω such that δ Ω (ζ ) < η 1 }.
First, we regularize the currents using Proposition 4.1 so that the regularized currents are smooth and closed in

V ε = {ζ ∈ V such that δ ∂V (ζ ) > ε}.
Thus, to finish the proof of Proposition 5.1 we assume the currents ϑ and ω supported and smooth in V ε and we will solve the equation dw = ϑ and

d f = ω in W ε = B e (p, r 2 -ε) ∩ {ζ ∈ Ω such that δ Ω (ζ ) < η 1 } (so that ∪ ε W ε = W ),
using Proposition 4.3, and the Proposition 5.1 will follow a standard weak limiting procedure.

Proof of Proposition 5.1 for smooth currents in V ε . By translation we assume A(p) = 0.

Let (P j ) j be a minimal covering of V ε by polydisks centered on Z j , P j = P (Z j , δ Ω (Z j )). Let (Φ j ) j be a smoth partition of 1 associated to the P j i.e. Φ j ≥ 0, Φ j = 1, Φ j identically zero outside 2P j , chosen so that

∂ Φ j ∂ v τ (•, v, δ Ω (Z j )) -1 . Let (ψ k ) k≥0 be a family of functions in C ∞ (R) with support in 2 -k-1 , 2 -k+1 and such that ψ k ≥ 0, ψ k = 1 on ]0, 1] and ψ ′ k (t) 2 k . Finally let ϕ ∈ C ∞ (R), 0 ≤ ϕ ≤ 1 such that ϕ(x) = 1 if x < 1 /2 and ϕ(x) = 0 if x > 1.
Let us denote D the unit disk of C. For Λ = (λ 1 , . . . , λ n ) ∈ D n and for c sufficiently small, to be precised, we consider the following function

(5.1) h Λ (t, z) = tz + ct k, j ψ k (1 -t)Φ j (tz) n i=1 A i, j,k (z)
with

A i, j,k (z) = ϕ Ç 2 -k -ρ(z) å 2 -k δ Ω (Z j ) λ i τ i (Z j , δ Ω (Z j )) e i (Z j , δ Ω (Z j )) + Ç 1 -ϕ Ç 2 -k -ρ(z) åå λ i τ i Ä Z j , 2 -k ä e i Ä Z j , 2 -k ä
where (e i (Z j , δ Ω (Z j ))) i is the (Z j , δ Ω (Z j ))-extremal basis used for the polydisk P j and e i Z j , 2 -k i a Z j , 2 -k -extremal basis. This function satisfies the following properties: Lemma 5.1.

(1) h λ (0, z) = 0, h λ (1, z) = z and d z h λ (1, z) = dz; (2) If Φ j (tz) = 0, then δ Ω (tz) ≃ δ Ω (Z j ) and tz ∈ P j (Z j , Kδ Ω (Z j ));

(3) If Φ j (tz) = 0, if c 0 ((3.2)) and c ((5.1)) are small enough,

(a) if ϕ Ä 2 -k -ρ(z) ä = 0, then δ Ω (h Λ (t, z)) ≃ δ Ω (tz) ≃ δ Ω (z) ≃ δ Ω (Z j ) and h Λ (t, z) ∈ P (Z j , Kδ Ω (Z j )), (b) If ϕ Ä 2 -k -ρ(z) ä = 0, then δ Ω (h Λ (t, z)) ≃ 2 -k and h Λ (t, z) ∈ P Z j , K2 -k ; (4) Denoting Q(t, z) = ® B(tz, 1 -t) if 1 -t ≥ -ρ(z) 1-t -ρ(z) B(tz, -ρ(z)) if 1 -t ≤ -ρ(z) and Q 1 (t, z) = {w = h Λ (t, z) for Λ ∈ D n }. (a) Then c in (5.1) being small enough, for z ∈ W and ∀t ∈ [0, 1], Q 1 (t, z) ⊂ Q(t, z), , Q 1 (t, z) ⊂ B e (p, r 1 ) ∩ Ω, (b) ∀t 0 > 0, there exists c 1 = c 1 (Ω, c,t 0 ) > 0 so that c 1 Q(t, z) ⊂ Q 1 (t, z), ∀t ≥ t 0 .
The homotopy operator H for d is then defined on smooth forms ϑ taking the average over D n of the integrals of the dt-component of h * Λ ϑ : for example, if ϑ is a smooth form of degree 2, H(ϑ ) = H ϑ is the form of degree 1

H ϑ (z) = H Λ (ϑ )(z)dΛ,
where H Λ (ϑ )(z) is defined on every vector v by

H Λ (ϑ )(z)(v) = 1 0 h * Λ ϑ dt = 1 0 ϑ (h Λ (t, z)) (Y t , Z t,v ) dt, where Y t = ∂ ∂t h Λ (t, z) and Z t,v = ∂ ∂ v h Λ (t, z).
Then H(ϑ ) is smooth and dH + Hd = Id. To get the required Carleson estimate for H(ϑ ), ϑ and ω must be zero around the origin A(p), so we follow a classical procedure (see [START_REF] Andersson | On varopoulos's theorem about zero sets of H p -functions[END_REF]). Let R > 0 such that B(0, 2R) ⊂ V ε \ W ε (ε > 0 small). Let ψ be a smooth cut of function equal to 0 in B(0, R) and 1 in V ε \ B(0, 2R). Then, with T = ϑ or ω, we have dH(ψT ) = ψT -H(dψ ∧ T ), and H(dψ ∧ T ) is closed in W ε and, if dτ = H(dψ ∧ T ) then d(H(ψT

) + τ) = ϑ in W ε .
Then the conclusion follows the next proposition because, using a standard Poincaré homotopy (reducing eventually W ε ), τ can be chosen satisfying

χ W ε δ 1/m-1 Ω |τ| W 1 (Ω) τ L ∞ (W ε ) ≤ H(dψ ∧ T ) L ∞ (W ε ) .
Proposition 5.2. There exists a constant C such that:

(1) with the above notation, H(dψ

∧ T ) L ∞ (W ε ) ≤ C T W 1 (Ω) ; (2) if ϑ is a smooth current of degree 2 identically zero in B (0, R), then χ W ε H(ϑ ) dλ W 1 (Ω) ≤ C δ Ω ϑ dλ W 1 (Ω) ;
(3) if ω is a smooth current of degree 1 identically zero in B (0, R), then χ W ε δ

1/m-1 Ω H(ω)dλ W 1 (Ω) ≤ C ω dλ W 1 (Ω) ;
Proof. We begin to prove (2). Note that there exists t 0 > 0 such that, ϑ (h Λ (t, z)) = 0 implies t > t 0 so that

H Λ (ϑ )(z)(v) = 1 t 0 ϑ (h Λ (t, z)) (Y t , Z t,v ) dt.
Lemma 1. Let T be the operator defined on non negative functions f on V ε , by

T ( f )(z) = δ Ω (z) 1/m-1 1 t 0 (1 -t) 1/m-1 Ç Q(t,z) f (w)δ 1-2/m Ω (w)dλ (w) å dt, z ∈ W ε . Then, for z ∈ W ε , H ϑ (z) T (δ Ω (•) ϑ (•) ) (z).
Proof of the lemma. By the definition of the "norm" • we have

|H ϑ (z)(v)| ≤ ñ 1 t 0 ϑ (h Λ (t, z)) k (h Λ (t, z),Y t ) k (h Λ (t, z), Z t,v ) dt ô dΛ.
By the definition of h Λ (t, z)

Y t = z + c k, j ψ k (1 -t)Φ j (tz) n i=1 A i, j,k (z)+ ct   k, j dΦ j (tz); z ψ k (1 -t) -Φ j (tz)ψ ′ k (1 -t)   i A i, j,k (z).
Assume Φ j (tz) = 0 and ψ k (1 -t) = 0. Let us first estimate k (h Λ (t, z),Y t ). We have dΦ j (tz); z δ Ω (Z j ) -1 ≃ δ Ω (tz) -1

1 1-t , ψ ′ k (1 -t)
τ Ä h Λ (t, z), e i Ä Z j , 2 -k ä , δ Ω (h Λ (t, z)) ä τ Ä Z j , e i Ä Z j , 2 -k ä , δ Ω (h Λ (t, z) ä . As s > 1 implies τ(p, v, sδ ) s 1/m τ(p, v, δ ) we get, if 2 -k ≤ -ρ(z), τ i Ä Z j , 2 -k ä k Ä h Λ (t, z), e i Ä Z j , 2 -k ää δ Ω (h Λ (t, z)) Å δ Ω (h Λ (t, z)) 2 -k ã -1/m . Similarly, if 2 -k ≥ -ρ(z) /2, τ i (Z j , δ Ω (Z j )) k (h Λ (t, z), e i (Z j , δ Ω (Z j ))) δ Ω (Z j ) ≃ δ Ω (h Λ (t, z)) ,
and (because δ Ω (h

Λ (t, z)) ≃ 2 -k ) 2 -k δ Ω (Z j ) τ i (Z j , δ Ω (Z j )) k (h Λ (t, z), e i (Z j , δ Ω (Z j ))) δ Ω (h Λ (t, z)) Å δ Ω (h Λ (t, z)) 2 -k ã -1/m .

These estimates give

k (h Λ (t, z),Y t ) Å δ Ω (h Λ (t, z)) 2 -k ã 1-1/m ≃ Å δ Ω (h Λ (t, z)) 1 -t ã 1-1/m .
The estimate k (h Λ (t, z), Z t,v ) is easy: recording that ((1) of Lemma 3.1)

∂ Φ j (•) ∂ v τ (•, v, δ Ω (Z j )) -1 , ∂ ∂ v (-ρ)(z) δ Ω (z) τ (z, v, δ Ω (z)) and that ϕ ′ Ä 2 -k -ρ(z) ä = 0 implies 2 -k ≃ -ρ(z), one easily gets k (h Λ (t, z), Z t,v ) δ Ω (z) 1/m δ Ω (h Λ (t, z)) 1-1/m τ (z, v, δ Ω (z)
) .

Then we obtain

H(ϑ )(z) δ Ω (z) 1/m-1 dΛ 1 0 ϑ (h Λ (t, z)) δ Ω (h Ω (t, z)) 2-2/m (1 -t) 1/m-1 ,
and the lemma is obtained making the change of variables Λ → h Λ (t, z), the jacobian being proportional to the volume of

Q 1 (t, z) which is equivalent to Q(t, z) because t ≥ t 0 implies c 1 Q(t, z) ⊂ Q 1 (t, z) ⊂ Q(t, z).
Lemma 2. The operator T of Lemma 1 satisfies the following estimate

χ W ε T ( f )dλ W 1 (Ω) χ V ε f dλ W 1 (Ω) .
Proof of the lemma. Let B(ξ , ε) be a pseudo-ball on ∂ Ω and B(ξ , ε) the tent over B(ξ , ε).

For z ∈ B(ξ , ε) we decompose T ( f ) into two pieces (to simplify the notation we write

f dλ W 1 instead of χ V ε f dλ W 1 (Ω) ): T 1 ( f )(z) = δ Ω (z) 1/m-1 1-ε t 0 (1 -t) 1/m-1 Ç Q(t,z) f (w)δ 1-2/m Ω (w)dλ (w) å dt, and
T 2 ( f )(z) = δ Ω (z) 1/m-1 1 1-ε (1 -t) 1/m-1 Ç Q(t,z) f (w)δ 1-2/m Ω (w)dλ (w) å dt. Consider first T 1 ( f ). As t < 1 -ε and δ Ω (z) ε, we have δ Ω (w) ≃ δ Ω (tz) ≃ 1 -t + δ Ω (z) ≃ 1 -t, d (w, ξ ) 1 -t and Q(t, z) ⊂ B(ξ , K(1 -t)). Then (note that B(ξ , 1 -t)⊂ KQ(t, z), because -ρ(z) ≃ δ Ω (z) ε 1 -t) Q(t,z) f (w)dλ (w) f dλ W 1 Vol Ä B(ξ , 1 -t) ä 1 -t f dλ W 1 Vol (Q(t, z)) 1 -t ,
and, using δ Ω (w) ≃ 1 -t, we get

T 1 ( f )(z) f dλ W 1 δ Ω (z) 1/m-1 1-ε t 0 (1 -t) -1/m-1 dt f dλ W 1 δ Ω (z) 1/m-1 ε -1/m and B(ξ ,ε) T 1 ( f )(z)dλ (z) f dλ W 1 ε -1/m B(ξ ,ε) δ Ω (z) 1/m-1 dλ (z) f dλ W 1 σ (B(ξ , ε)).
Consider now T 2 ( f ). B(ξ , ε) is equivalent to the set {rη such that 1 -ε ≤ r ≤ 1 and η ∈ B(ξ , ε)} , and if z = rη, δ Ω (z) ≃ 1 -r, and for w ∈ Q(t, z), w ∈ B(ξ , Kε) and δ Ω (w) ≃ δ Ω (tz) ≃ 1 -tr. Then

I = B(ξ ,ε) T 2 ( f )(z)dλ (z) B(ξ ,Kε) f (w)δ Ω (w) -1/m Ç D w (1 -r) 1/m-1 (1 -t) 1/m-1 (1 -tr) 1-1/m Vol (Q(t, rη)) drdtdσ (η) å dλ (w)
where w , δ Ω (trη) ≃ 1t and δ Ω (w) ≃ 1 -t. Let w 1 be the intersection of ∂ Ω with the half real line passing through 0 and w. Then η ∈ B (w 1 , K 1 (1t)), t ∈ [1 -K 2 δ Ω (w), 1 -c 2 δ Ω (w)] and r ≥ t ≥ 1 -K 2 δ Ω (w). As (by (2) of Proposition 4.3 and (3.7))

D w = ¶ (t, r, η) such that w ∈ Q(t, rη), (t, r) ∈ [1 -ε, 1] 2 and η ∈ B(ξ , ε) © . Note that 1 -tr ≃ max {(1 -t), (
Vol (Q(t, rη)) ≃ Vol (B(w, 1 -t)) ≃ (1-t)σ (B (w 1 , 1 -t)) ≃ (1-t)σ (B (w 1 , K 1 (1 -t))) , we get I 1 B(ξ ,Kε) f (w)δ Ω (w) -1/m Ç D 1 w (1 -r) 1/m-1 (1 -t)σ (B (w 1 , K 1 (1 -t))) drdtσ (η) å dλ (w) B(ξ ,Kε) f (w)δ Ω (w) -1/m 1-c 2 δ Ω (w) 1-K 2 δ Ω (w) Ç B(w 1 ,K 1 (1-t)) dσ (η) σ (B (w 1 , K 1 (1 -t))) 1 1-K 2 δ Ω (w) (1 -r) 1/m-1 dr å dt 1 -t dλ (w) B(ξ ,Kε) f (w)dλ (w) f dλ W 1 σ (B(ξ , Kε)) f dλ W 1 σ (B(ξ , ε)) . Finally, if (t, r, η) ∈ D 2 w , δ Ω (w) ≃ 1 -r and • if 1-t -ρ(rη) ≤ 1, then w ∈ 1-t -ρ(rη) B(trη, -ρ(rη)) ⊂ 1-t 1-r B(trη, 1 -r) so trη ∈ K 1 1-t 1-r B(w, 1 -r), and, moreover Vol Å 1 -t -ρ(rη) B(trη, -ρ(rη)) ã ≃ Vol Å 1 -t 1 -r B(w, 1 -r) ã ; • if 1-t -ρ(rη) ≥ 1, then w ∈ B(trη, 1 -t) ⊂ K ′ 1-t 1-r B(w, 1 -r), because in this case 1 -t ≃ 1 -r, so trη ∈ K 1 1-t 1-r B(w, 1 -r), and Vol(B(trη, 1 -r)) ≃ Vol (B(w, 1 -r)).
Now, for t, r and w fixed, σ η such that (t, r, η) ∈ D 2 w 1 1-t Vol Ä 1-t 1-r B(w, 1r) ä , and, 1-t 1-r B(w, 1r) ⊂ K 2 B(w, 1t) by (3.7) and (3.8), and, by the last property of W , for w and t fixed, the lengh of the set of r such that there exists η such that trη ∈ D 2 w is 1t. Then (2) of the proposition follows immediately the lemmas. Assertion (3) of the proposition is proved in a similar and easier way. We will not give more details.

We finish giving briefly the proof of (1) of the proposition.There exists δ > 0 such that t < δ or t > 1 -δ implies dψ ∧ ϑ (h Λ (z,t)) = 0 so It is easy to see that, if ω is smooth, then, the proof of Proposition 4.3 shows that ω W 1 s (D) ≃ ω dλ W 1 s (D) . We start the proof of the theorem regularizing the current ω using Proposition 4.2: then (simplifying the notations), for ε > 0 small enough, the regularized current ω ε is smooth ∂ -closed in Ω ε = {ρ < -Cε} and satisfies ω ε W 1 s (Ω ε ) ω W 1 (Ω) . Now we solve the equation ∂ u ε = ω ε using the method described in the proof 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]Theorem 2.4]: u ε is given by the formula

H(dψ ∧ ϑ )(z) (v 1 , v 2 ) = Λ 1-δ δ dψ ∧ ϑ (h Λ (z,t)) (Y t , Z t,v 1 , Z t,v 2 ) dt.
u ε (z) = Ω ε K 1 ε (z, ζ ) ∧ ω ε (ζ ) -∂ * N ε Å Ω ε P ε (z, ζ ) ∧ ω ε (ζ )
ã the kernels K 1 ε and P ε are associated to the defining function ρ + Cε as described 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]. We proceed as H Skoda in [Sko76, Section 8]. Clearly, the C ∞ -smooth kernels P ε , as well as their derivatives, converge uniformly to the corresponding kernel P of Ω so that Ω ε P ε (z, ζ ) ∧ ω ε (ζ ) converges in every Sobolev norm to a ∂ -closed form g on Ω such that, for all integer k, g H k is bounded by the total mass of ω. Then

∂ * N ε Å Ω ε P ε (z, ζ ) ∧ ω ε (ζ ) ã converges in C 1 Ω to a function h.
Let Φ ε : ∂ Ω → ∂ Ω ε be a family of C ∞ diffeomorphisms such that Φ ε converges to the identity uniformly in C ∞ norm on ∂ Ω.

As ∂ u ε = ω ε , denoting

v ε = Ω ε K 1 ε (z, ζ ) ∧ ω ε (ζ ), if v ε • Φ ε converges in L 1 (∂ Ω) to v = Ω K 1 (z, ζ ) ∧ ω(ζ ), for z ∈ ∂ Ω, the function u(z) = Ω K 1 (z, ζ ) ∧ ω(ζ ) -h

  is a smooth function. The pseudo-balls B ε (ζ ) = B(ζ , ε) (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)}

  and, by (3) (a) of Lemma 5.1,

  1r)} and let us cut D w into two parts D 1 w = D w ∩ {r ≥ t} and D 2 w = D w ∩ {r < t} , and define I i , i = 1, 2, replacing in the definition of I D w by D i w . If (t, r, η) ∈ D 1

I 2 B

 2 (ξ ,Kε) f (w)δ Ω (w) -1/m 1 1-K 3 δ Ω (w) Ñ {(r,η) s. t. trη∈D 2 w } dσ (η)dr Vol Ä 1-t 1-r B (w, 1t) ä é (1t) 1/m-1 dtdλ (w),and we get I 2 B(ξ ,Kε) f (w)dλ (w) f dλ W 1 σ (B(ξ , ε)) finishing the proof of the lemma.

  Note that, for t ∈ [δ , 1 -δ ] and |v i | ≤ 1, |Z t,v i | and |Y t | are bounded by C = C(δ ), i = 1, 2. Then after the change of variables w(Λ) = h Λ (z,t) we get|H(dψ ∧ ϑ )(z) (v 1 , v 2 )| w) |dψ ∧ ϑ | (w)dλ (w)dt B δ Ω (w) |dψ ∧ ϑ | (w)dλ (w) δ Ω dψ ∧ ϑ W 1 δ Ω ϑ W 1the first inequality coming from the fact that Vol (Q(t, z)) and δ Ω (w) are bounded from below, B in the third inequality being a tent containing Q(t, z) and the last because σ (B) is bounded from below.The proof of Proposition 5.1 is now complete.6. PROOF OF THEOREM 2.2Let us introduce the notion of BMO s functions, similarly to s-Carleson measures and currents defined before Proposition 4.2:f is in BMO s (∂ D) if f BMO s (∂ D) := supz∈∂ D, s<t<ε 0 P t (z)∩∂ D f -P t (z)∩∂ D f < +∞.

  By (2) of Lemma 3.1, k (ξ , e ′ i ) ≃ k (ξ , e i ), and ψ; e ′ i , e ′ j = ψ; e i , e j +

	O	Ä	δ 2 Ω (w) ä	ψ; e s , e t
	s,t

Lemma 2. Under the conditions of the previous lemma, the inequalities (4.2) and (4.1) are still true replacing the basis (e i ) i by the basis (e ′ i ) i where e ′ i = e i + O δ 2 Ω (w) .

Proof.

is a solution of the equation ∂ b u = ω. By the properties of the kernels K 1 ε and K 1 (K 1 ε converges uniformly on ∂ Ω × Ω η (η > 0 fixed) to K 1 ) this convergence follows exactly the proof made by H Skoda in [START_REF] Skoda | Valeurs au bord pour les solutions de l'opérateur ∂ , et caractérisation des zéros des fonctions de la classe de Nevanlinna[END_REF]p. 272].

To conclude the proof of Theorem 2.2 we have to show that v BMO(∂ Ω) ω W 1 (Ω) . The proof of [CD, Theorem 2.4] gives

with a constant C 1 uniform in ε (small enough) because the estimates of [CD, lemmas 3.4, 3.5 and 3.6] are uniform in a neighborhood of ∂ Ω. Thus the end of the proof is the following lemma Lemma 6.1. With the previous notations u BMO(∂

Proof. Let ξ ∈ ∂ Ω and let B(ξ ,t) be a pseudo-ball on ∂ Ω. Then σ ε (B (Φ ε (ξ ),t)) converges to σ (B(ξ ,t)) and

u.

The lemma follows easily.


Marcel Kuper
email: kuper@cirad.fr

Zaigham Habib
email: zaihabib@yahoo.com

Containing

Serge Marlet

Pierre Ruelle

Containing salinity through irrigation management: the case of the Fordwah area in Pakistan

come

Introduction

Large-scale irrigation in the Indus plains: a conjunctive use environment

The Indus Basin irrigation system, fed by the snow-covered peaks of the Trans-Himalayan ranges of Pakistan, China, India and Kashmir, irrigates an area of about 16 million ha, diverting annually about 128 billion m 3 surface water to 43 canal systems. Groundwater has traditionally been exploited in the Indus plains, mainly through Persian wheels. At the turn of the century, 350,000 wells existed in the Punjab and the North West Frontier Province, contributing about 40% of the total irrigation supplies (Indian Irrigation Commission, 1903). [START_REF] Lieftinck | Water and Power resources of West Pakistan, a study in sector planning[END_REF] considered that about 200,000 wells were still operational in the 1960s, estimating the groundwater use in the Indus Basin canal commands at 11.9 billion m 3 , amounting to 15 % of the total irrigation supplies. In the past 35 years extensive development of Pakistan's groundwater resources has taken place through the installation of 16.000 public tubewells. These tubewells were meant to lower the groundwater tables, but also provided additional water supplies for irrigation. At the same time, farmers have installed a large number of shallow tubewells, having a capacity of about 30 ls -1 . Presently, 500,000 tubewells provide 30-40 % of the irrigation water at the farm gate (Nespak/Sgi, 1991). Studies in the 1990s showed that almost all farmers have now access to groundwater either through ownership or through water markets [START_REF] Strosser | Water markets in the Fordwah/Eastern Sadiqia area[END_REF]. The conjunctive use of surface and groundwater for irrigation is, therefore, a reality for the majority of the farmers in the Indus Basin, especially in the Punjab, allowing them a greater flexibility and better access to irrigation water. Traditionally, salinity has been associated with irrigated agriculture in the Indus Basin (Framji et al., 1984, Ahmed and[START_REF] Chaudry | Irrigated agriculture of Pakistan[END_REF]. Salinity was considered linked with waterlogging, related to the introduction of large-scale perennial irrigation in the Indus Basin. However, the Soil Survey of Pakistan showed in the 1970s that there were more causes of salinity. Genetic salinity, due to weathering of parent material, was thought to affect some 4.8 million ha of land [START_REF] Choudry | Secondary salinization in the Indus plains, Pakistan. Expert consultation on identification and reclamation of salt-affected soils[END_REF]. A second source of salinization was the rise in groundwater tables in the doabs (land between two rivers), displacing salts and bringing them into the root zone through capillary rise. Finally, the Soil Survey of Pakistan warned about the threat of salinization through the use of poor quality groundwater by tubewells. The often doubtful quality of groundwater was known for much longer (Council of the Bahawalpur State, 1900), but has become an important issue due to the massive deployment of tube wells in the Indus Basin.

Soil salinity: status and measures taken

Measures in Pakistan to control salinity have focused on controlling the groundwater table. This included the prevention of seepage through canal lining (from 1895 onwards), surface and interceptor drains (from 1930 onwards), irrigation management (canal closures) and vertical drainage through tubewells from 1940 onwards [START_REF] Chaudry | Irrigated agriculture of Pakistan[END_REF]. The Directorate for Land Reclamation, Punjab, conducts an annual salinity survey since 1943 [START_REF] Muhammed | Salt-affected soils of Pakistan[END_REF]. On this basis, it advises the irrigation agency on the installation of reclamation shoots (pipe outlets installed for 3 months during the flood season), thus providing extra water to those areas affected by salinity. In recent years, due to the tremendous pressure on canal water, very few reclamation shoots have been sanctioned. The Water and Power Development Authority (WAPDA), created in 1958, was entrusted with the responsibility of tackling waterlogging and salinity in the Indus Basin, through large-scale vertical and horizontal drainage schemes, referred to as Salinity Control And Reclamation Projects (SCARP). These had a dual aim of lowering groundwater tables by installing public tubewells and increasing agricultural production, making additional irrigation supplies available at the farm gate. 32 SCARP projects of a total cost of 1.2 billion dollars were implemented covering 3.8 million ha, resulting in the installation of about 13,000 public tubewells, 5000 km of surface drains and 2000 km of sub-surface drains [START_REF] Ghassemi | Salinisation of land and water resources: human causes, management and case studies[END_REF]. In addition, the Left Bank Outfall Drain in Lower Sindh drains an area of 600,000 ha with a main drain of 270 km and 2000 public tubewells. Public tubewells were later transferred to farmers due to excessive operation and maintenance costs. In most cases, farmers preferred installing small private tubewells, further encouraged by government subsidies. In the seventies, researchers recognized the existence of sodic soils, evidenced in the WAPDA surveys, and realized the adverse effects of (bi)carbonates and sodium in irrigation water hailing from the aquifer. Recommendations were made regarding safe limits of irrigation water, gypsum application, and salinity-tolerant crops [START_REF] Ruelle | Country report -Pakistan[END_REF][START_REF] Bakhsh | Prospects of irrigated agriculture with saline water in Pakistan[END_REF]Muhammed, 1975). Their recommendations went largely unheeded in the large scale development projects in the country, which continued to focus on the prevention of seepage through the lining of about 20,000 thousand watercourses and the implementation of large-scale drainage projects based on the waterlogging and salinity paradigm [START_REF] Muhammed | Salt-affected soils of Pakistan[END_REF]. This signaled the inception of a dispute between researchers who understood the complexity of the salinity issue and engineers who had to come up with practical solutions for a problem affecting millions of hectares. Several surveys have been undertaken in the Indus Basin to qualify the extent of soil salinity and the impact of government interventions, including the 1953-54 WASID survey and the WAPDA survey of 1977-79. Ahmad andChaudhry (1988) show that the overall salinity status has improved over these years. While in 1963 only 53% of the irrigated area was salt free, about 67% of the area was free of salts in 1978. The 1977-79 survey showed that 4,2 million ha were affected by salinity. It further showed the importance of sodicity, present in more than 25% of the samples (n = 63866). The decreasing trend in soil salinity, which can be partly attributed to the lowering of the groundwater tables due to the large-scale interventions of the government and the groundwater exploitation of farmers, has also been favoured by the increased availability of irrigation water. Groundwater resources indeed provided an additional 50 billion m 3 of irrigation water. At the same time, the surface water diversions increased from 75 billion m 3 in the 1950s to a maximum of about 135 billion m 3 with the extension of the irrigation schemes (Kotri, Taunsa, Guddu) and the achievement of Mangla and Tarbela dams (6.58 and 11.59 billion m 3 storage respectively) in 1965 and 1978. Farmers were thus able to reclaim large tracts of land in their conjunctive use environment. Interestingly, farmers have been very diverse in their measures, bringing large areashitherto fore affected by salinity -under cultivation. These range from the application of good quality water and taking various biotic, mechanical and chemical reclamation measures. Farmers are also unremittingly mitigating the effects of high groundwater tables and poor quality irrigation water on soils and crop yields, by adapting the quantity, frequency and ratio of application of surface and tubewell water, the crop choice and the application of chemical amendments [START_REF] Kielen | Farmers' ability to cope with salinity and sodicity[END_REF][START_REF] Kuper | Irrigation management strategies for improved salinity and sodicity control[END_REF]. This paper investigates the impact of farmers' measures on soil salinity and determines the physical constraints they are confronted with in order to formulate recommendations on government interventions aimed at reducing the impact of soil salinity on agricultural production.

Methods and materials

Study area

The research was carried out in the Chishtian sub-division, a 75.000 ha hydraulic unit of the Fordwah canal command offtaking from Suleimanke Headworks on Sutlej river in Pakistan's Punjab (Figure 1).

Figure 1. Map of the Chishtian sub-division, Punjab, Pakistan

Chishtian is fed by the Fordwah Branch (primary canal), serving 14 distributaries (secondary canals) who in turn distribute water to a total of 503 watercourses (tertiary units). The area receives about 0.51 billion m 3 of surface water annually. In addition, a total number of 4450 tubewells (according to a 1996 census), corresponding to a density of 6.4 tubewells per 100 ha, represent a total pumping capacity more than three times the maximum discharge of the surface water irrigation system. The cropping pattern in the area is mainly based on a cotton (summer) -wheat (winter) rotation and annual cropping intensities reach 150%. The crop yields are below the national average with an average wheat yield of almost 1.9 t ha -1 (2.3 at the national level). The area operated by farmers is on average 6.4 ha, but varies considerably. The climate is semi-arid continental with annual potential evaporation (class A pan) at 2400 mm far exceeding the annual rainfall of 200 mm.

Research methodology

A water and salt balance was elaborated for 8 watercourses of Fordwah and Azim distributaries, two major secondary canals in the Chishtian sub-division. The equations used are classical and documented for the study area by [START_REF] Waijjen | Study of water and salt balances for 8 sample watercourse commands in Chishtian Sub-division, Punjab, Pakistan[END_REF]. For these watercourses, irrigation application data were collected weekly and soil and water samples (n=600) seasonally. Interviews with farmers (n=278) allowed to determine the farm characteristics for all landholdings in these watercourses. To determine the effect of irrigation on soil salinity, a one-dimensional soil water flow-solute transport model, SWAP93 was used [START_REF] Feddes R | Modeling soil water dynamics in the unsaturated zone -state of the art[END_REF][START_REF] Dam | Modeling of water flow and solute transport for irrigation and drainage[END_REF]. The model simulates water, solute and heat transport in the air-plant-soil environment. It includes hysteresis, the possibility of simulating preferential flow and adsorption and decomposition processes as described by [START_REF] Boesten | Modeling the influence of sorption and transformation on pesticide leaching and persistence[END_REF]. The model calculates the soil water flow with Richards' equation, based on Darcy's law and the principle of mass conservation. To solve the equation, the soil hydraulic functions are described by the van Genuchten-Mualam parameters (van Genuchten, 1987;[START_REF] Wösten | Using texture and other soil properties to predict the unsaturated soil hydraulic properties[END_REF]. The root water uptake is described semi-empirically by a sink term as a function of the maximum root water uptake, the soil water pressure head and the salt concentration [START_REF] Feddes R | Modeling soil water dynamics in the unsaturated zone -state of the art[END_REF][START_REF] Prasad R | A linear root water uptake model[END_REF]. The transport of conservative solutes is described with the convection-dispersion equation. Saline soil moisture conditions reduce the root water uptake due to an increased osmotic head of the soil water. The osmotic head is added to the matrix head, and the total head is used to derive the reduction factor for root water uptake. The model reduces crop transpiration due to water and salinity stress only, while all other factors (e.g. nutrient availability) are considered optimal.

Groundwater use and salinity in the Chishtian sub-division

Following earlier studies of the Soil Survey of Pakistan (e.g. Choudhry, 1979), [START_REF] Kijne | Salinity in Punjab watercourse commands and irrigation system operations[END_REF] demonstrated the impact of the application of poor quality groundwater through tubewells on soil salinity and sodicity in the Mananwala distributary command area in central Punjab. In subsequent publications they referred to this phenomenon as the process of "secondary salinization" as opposed to a primary salinization caused by the capillary rise of salts in the rootzone in the presence of shallow groundwater tables. They found an increasing trend of soils salinity and sodicity towards the tail of the Mananwala distributary, which they attributed to a decreasing availability of canal water from head to tail, as well as a deteriorating groundwater quality, impacting on the quality of available irrigation water. A redistribution of (good quality) canal water, bringing it there where it is most needed, i.e. to those areas suffering from poor quality groundwater, would then be a means of sustaining agricultural production with poor quality groundwater [START_REF] Kijne | Salinity in Punjab watercourse commands and irrigation system operations[END_REF]. In further studies the impact of irrigation with poor quality groundwater on soil salinity and sodicity in the Indus Basin was confirmed ( [START_REF] Waijjen | Study of water and salt balances for 8 sample watercourse commands in Chishtian Sub-division, Punjab, Pakistan[END_REF][START_REF] Smets | Salinization and crop transpiration of irrigated fields in Pakistan's Punjab[END_REF][START_REF] Condom | Salinisation, alkalinisation and sodification in Punjab, Pakistan: Characterisation of the geochemical and physical processes of degradation[END_REF]. It was further shown that the physical environment inducing problems of salinity and sodicity is very diffuse and complex. Along an irrigation canal soil types, depth to groundwater tables and groundwater quality are generally quite heterogeneous, impacting on the soil salinity and sodicity status of these units. This is shown for eight tertiary units of the Fordwah and Azim distributaries in Table 1. The table shows indeed a contrasting physical environment, linked to the geological formation of the alluvial plains of the Sutlej and Hakra rivers [START_REF] Survey | Semi-detailed soil survey of the Chishtian sub-division[END_REF], where especially farmers in Fordwah 14 are faced with adverse conditions combining a poor groundwater quality with a groundwater table close to the surface. The proximity of the groundwater table seems a determinant factor in explaining the adverse salinity and sodicity conditions in Fordwah 14 and 62. The physical conditions are more favourable for the Azim tertiary units. However, the surface water availability for these units is much inferior to that of the Fordwah units, which explains the high groundwater use in the Azim units and thus the relatively poor irrigation water quality (EC iw , SAR iw ). The irrigation water quality is here practically of the same quality as the groundwater inducing high levels of soil sodicity. The electrical conductivity is presented in dS m -1 , the Sodium Adsorption Ratio in (mmol l -1 ) 0.5 . The number after the name of the distributary indicates the distance to the head of the canal in thousand feet. The time-related data cover a one-year period from May 1994 to April 1995.

On the basis of these data, a salt and water balance at the level of the watercourse was established by [START_REF] Waijjen | Study of water and salt balances for 8 sample watercourse commands in Chishtian Sub-division, Punjab, Pakistan[END_REF], showing the relative water stress imposed by farmers on their crops with a ratio ET a /ET pot of less than 0.9 and an average leaching fraction of a little over 0.1. This leaching fraction is sufficient to keep the average predicted salinity levels below 4 dS m -1, except in Fordwah 14 due to its physical constraints (poor groundwater quality, superficial groundwater table) and Azim 63 due to under-irrigation. Interestingly, the capillary rise in Fordwah 14 has a positive impact on the soil salinity. With capillary rise the soil is more wet, resulting in a higher leaching fraction. The salts removed from the rootzone through percolation exceed the salt inflow through capillary rise.

The importance of the availability of good quality surface water for the salinity status of irrigated lands was also demonstrated in other canal commands in the Punjab [START_REF] Kijne | Salinity in Punjab watercourse commands and irrigation system operations[END_REF][START_REF] Kuper | Farmers' irrigation practices and their impact on soil salinity: is salinity here to stay? Paper presented at the ninth internal program review[END_REF]. The surface water availability is very heterogeneous between the different canal commands. The water allocation to different canal commands at the watercourse head varies from 500 to more than 2000 mm for different canal commands. In addition, the surface water availability along irrigation canals is not always uniform [START_REF] Bhutta | Equity of water distribution along secondary canals in the Punjab[END_REF][START_REF] Kuper | Farmers' irrigation practices and their impact on soil salinity: is salinity here to stay? Paper presented at the ninth internal program review[END_REF][START_REF] Tareen | Water distribution at the secondary level in the Chishtian sub-division[END_REF]. In addition, certain watercourses are privileged, usually through an increased size of the outlet (breadth, width), irrespective of their location along the canal. In other words, in addition to a considerable heterogeneity in physical constraints (groundwater quality, depth to groundwater table, soils), there is also a great diversity in the access to surface water, which explains the wide variety of situations to which farmers are confronted.

Farmers' salinity management

"Owners are reluctant to give up cultivation until the process of deterioration makes germination of seed impossible" [START_REF] Mehta | The formation and the reclamation of thur lands in the Punjab[END_REF].

There is a need to understand and analyse the salinity management of farmers. Firstly, the diverse physical constraints farmers face need to be understood. Only then, appropriate interventions can be formulated, which effectively help farmers to cope with the adverse effects of irrigation. Secondly, the experiences farmers have in coping with salinity and sodicity can be beneficial for devising interventions. Finally, there is a large range in farmers' socio-economic background, determining whether farmers can or want to take advantage of the opportunities that are offered. This latter aspect is not developed in this paper.

Farmers use a vernacular terminology to define salinity phenomena (Table 2). The classification is based on visual characteristics, such as the white efflorescence on soil surfaces or the dark film caused by a dispersion of organic matter, the physical degradation (reduced intake rate, surface crust or hard layers), and the effects on crop growth (germination problems). The Central Board of Irrigation (1941) investigated this vernacular classification. White salinity or kallar was found to be associated with sodium and magnesium chlorides, while black kallar was found in the presence of sodium carbonates.

Farmers in the study area distinguish mainly between white or chitta kallar and black or kala kallar. A surface crust is mainly associated with white salinity, while black salinity is often accompanied by hard layers at the surface or in the profile (zacht). Kallar shor, mentioned in the 1941 classification as "impregnated with salts", is a soil difficult to cultivate due to poor physical properties. Hard layers in the profile Kallar shor White salts at the surface, extremely difficult to cultivate When relating farmers' observations to the results of soil samples, it appeared that farmers observe chitta kallar at EC e levels of 2.4 dS m -1 and higher, while zacht happens at SAR levels as low as 6 [START_REF] Kielen | Farmers' ability to cope with salinity and sodicity[END_REF], which corresponds to an ESP of only 7 when using the relationship developed by the USDA (Richards, 1954). The values that are thus obtained, are much lower than the criteria defined in the same publication, where an EC e of 4 dS m -1 and an ESP of 15 are assumed to distinguish between non-saline and saline, and non-sodic and sodic soils. This is an important observation, as these criteria are often used in Pakistan as a reference. Farmers recognize the different origins of salinity and/or sodicity status, relating them to the presence of high groundwater tables, to genetic salinity and to the use of poor quality irrigation water. Farmers appreciate the potential contribution of high groundwater tables to salinity and sodicity, even though the groundwater can provide considerable amounts of water to the crop. Genetic salinity covered substantial parts of the Chishtian Sub-division, but farmers have reclaimed large tracts using canal water. Some of these soils, e.g. the dense sodic soils, have physical properties, which make them difficult to cultivate, and pose lasting limitations to farmers (Soil Survey of Pakistan, 1998). The use of poor quality irrigation water, pumped by tube wells, in areas away from the river is a relatively recent phenomenon, from 1985 onwards. Farmers differentiate the impact of various tube well waters on soil and plants conditions. The importance of the quality of irrigation water can be appreciated from the farmers' classification of irrigation water. Generally, irrigation water is evaluated for its effect on soils and crops: the water of a certain tube well causes zacht or a hard layer in the profile. The classification is not entirely in line with the FAO classification of [START_REF] Ayers R | Water quality for agriculture[END_REF], who emphasize the risk of reduced infiltration rates with waters of low salt concentration and a relatively high amount of sodium, see Figure 2. The greatest disagreement relates perhaps to canal water with an EC of 0.19 and an SAR of 0.2, which poses according to the FAO classification a moderate sodicity hazard but is judged to be of excellent quality by farmers. According to farmers, tube well water with an SAR greater than 5 and an EC greater than 1.0 causes hard layers in the soil. Farmers experience the adverse effects of poor quality irrigation water rapidly. After 2-3 irrigations with such water, a surface crust develops, while hard layers in the soil can occur within an irrigation season. Their views were confirmed by [START_REF] Condom | Salinisation, alkalinisation and sodification in Punjab, Pakistan: Characterisation of the geochemical and physical processes of degradation[END_REF], who used a geo-chemical model in conjunction with a solute transfer model and provided evidence for a rapid sodification of soils. From Figure 2 it appears that the logic of the FAO classification does not correspond with farmers perceptions. The FAO emphasizes the physical logic. When a soil is irrigated with water having a low concentration in salts, the Diffuse Double Layer tends to increase in size, degrading the soil structure and reducing the hydraulic conductivity. This increase is favoured by the presence of the mono valent cation Na + as opposed to the divalent cations Ca 2+ and Mg 2+ , i.e. high values of SAR. Farmers have a long term perspective. Irrigation waters with high salt concentrations are more likely to cause soil salinity and sodicity, especially when dominated by sodium bicarbonates. This was confirmed in a modelling exercise where the ESP of a soil was determined while concentrating the soil water [START_REF] Condom | Salinisation, alkalinisation and sodification in Punjab, Pakistan: Characterisation of the geochemical and physical processes of degradation[END_REF]). An ESP of 15 was attained with a 5-fold concentration of irrigation water with an EC of 1.4 dS m -1 , while a 10fold concentration was required to obtain an ESP of 15 for an irrigation water with a lower EC (0.8 dS m -1 ), but with a similar chemical composition.

Simulating the impact of irrigation application on soil salinity

The soil water flow -solute transport model SWAP93 was calibrated and validated for four fields of 0.4 ha each, cultivated by farmers with a cotton-wheat rotation, and representing the dominant soil types in the area, i.e. a loamy sand (LS), a sandy loam (SL), a loam (L) to silty clay loam (SiCL), and a loam to silt loam (SiL). The calibration and validation of the model was documented by [START_REF] Smets | Salinization and crop transpiration of irrigated fields in Pakistan's Punjab[END_REF].

Simulating the effect of the irrigation application on the water and salt balance

The long term effect of different quantities and qualities of irrigation on salinity and T act was assessed for one of the four fields, a sandy loam. A total of 16 irrigations were provided during a single year for a cotton and wheat crop. The groundwater table was assumed to be sufficiently deep in order to have free drainage. The water and solute transfer was simulated for a soil profile of 2 m. Since small quantities of water were applied in some of the scenarios, which influences the rapidity with which an equilibrium is reached, a total simulation period of 10 years was adopted. The results at the end of this period were compared. This is at the end of the cotton season, just before the pre-sowing irrigation for wheat. The simulation results are shown in Figures 3 and4 for the relative transpiration and S, respectively. The total yearly irrigation application of the different scenarios were as follows: 133, 120, 107, 93, 80 cm. The irrigation water quality was 1, 2, 3, 4 dS m -1 respectively. Figure 3 shows the direct relationship between the irrigation quantity and the amount of water that can be extracted by plants. This relationship is curvi-linear with a tangent that decreases with the amount of water that is applied. A reduction in irrigation quantity leads also directly to a reduction in the leaching fraction. When irrigating with an EC of 4 dS m -1 , for example, the leaching fraction decreases from 0.22 to 0.09 when comparing the highest with the lowest irrigation quantity. This leads to increases in S with decreasing irrigation amounts, as shown in Figure 4. Increases in the EC of the irrigation water lead to slightly higher leaching fractions, due to the fact that in dry conditions the plants are more restricted in their water uptake when the soil water is saline. This positive effect attenuates to a certain extent the extra input of salts when irrigating with more saline water, although it cannot prevent the increase in S for irrigation waters with higher concentrations. The relative transpiration is adversely affected by the increase of concentration of the irrigation water.

Simulating the effect of the groundwater table on the water and salt balance

In a limited part of the area groundwater tables are within 2-3 m of the soil surface. In the section on calibration/validation, it was already shown that this affects soil salinity. Long term simulations were carried out to assess the effect of a groundwater table at 2 m of the soil surface on the water and salt balance for four soil types, i.e. loamy sand, sandy loam, loam to silty clay loam and silt loam. The EC of the irrigation water is 3 dS m -1 . The results are presented in Figure 5 for the soil salinity. The effect of the groundwater table is very pronounced. When comparing with Figure 4, the salinity has in some cases more than doubled. This is related to the capillary rise that occurs, especially in the first 2-3 years of the simulations. The cumulative leaching fraction is very low in this period. At the same time, the soil profile is wetter due to the presence of the groundwater table, resulting in lower salt concentrations, and thus a smaller amount of salts that is leached. Soon, a relatively high salinity level is reached. Due to the fact that the plants are constrained in their root water uptake, the leaching fraction increases. At the end of the simulation period, the leaching fraction is only slightly smaller than in the case of free drainage. However, the salinity levels remain relatively high. In case of small irrigation quantities, the effect of the groundwater table is more important as is evidenced by the tangent of the curve in Figure 5, which is steeper than that in Figure 4. The presence of a groundwater table has also an impact on the transpiration. Two effects can be discerned. Firstly, the higher salinity levels reduce the transpiration. Secondly, the capillary rise contributes positively to the transpiration. For the medium textured soils, the two phenomena appear to have about the same importance with a relative transpiration that is the same with or without the presence of a groundwater table. However, for the coarse textured soils, which have a lower salinity level, the capillary rise is clearly more important and T act increases by about 2 cm per year, as compared to a situation with a deep groundwater table.

The effect of farmers' irrigation practices on soil salinity and transpiration

Farmers were shown in an earlier section to dispose of a large array of measures to deal with salinity. In this section, the effect of some of these measures in the present physical environment will be evaluated. This will enable an assessment of how much of the salinity problem can be overcome by farmers themselves, and how the effectiveness of their measures compare with the proposed interventions. Two measures are investigated, i.e. the frequency of irrigation and the quality of the pre-sowing irrigation water, since farmers have expressed to use both techniques in order to mitigate the effect of irrigation water quality on crop production. The existing irrigation regime consists generally of a large pre-sowing application, in one or two applications, and a number of subsequent irrigations. In the study area, the number of applications for wheat is generally recommended to be 1 pre-sowing irrigation and 4-6 subsequent applications, while for cotton these are usually 2 and 6-8, respectively. Farmers generally prefer to apply good quality canal water (0.2 dS m -1 ) for the pre-sowing irrigations. The reference scenario will follow the recommended practices. The first alternative scenario consists of applying the same amount of water in a higher number of applications, see Table 3. The second alternative scenario follows the frequency of the reference scenario, but the pre-sowing irrigations are done with poor quality water, i.e. an EC of 1.5 dS m -1 . Both alternative scenarios will be compared with the irrigation management intervention, i.e. a reduction of 40% in the quantity of water that is available. Also, a comparison will be made with a scenario showing a physical constraint in the form of the presence of a shallow groundwater table. An analysis of the evolution of salinity and T act will be carried out inside the crop seasons, because the interventions of farmers occur punctually during the season. The irrigation regime of the reference scenario was derived from the recommendations of the Agricultural Department in the study area and consist of 6 irrigations for wheat and 10 for cotton, including the pre-sowing irrigations. The total application is 132 cm. The initial soil moisture content is 15.7 cm for the profile, while the initial value of S is 153.2 mg cm -2 . The simulations are carried out for a three year period with cultivation of cotton and wheat under the same irrigation practices. The results of the third year are then analyzed. The first results show that the leaching fractions for the high frequency irrigation and for the poor quality pre-sowing irrigation are very similar to that of the reference scenario [START_REF] Kuper | Irrigation management strategies for improved salinity and sodicity control[END_REF]. This is certainly not the case for the other two scenarios. In case of a presence of a superficial groundwater table, the leaching fraction increases considerable with values between 0.2 and 0.6. The under-irrigation scenario decreases the leaching fraction, as expected, with values below 0.05. The effect of these scenarios on the relative transpiration is limited for most of the scenarios. Only in case of under-irrigation there is a reduction by about 10% of the fraction T a /T pot .

Table III. Definition of scenarios to evaluate the intra-seasonal effect of irrigation practices and a shallow groundwater table on soil salinity and transpiration

Scenarios

The simulation results show a high leaching fraction in the period January-March for most of the scenarios. Interesting is that the leaching fraction for the under-irrigation scenario is almost zero in that same period, resulting in a substantial salinization. Finally, the effect of a groundwater table at 2 m is shown to impact greatly on the leaching fraction. However, the concentration of the water that is leached is much lower than for other scenarios due to the wetter profile. Thus, only in case of under-irrigation there is a considerable impact on T act . This applies both to cotton and wheat. In the case of wheat, the fact that the pre-sowing irrigation in December as well as the first subsequent irrigation are 40% lower than in the reference scenario is shown to especially affect adversely T act .

The effect of the different irrigation practices on the salt accumulation for the same soil (silt loam) is presented in Figure 6. Generally, the salinity tends to be lower in Kharif than in Rabi. This is due to the fact that in the period January to March before the beginning of Kharif overirrigation takes place. The temperatures are not so high yet resulting in low values of ET pot , which makes leaching possible. In Kharif farmers have difficulty to even keep the profile moist and leaching reduces to a minimum. Thus, salinity gradually increases and is at its peak at the onset of Rabi.

Figure 6 shows that applying the same amount of irrigation water in a higher frequency does not affect much the salinity as compared to the reference scenario, a finding which was also established for irrigation with a lower frequency, i.e. 12 irrigations yearly, for the same soil [START_REF] Smets | Salinization and crop transpiration of irrigated fields in Pakistan's Punjab[END_REF]. This may well be related to the soil type. [START_REF] Smets | Salinization and crop transpiration of irrigated fields in Pakistan's Punjab[END_REF] found for a loamy sand that the frequency of irrigation is very important due to its lower soil moisture retention. Moisture stress more easily occurs in this case when the irrigations are too far apart. More impact has the decrease in water quality for the pre-sowing irrigations for wheat and cotton, resulting in a higher soil salinity. The impact of the pre-sowing irrigations, one in December and two in May/June can be detected in Figure 6. A slightly different pattern is observed only in case of a groundwater table at 2m. In this case, capillary rise contributes to a salinization in April/May, when farmers are not irrigating and the temperatures are high. A shallow groundwater table contributes considerably to salinization.

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An interesting observation was made by [START_REF] Smets | Salinization and crop transpiration of irrigated fields in Pakistan's Punjab[END_REF], who showed that in case of over-irrigation soil salinity may actually decrease in the presence of a shallow groundwater table. This was attributed to a higher leaching fraction, induced by a higher soil moisture content. However, in case of under-irrigation, a shallow groundwater table clearly increases S in the soil profile as shown in Figure 6. The under-irrigation scenario is shown to affect salinity very much. It is also the only scenario where after 3 years of irrigation, still no equilibrium is attained. The most important impact of a reduction of irrigation quantity is that farmers are no longer able to reduce salinity in the period January-March. The remainder of the year the salinity trend is very similar to the reference scenario.

There is considerable variation in soil salinity during the year with higher salinity levels in Rabi than in Kharif. The period January-March is shown to be of utmost importance for salinity management. In winter time farmers can leach salts due to low values of ET pot . The effect on transpiration due to high salinity levels seems limited, but is highest towards the end of Kharif for cotton as well as the beginning of Rabi for wheat. Farmers' practices, within the existing limits of irrigation water availability and physical conditions, were shown to influence soil salinity. However, the effects of farmers' practices are certainly much smaller than a change in the irrigation quantity or quality.

Conclusion

Farmers have shown a tremendous ability to manage salinity in the Indus Basin irrigation system. Firstly, they have reclaimed large tracts of hitherto fore salinized lands through a series of measures. Indeed, in the Chishtian sub-division the area moderately or severely affected by salinity decreased by 7,000 ha from 16,950 to 10,275 ha (-40%) from 1953 to 1978. Secondly, they have been able to maintain cropping intensities at around 150% even on salt affected lands over a long period of time, juggling between different sources of irrigation water (surface and groundwater) and adapting their irrigation practices to a wide diversity of physical constraints. Laboratory results indicate that for different watercourses of the Fordwah and Azim distributaries almost 30% of the cultivated fields that were sampled had an SAR greater than 8. These fields are confronted with severe problems of sodicity (hard layers, infiltration problems), but continue to produce. The government interventions, mainly focused on lowering groundwater tables, have supported farmers in their salinity management. The greater availability of good quality surface water to the Indus Basin canal commands has also contributed significantly in this regard. This study shows indeed the importance of an access for farmers to an irrigation water of sufficient quality, be it surface or groundwater. Farmers in Azim distributary have thus been able to replace a diminishing surface water supply by using more tubewell water. Farmers in Fordwah distributary, whose poor quality groundwater can only be used when combined with surface water, have made sure that their access to canal supplies remains ensured. In those areas where the groundwater is of poor quality, farmers rely indeed entirely on the supply of surface water and any problem related to this supply impacts directly on the crop production. Extra water supplies for reclaiming portions of land affected by salinity are no longer sanctioned due to the severe water stress in the Indus Basin. A reallocation of water to canal commands, taking the salinity status and the present conjunctive use environment into account, could be envisaged in order to maximise the extent of the irrigated land and the number of farmers that participate in irrigated agriculture. The water allocation in the Indus Basin is a patchwork of water rights that have been fixed at the design stage of the different canal commands, defining the extent of the irrigation schemes (the so-called canal command areas, CCA) and the entitlement per unit of land. In addition, certain canal commands receive year-round supplies (perennial systems), while others receive water only during the flood season, Kharif (non-perennial systems). Climate, soil and crop type, depth to groundwater table, historical water rights and a negotiation process between the different riparian provinces and independent states to demarcate the share of water of a proposed irrigation scheme were important factors in this process. The water allocation is revised regularly at a local level through the inclusion of hitherto fore unirrigated land into the CCA. However, these revisions are generally not accounted for in the authorised discharges at higher levels of the irrigation system. The water allocation has also been discussed regularly at a regional level through a series of high level committees culminating in the 1991 Water Apportionment Accord, specifying the shares of each of the provinces. The accord is much less specific at the canal command level. A reallocation of water in the Indus Basin canal commands is, of course, a very sensitive and extremely important issue, and is definitely not related to salinity control only. However, the salinity of irrigated lands threatens the livelihood of those farmers located in physically unfavourable conditions and should be taken into account when discussing the water allocation in the Indus Basin.

The spatial heterogeneity of physical parameters and temporal variability of different processes, as well as the diversity of farms are inherent in the analysis of irrigation systems. This is a disadvantage, because it implies that the spatial and temporal structure of information needs to be analyzed, requiring a more substantial data set, and advanced geostatistical techniques to classify and extrapolate. For policy makers and irrigation managers, this poses also a serious problem as the effectiveness of global interventions are reduced. However, heterogeneity/variability can also be seen as an important strength and opportunity. Strength because the system is better adapted to external shocks, and opportunity because the heterogeneity and variability offer possibilities for redistribution of resources. [START_REF] Kuper | Irrigation management strategies for improved salinity and sodicity control[END_REF] showed that due to a heterogeneity in groundwater quality and soil types, a redistribution of water can lead to a considerable decrease in the area affected by salinity and sodicity. The challenge is thus in understanding and quantifying the existing heterogeneity and use it for defining policy and management interventions. These interventions will need to be more diverse and better targeted to specific local conditions in order to be more effective.

Figure 2 .

 2 Figure 2. Farmers' perceptions of the irrigation water quality (good, marginal and poor) in comparison with the FAO classification (Ayers and Westcot, 1985). The farmers' perceptions of the water quality are obtained from[START_REF] Kielen | Farmers' ability to cope with salinity and sodicity[END_REF]

Figure 3 .Figure 4 .

 34 Figure 3. Simulation results of the effect of irrigation quantity and quality on the relative transpiration T act /T pot , of cotton and wheat for a sandy loam (after Smets et al., 1997)

Figure 5 .

 5 Figure 5. Simulation results of the effect of the irrigation quantity on the soil salinity S, in the presence of a groundwater table at 2 m depth for a loamy sand (LS), sandy loam (SL), loam to silty clay loam (SiCL) and a silt loam (SiL). The EC of the irrigation water is 3 dS m -1

Figure 6 .

 6 Figure 6. Simulation results of the impact of irrigation practices on soil salinity S for a silt loam

Table I . Physical environment, irrigation practices and their impact on soil salinity and sodicity for eight watercourses of the Fordwah and Azim distributaries (Kuper, 1997)

 I

	Tertiary	Soil type	Groundwater	EC gw V cw	V tw	EC iw SAR iw EC e SAR e
	units		table depth (m)		(mm)	(mm)
	Azim 20	silt loam	3	0,65 831	203	0.3	4.0	1.2	4.6
	Azim 43	silt loam	4	0,80 590	368	0.4	3.7	2.7	7.6
	Azim 63	silt loam	4	0,76 292	549	0.5	4.2	2.3	7.7
	Azim 111	clay loam	6	1,07	9	1107 1.0	9.5	2.6 11.1
	Fordwah 14 sandy loam	2	1,76 508	273	0.9	9.2	2.8	7.0
	Fordwah 46 sandy loam	3	0,86 784	130	0.3	4.1	1.1	3.4
	Fordwah 62 sandy loam	2.5	0,96 761	266	0.4	6.6	1.6	5.9
	Fordwah	loamy sand	3	1,34 400	655	0.8	11.5 1.6	9.4
	130

Table II . Vernacular soil salinity classification (Central Board of Irrigation, 1941; Kielen, 1996)

 II

	Classes	Characteristics
	Chitta	White (chitta) efflorescence, surface crust
	Kala kallar Black (kala) appearance with hard upper soil layer
	Zacht

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