Annalisa Baldi

Bruno Franchi

Pierre Pansu

Gagliardo-Nirenberg Inequalities

GAGLIARDO-NIRENBERG INEQUALITIES FOR DIFFERENTIAL FORMS IN HEISENBERG GROUPS

Keywords: 1991 Mathematics Subject Classification. 58A10, 35R03, 26D15, 43A80, 46E35, 35F35 Heisenberg groups, differential forms, Gagliardo-Nirenberg inequalities, div-curl systems, symbols of differential operators

published or not. The documents may come

Introduction

The L 1 -Sobolev inequality (also known as Gagliardo-Nirenberg inequality) states that for compactly supported functions u on the Euclidean nspace, [START_REF] Abraham | Manifolds, tensor analysis, and applications[END_REF] u

L n/(n-1) (R n ) ≤ c ∇u L 1 (R n ) .
The generalization to differential forms is recent (due to Bourgain & Brezis and Lanzani & Stein), and states that the L n/(n-1) -norm of a compactly supported differential h-form is controlled by the L 1 -norm of its exterior differential du and its exterior codifferential δu (in special cases the L 1 -norm must be replaced by the H 1 -Hardy norm). We shall extend this result to Heisenberg groups in the framework of an appropriate complex of differential forms.

1.1. The Euclidean theory. In a series of papers ( [START_REF] Bourgain | On the equation div Y = f and application to control of phases[END_REF], [START_REF] Bourgain | New estimates for the Laplacian, the div-curl, and related Hodge systems[END_REF], [START_REF] Bourgain | New estimates for elliptic equations and Hodge type systems[END_REF]), Bourgain and Brezis establish new estimates for the Laplacian, the div-curl system, and more general Hodge systems in R n and they show in particular that if → F is a compactly supported smooth vector field in R n , with n ≥ 3, and if curl → F = → f and div → F = 0, then there exists a constant C > 0 so that (2)

→ F L n/(n-1) (R n ) ≤ → f L 1 (R n ) .
This result does not follow straightforwardly from Calderòn-Zygmund theory and Sobolev inequality. Indeed, suppose for sake of simplicity n = 3 and let → F be a compactly supported smooth vector field, and consider the system [START_REF] Baldi | A recursive basis for primitive forms in symplectic spaces and applications to Heisenberg groups[END_REF]. Then, by Calderón-Zygmund theory we can say that

(3)    curl → F = → f div → F = 0 .

It is well known that

→ F = (-∆) -1 curl → f is a solution of
∇ → F L p (R 3 ) ≤ C p → f L p (R 3 ) , for 1 < p < ∞.
Then, by Sobolev inequality, if 1 < p < 3 we have:

→ F L p * (R 3 ) ≤ → f L p (R 3 ) ,
where 1 p * = 1 p -1 3 . When we turn to the case p = 1 the first inequality fails. The second remains true. This is exactly the result proved by Bourgain and Brezis.

In [START_REF] Lanzani | A note on div curl inequalities[END_REF] Lanzani & Stein proved that (1) is the first link of a chain of analogous inequalities for compactly supported smooth differential h-forms in R n , n ≥ 3,

u L n/(n-1) (R n ) ≤ C du L 1 (R n ) + δu L 1 (R n ) if h = 1, n -1; (4) u L n/(n-1) (R n ) ≤ C du L 1 (R n ) + δu H 1 (R n ) if h = 1; (5) u L n/(n-1) (R n ) ≤ C du H 1 (R n ) + δu L 1 (R n ) if h = n -1, ( 6
)
where d is the exterior differential, and δ (the exterior codifferential) is its formal L 2 -adjoint. Here H 1 (R n ) is the real Hardy space (see e.g. [START_REF] Elias | Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals[END_REF]). In other words, the main result of [START_REF] Lanzani | A note on div curl inequalities[END_REF] provides a priori estimates for a div-curl systems with data in L 1 (R n ). We stress that inequalities [START_REF] Baldi | Compensated compactness for differential forms in Carnot groups and applications[END_REF] and ( 6) fail if we replace the Hardy norm with the L 1 -norm. Indeed (for instance), the inequality [START_REF] Baldi | Gagliardo-Nirenberg inequalities for horizontal vector fields in the Engel group and in the 7-dimensional quaternionic Heisenberg group[END_REF] u

L n/(n-1) (R n ) ≤ C du L 1 (R n ) + δu L 1 (R n )
is false for 1-forms. The counterexample is given by E.M. Stein in [START_REF] Elias | Singular integrals and differentiability properties of functions[END_REF], p. 191. Indeed, take f k ∈ D(R n ) such that f L 1 (H n ) = 1 for all k ∈ N and such that (f k ) k∈N tends to the Dirac δ in the sense of distribution. Set now v k := ∆ -1 f k . Then estimate [START_REF] Baldi | Gagliardo-Nirenberg inequalities for horizontal vector fields in the Engel group and in the 7-dimensional quaternionic Heisenberg group[END_REF] would yield that {|∇v k | ; k ∈ N } is bounded in L n/(n-1) (R n ), and then, taking the limit as k → ∞ that |x| -n ∈ L 1 (R n ).

1.2. The Heisenberg setting. Recently, in [START_REF] Chanillo | Subelliptic Bourgain-Brezis estimates on groups[END_REF], Chanillo & Van Schaftingen extended Bourgain-Brezis inequality to a class of vector fields in Carnot groups. Some of the results of [START_REF] Chanillo | Subelliptic Bourgain-Brezis estimates on groups[END_REF] are presented in Theorems 4.2 and 4.3 below in the setting of Heisenberg groups. These are the main tool that allows us to give a Heisenberg version of Lanzani & Stein's result. We describe now the operators that will enter our theorem.

We denote by H n the n-dimensional Heisenberg group. It is well known that the Lie algebra h of the left-invariant vector fields admits the stratification h = h 1 ⊕ h 2 . We shall refer to the elements of h 1 as to the horizontal derivatives on H n .

Heisenberg groups admit a one-parameter group of automorphisms called dilations. Whereas, in Euclidean space, all exterior forms of degree h have homogeneity h under Euclidean dilations, on the contrary, because of the stratification of h, h-forms on Heisenberg groups split into two weight spaces, with weights h and h + 1. This leads to a modification (E * 0 , d c ) of the de Rham complex introduced by Rumin. Bundles of covectors are replaced by subbundles E h 0 and the exterior differentials by differential operators d c on spaces Γ(E h 0 ) of smooth sections of these subbundles. It turns out that this complex, which is both invariant under left-translations and dilations, is easier to work with that ordinary differential forms.

The core of Rumin's theory relies on the following result.

Theorem 1.1. If 0 ≤ h ≤ 2n + 1 there exists a linear map

d c : Γ(E h 0 ) → Γ(E h+1 0 ) such that i) d 2 c = 0 (i.e. E 0 := (E * 0 , d c ) is a complex); ii) the complex E 0 is exact; iii) d c : Γ(E h 0 ) → Γ(E h+1 0
) is an homogeneous differential operator in the horizontal derivatives of order 1 if h = n, whereas

d c : Γ(E n 0 ) → Γ(E n+1 0
) is an homogeneous differential operator in the horizontal derivatives of order 2;

iv) if 0 ≤ h ≤ n, then * E h 0 = E 2n+1-h 0 ; v) the operator δ c := (-1) h(2n+1) * d c * is the formal L 2 -adjoint of d c . Definition 1.2. If 0 ≤ h ≤ 2n + 1, 1 ≤ p ≤ ∞, we denote by L p (H n , E h 0 )
the space of all sections of E h 0 such that their components with respect a given left-invariant basis belong to L p (H n ), endowed with its natural norm. Clearly, this definition is independent of the choice of the basis itself. If h = 0, we write

L p (H n ) for L p (H n , E 0 0 ). The notations D(H n , E h 0 ), S(H n , E h 0 ), E(H n , E h 0 )
, and H 1 (H n , E h 0 ) have an analogous meaning (here H 1 is the Hardy space in H n defined in [START_REF] Folland | Hardy spaces on homogeneous groups[END_REF], p.75). Now can state our main result that generalizes the results of [START_REF] Baldi | Sharp a priori estimates for div-curl systems in Heisenberg groups[END_REF] to all Heisenberg groups. Theorem 1.3. Denote by (E * 0 , d c ) the Rumin's complex in H n , n > 2 (for the cases n = 1, 2 we refer to [START_REF] Baldi | Sharp a priori estimates for div-curl systems in Heisenberg groups[END_REF]). Then there exists C > 0 such that for any

h-form u ∈ D(H n , E h 0 ), 0 ≤ h ≤ 2n + 1, such that d c u = f δ c u = g
we have:

i) if h = 0, 2n + 1, then u L Q/(Q-1) (H n ) ≤ C f L 1 (H n ,E 1 0 ) ; u L Q/(Q-1) (H n ,E 2n+1 0 ) ≤ C g L 1 (H n ,E 2n 0 ) ; ii) if h = 1, 2n, then u L Q/(Q-1) (H n ,E 1 0 ) ≤ C f L 1 (H n ,E 2 0 ) + g H 1 (H n ) ; u L Q/(Q-1) (H n ,E 2n 0 ) ≤ C f H 1 (H n ,E 2n+1 0 ) + g L 1 (H n ,E 2n-1 0 ) ; iii) if 1 < h < 2n and h = n, n + 1, then u L Q/(Q-1) (H n ,E h 0 ) ≤ C f L 1 (H n ,E h+1 0 ) + g L 1 (H n ,E h-1 0 ) ; iv) if h = n, n + 1, then u L Q/(Q-2) (H n ,E n 0 ) ≤ C f L 1 (H n ,E n+1 0 ) + d c g L 1 (H n ,E n 0 ) ; u L Q/(Q-2) (H n ,E n+1 0 ) ≤ C δ c f L 1 (H n ,E n+1 0 ) + g L 1 (H n ,E n 0 ) ; u L Q/(Q-1) (H n ,E n 0 ) ≤ C g L 1 (H n ,E n-1 0 ) if f = 0; u L Q/(Q-1) (H n ,E n+1 0 ) ≤ C f L 1 (H n ,E n+2 0 ) if g = 0.
The proof of Theorem 1.3 follows the lines of the proofs in [START_REF] Baldi | Sharp a priori estimates for div-curl systems in Heisenberg groups[END_REF] of the corresponding results for H 1 and H 2 . The new crucial contribution of the present paper in contained in Theorem 5.1 that, roughly speaking, states that the components with respect to a given basis of closed forms in E h 0 are linear combinations of the components of a horizontal vector field with vanishing "generalized horizontal divergence". This is obtained by proving that the symbol of the intrinsic differential d c is left-invariant and invertible (see Corollary 5.5 and Proposition 5.6).

In Section 2 we fix the notations we shall use throughout this paper. In Section 3 we gather some more or less known results about tensor analysis in Heisenberg groups. Section 4 recalls results borrowed to Chanillo and Van Schaftingen. Section 5 contains Theorem 5.1 together with several auxiliary results. The proof of Theorem 1.3 is completed in Section 6. Section 7 contains a variant of Theorem 1.3 where no differential operator occurs on the right hand side.

Notations and definitions

As above, we denote by H n the n-dimensional Heisenberg group identified with R 2n+1 through exponential coordinates. A point p ∈ H n is denoted by p = (x, y, t), with both x, y ∈ R n and t ∈ R. If p and p ′ ∈ H n , the group operation is defined as

p • p ′ = (x + x ′ , y + y ′ , t + t ′ + 1 2 n j=1 (x j y ′ j -y j x ′ j )).
In particular for any p ∈ H n there is a familiy of (left) translations τ p : H n → H n defined by τ p q := p • q, q ∈ H n .

For a general review on Heisenberg groups and their properties, we refer to [START_REF] Elias | Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals[END_REF], [START_REF] Gromov | Carnot-Carathéodory spaces seen from within[END_REF], [START_REF] Bonfiglioli | Stratified Lie groups and potential theory for their sub-Laplacians[END_REF], and to [START_REF] Th | Analysis and geometry on groups[END_REF]. We limit ourselves to fix some notations, following [START_REF] Franchi | Regular submanifolds, graphs and area formula in Heisenberg groups[END_REF].

We denote by h the Lie algebra of the left invariant vector fields of H n . As customary, h is identified with the tangent space T e H n at the origin. The standard basis of h is given, for i = 1, . . . , n, by

X i := ∂ x i - 1 2 y i ∂ t , Y i := ∂ y i + 1 2 x i ∂ t , T := ∂ t .
The only non-trivial commutation relations are [X j , Y j ] = T , for j = 1, . . . , n.

The horizontal subspace h 1 is the subspace of h spanned by X 1 , .

h = h 1 ⊕ h 2 .
The stratification of the Lie algebra h induces a family of non-isotropic dilations δ λ , λ > 0 in H n . The homogeneous dimension of H n with respect to δ λ , λ > 0 is Q = 2n + 2. The vector space h can be endowed with an inner product, indicated by •, • , making X 1 , . . . , X n , Y 1 , . . . , Y n and T orthonormal.

Throughout this paper, to avoid cumbersome notations, we write also (8)

W i := X i , W i+n := Y i , W 2n+1 := T, for i = 1, • • • , n.
The dual space of h is denoted by 1 h. The basis of 1 h, dual to the basis {X 1 , . . . , Y n , T }, is the family of covectors {dx 1 , . . . , dx n , dy 1 , . . . , dy n , θ} where

θ := dt - 1 2 n j=1 (x j dy j -y j dx j ) is called the contact form in H n .
We indicate as •, • also the inner product in 1 h that makes (dx 1 , . . . , dy n , θ) an orthonormal basis. The same notation will be used to denote the scalar product in 1 h that makes (X 1 , . . . , X n , T ) an orthonormal basis.

Coherently with the previous notation (8), we set

ω i := dx i , ω i+n := dy i , ω 2n+1 := θ, for i = 1, • • • , n.
We put 0 h := 0 h = R and, for 1

≤ k ≤ 2n + 1, k h := span{W i 1 ∧ • • • ∧ W i k : 1 ≤ i 1 < • • • < i k ≤ 2n + 1} =: span Ψ k , k h := span{ω i 1 ∧ • • • ∧ ω i k : 1 ≤ i 1 < • • • < i k ≤ 2n + 1} =: span Ψ k .
The inner product •, • extends canonically to k h and to k h making both bases Ψ k and Ψ k orthonormal.

If 1 ≤ k ≤ 2n + 1, we denote by * the Hodge isomorphism * :

k h ←→ 2n+1-k
h associated with the scalar product •, • and the volume form

dV := ω 1 ∧ • • • ∧ ω 2n ∧ θ.
The same construction can be performed starting from the vector subspace h 1 ⊂ h, obtaining the horizontal k-vectors and horizontal k-covectors

k h 1 := span{W i 1 ∧ • • • ∧ W i k : 1 ≤ i 1 < • • • < i k ≤ 2n} k h 1 := span{ω i 1 ∧ • • • ∧ ω i k : 1 ≤ i 1 < • • • < i k ≤ 2n}.
It is well known that the Lie algebra h can be identified with the tangent space at the origin e = 0 of H n , and hence the horizontal layer h 1 can be identified with a subspace of T H n e that we can still denote by 1 h 1 . In addition, the symplectic 2-form

-dθ = n i=1 dx i ∧ dy i
induces on h 1 a symplectic structure. We point out that this symplectic structure is compatible with our fixed scalar product •, • and with the canonical almost complex structure on h 1 ≡ C n .

Horizontal k-vectors can be identified with skew-symmetric k-tensor in ⊗ k h 1 .

To fix our notations, we remind the following definition.

Definition 2.1. If V, W are finite dimensional linear vector spaces and S : V → W is a linear isomorphism, we define a map

⊗ r S : ⊗ r V → ⊗ r W
as the linear map defined by

(⊗ r S)(v 1 ⊗ • • • ⊗ v r ) = S(v 1 ) ⊗ • • • ⊗ S(v r ),
and a map

⊗ s S : ⊗ s W * → ⊗ s V *
as the linear map defined by

(⊗ s S)(α)|v 1 ⊗ • • • ⊗ v s = α|(⊗ s S)(v 1 ⊗ • • • ⊗ v s ) for any α ∈ ⊗ s W * and any s-tensor v 1 ⊗ • • • ⊗ v s ∈ ⊗ s V . Finally, we define (⊗ s r S) : (⊗ r V ) ⊗ (⊗ s V * ) → (⊗ r W ) ⊗ (⊗ s W * ) as follows: (⊗ s r S)(v ⊗ w) := (⊗ r S)(v) ⊗ (⊗ s S -1
)(w). Throughout this paper, we shall deal with (r, s)-tensors in ⊗ r h ⊗ ⊗ s h * , with r, s ∈ Z, r, s ≥ 0, that, in turn define a left-invariant fiber bundle over H n , that we still denote by ⊗ r h ⊗ ⊗ s h * as follows: first we identify

⊗ r h ⊗ ⊗ s h * with a subspace of ⊗ r T e H n ⊗ ⊗ s T * e H n that we denote by ⊗ r h e ⊗ ⊗ s h * e .
Then the fiber of

⊗ r h p ⊗ ⊗ s h * p of ⊗ r h ⊗ ⊗ s h * over p ∈ H n is ⊗ r h p ⊗ ⊗ s h * p := (⊗ h dτ p (e)) ⊗ r h e ⊗ (⊗ h dτ p -1 (p)) ⊗ s h * e .
The elements of the space of smooth sections of this bundle, i.e.

Γ H n , ⊗ r h ⊗ ⊗ s h * ,
are called (r, s)-tensors fields.

A special instance will be the horizontal tensors belonging to

⊗ r h 1 ⊗ ⊗ s h * 1 .
The horizontal (r, 0)-tensors fields will be called also horizontal r-vector fields. The skew-symmetric horizontal (0, s)-tensors fields are identified with the horizontal differential forms. Moreover, to avoid cumbersome notations, from now on, when dealing with a vector bundle N over H n , if there is no way to misunderstand we shall write also

Γ(N ) for Γ(H n , N ). Finally, a subbundle N of ⊗ r h ⊗ ⊗ s h * is said left invariant if (9) N p = (⊗ s r dτ p (e))N e .
It is customary (see e.g. [START_REF] Weil | Introduction à l'étude des variétés kählériennes[END_REF], Chapter I) to denote by

L : h h 1 → h+2 h 1
the Lefschetz operator defined by Lα := dθ ∧ α, and by Λ its dual operator with respect to •, • . If 2 ≤ h ≤ 2n, we denote by P h ⊂ h h 1 the space of primitive h-covectors defined by ( 10)

P 1 := 1 h 1 and P h := ker Λ ∩ h h 1 , 2 ≤ h ≤ 2n.
Following [START_REF] Michel Rumin | Formes différentielles sur les variétés de contact[END_REF], [START_REF] Michel Rumin | Sub-Riemannian limit of the differential form spectrum of contact manifolds[END_REF], for h = 0, 1, . . . , 2n + 1 we define a linear subspace E h 0 , of h h as follows:

Definition 2.2. We set

• if 1 ≤ h ≤ n then E h 0 = P h ; • if n < h ≤ 2n + 1 then E h 0 = {α = β ∧ θ, β ∈ h-1 h 1 , Lβ = 0}.
Remark 2.3. Definition 2.2 is not the original definition due to M. Rumin, but is indeed a characterization of Rumin's classes that is proved in [START_REF] Michel Rumin | Formes différentielles sur les variétés de contact[END_REF] (see also [START_REF] Pansu | Differential forms and connections adapted to a contact structure, after M. Rumin[END_REF], [START_REF] Baldi | Sharp a priori estimates for div-curl systems in Heisenberg groups[END_REF] and [START_REF] Baldi | A recursive basis for primitive forms in symplectic spaces and applications to Heisenberg groups[END_REF]).

By [START_REF] Bourbaki | Éléments de mathématique[END_REF], the spaces E * 0 define a family of left-invariant subbundles (still denoted by E h 0 , h = 0, . . . , 2n + 1). It turns out that we can identify E h 0 and (E h 0 ) e .

Basic facts on tensor analysis in Heisenberg groups

The following decomposition theorem holds.

Proposition 3.1. The space of 2-contravariant horizontal tensors ⊗ 2 h 1 can be written as a direct (orthogonal) sum

⊗ 2 h 1 = Sym (⊗ 2 h 1 ) ⊕ 2 h 1
of the space Sym (⊗ 2 h 1 ) of the symmetric 2-tensors and of the space 2 h 1 of the skew-symmetric 2-tensors.

An orthonormal basis of Sym (⊗ 2 h 1 ) with respect to the canonical scalar product in

⊗ 2 h 1 is 1 2 (W i ⊗ W j + W j ⊗ W i ) ; i ≤ j ,
whereas the canonical orthonormal basis Ψ 2 of 2 h 1 can be identified with

1 2 (W i ⊗ W j -W j ⊗ W i ) ; i < j . Definition 3.2 (See [2], Definition 1.7.16
). If φ : H n → H n is a diffeomorphism, then the push-forward of a tensor field t := v ⊗ α, with v ∈ ⊗ r h and α ∈ ⊗ s h * , then its push-forward φ * t at a point p ∈ H n is defined as

φ * t(p) := ⊗ s r dφ(p) ((v ⊗ α)(p))
, where p := φ -1 (p). Moreover, the pull-back φ * t of t is defined by

φ * t = (φ -1 ) * . A tensor field v ⊗ α ∈ Γ H n , ⊗ r h ⊗ ⊗ s h * is said left invariant if (τ q ) * v ⊗ α = v ⊗ α for all q ∈ H n . Lemma 3.3. Let v ⊗ ξ ∈ ⊗ r h e ⊗ ⊗ s h * e be given. If p ∈ H n , we set T p (v ⊗ ξ) := (τ p ) * (v ⊗ ξ) ∈ ⊗ r h p ⊗ ⊗ s h * p . Then the map p → T p (v ⊗ ξ) is left-invariant. Thus, if V ⊗ W is a linear subspace of ⊗ r h e ⊗ ⊗ s h * e , then {T p (v ⊗ ξ), v ⊗ ξ ∈ V ⊗ W, p ∈ H n } defines a left-invariant subbundle of ⊗ r h ⊗ ⊗ s h * . In addition, if {v i ⊗ ξ j , i = 1, . . . N, j = 1, . . . M } is a basis of V ⊗ W , then {T p (v ⊗ ξ), i = 1, . . . N, j = 1, . . . M } is a left-invariant basis of the fiber over p ∈ H n .
We remind the following well-known identity (see e.g. [START_REF] Abraham | Manifolds, tensor analysis, and applications[END_REF], Proposition 6.16): Remark 3.4. If W ∈ h 1 is identified with a first order differential operator and φ :

H n → H n is a diffeomorphism, then W (u • φ)(x) = [(φ * W )u] (φ(x)). Moreover, if W, Z ∈ h 1 , then φ * (W ⊗ Z) = φ * W ⊗ φ * Z. Set N h := dim E h 0 . Given a family of left-invariant bases {ξ h k , k = 1, . . . N h } of E h 0 , 1 ≤ h ≤ n as in Lemma 3.
3, the differential d c can be written "in coordinates" as follows.

Proposition 3.5. If 0 ≤ h ≤ 2n and α = k α k ξ h k ∈ Γ(E h 0 ), then d c α = I,k P I,k α k ξ h+1 I ,
where i) if h = n, then the P I,k 's are linear homogeneous polynomials in W 1 , . . . , W 2n ∈ h 1 (that are identified with homogeneous with first order left invariant horizontal differential operators), i.e.

P I,k = i F I,k,i W i ,
where the F I,k,i 's are real constants; ii) if h = n, then then the P I,k 's are linear homogeneous polynomials in W i ⊗ W j ∈ ⊗ 2 h 1 , i, j = 1, . . . , 2n (that are identified with homogeneous second order left invariant differential horizontal operators), i.e.

P I,k = i,j F I,k,i,j W i ⊗ W j ,
where the F I,k,i,j 's are real constants.

Definition 3.6. If 0 ≤ h < n we denote σ(d c ) the symbol of the intrinsic differential d c that is a smooth field of homomorphisms If h = n, d c is now a second order differential operator in the horizontal vector fields and then its symbol σ(d c ) can be identified with a section

σ(d c ) ∈ Γ(Hom (E h 0 , h 1 ⊗ E h+1 0 )) defined as follows: if p ∈ H n , ᾱ = k ᾱk ξ h k (p) ∈ (E h 0 ) p , then we can assume there exists a smooth differential form α = k α k ξ h k ∈ Γ(E h 0 ) such that ᾱ = α(p). Thus, if u ∈ E(H n ) satisfies u(p) = 0,
σ(d c ) ∈ Γ(Hom (E n 0 , ⊗ 2 h 1 ⊗ E n+1 0
))

as follows: if p ∈ H n , ᾱ = k ᾱk ξ n k (p) ∈ (E n 0 )
p , then we we can assume there exists a smooth differential form On the other hand, the canonical projection

α = k α k ξ n k ∈ E n 0 such that ᾱ = α(p). Thus, if u ∈ E(H n ) satisfies u(p) = 0 and W i u(p) = 0, i = 1, . . . ,
p : ⊗ 2 h 1 ⊗ E n+1 0 → ⊗ 2 h 1 2 h 1 ⊗ E n+1 0 given by p(W i ⊗ W j ⊗ ξ) := [W i ⊗ W j ] ⊗ ξ defines a new symbol (the symmetric part of the symbol) Σ(d c ) := p • σ(d c ) ∈ Hom (E n 0 , ⊗ 2 h 1 2 h 1 ⊗ E n+1 0 ).
Clearly, since we are dealing with 2-tensors,

[W i ⊗ W j ] can be represented by 1 2 (W i ⊗ W j + W j ⊗ W i ) . Thus Σ(d c )ᾱ = 1 2 I i,j,k ᾱk (F I,k,i,j + F I,k,j,i ) (W i ⊗ W j + W j ⊗ W i ) ⊗ ξ n+1 I =: I i,j,k ᾱk FI,k,i,j (W i ⊗ W j + W j ⊗ W i ) ⊗ ξ n+1 I . (11)
Remark 3.7. If 0 ≤ h < n, mimicking the usual definition of the principal symbol σ(P ) of a differential operator P (see e.g. [START_REF] Narasimhan | Analysis on real and complex manifolds[END_REF], Definition 3.3.13 or [START_REF] Palais | Seminar on the Atiyah-Singer index theorem[END_REF], IV.3), one can cook up a notion of horizontal principal symbol that takes into account Heisenberg homogeneity. For d c , this map would belong to Γ(Hom (

E h 0 ⊗ h * 1 , E h+1 0
)). However, our notation is not misleading, since, by [START_REF] Dieudonné | Éléments d'analyse. Tome III: Chapitres XVI et XVII. Cahiers Scientifiques[END_REF] (16.8.2.3) and (16.18.3.4),

Hom (E h 0 ⊗ h * 1 , E h+1 0 ) ∼ = Hom (E n 0 , Hom (h * 1 , E h+1 0 ) ) ∼ = Hom (E h 0 , h 1 ⊗ E h+1 0
).

An analogous comment applies when h = n. In this case, only the projection Σ(d c ) of the symbol onto symmetric tensors will be used.

Since h 1 , d c and E * 0 are invariant under left translations, then the symbol σ(d c ) is uniquely determined by its value at the point p = e. More precisely, we have: Proposition 3.8. If 1 ≤ h < n and p ∈ H n , then the following diagram is commutative:

(12) (E h 0 ) p σ(dc)(p) -----→ (h) p ⊗ (E h+1 0 ) p ⊗ h τp(e)   Tp   (E h 0 ) e σ(dc)(e) -----→ (h) e ⊗ (E h+1

0

) e An analogous comment applies when h = n.

Analytic facts

The proof of Theorem 1.3 consists in applying the following two results due to Chanillo & Van Schaftingen, after an algebraic reduction that will be performed in the next section. Definition 4.1. If f : H n → R, we denote by ∇ H f the horizontal vector field

∇ H f := 2n i=1 (W i f )W i , whose coordinates are (W 1 f, ..., W 2n f ). If Φ is a horizontal vector field, then ∇ H Φ is defined componentwise. Theorem 4.2 ([14], Theorem 1). Let Φ ∈ D(H n , h 1 ) be a smooth compactly supported horizontal vector field. Suppose G ∈ L 1 loc (H n , h 1 ) is H-divergence free, i.e. if G = i G i W i , then i W i G i = 0 in D ′ (H n ). Then G, Φ L 2 (H n ,h 1 ) ≤ C G L 1 (H n ,h 1 ) ∇ H Φ L Q (H n ,h 1 ) .
We notice that a stronger version of this result can be found in [START_REF] Yung | A subelliptic Bourgain-Brezis inequality[END_REF], Theorem 1.9.

As in the Euclidean case, estimates similar to Theorem 4.2 still hold when the condition on the divergence is replaced by a condition on higher-order derivatives [START_REF] Van Schaftingen | Estimates for L 1 vector fields under higher-order differential conditions[END_REF]. Similar ideas have been applied in nilpotent homogeneous groups by S. Chanillo and J. Van Schaftingen as follows.

Let k ≥ 1 be fixed, and let G ∈ L 1 (H n , ⊗ k h 1 ) belong to the space of horizontal k-tensors. We can write

G = i 1 ,...,i k G i 1 ,...,i k W i 1 ⊗ • • • ⊗ W i k .
We remind that G can be identified with the differential operator

u → Gu := i 1 ,...,i k G i 1 ,...,i k W i 1 • • • W i k u.
Denoting by D(H n , Sym(⊗ k h 1 )) the subspace of compactly supported smooth symmetric horizontal k-tensors, we have:

Theorem 4.3 ([14], Theorem 5). Let k ≥ 1 and G ∈ L 1 (H n , ⊗ k h 1 ), Φ ∈ D(H n , Sym(⊗ k h 1 )). Suppose that i 1 ,...,i k W i k • • • W i 1 G i 1 ,...,i k = 0 in D ′ (H n ). Then H n Φ, G dp ≤ C G L 1 (H n ,⊗ k h 1 ) ∇ H Φ L Q (H n ,⊗ k h 1 ) .

Main algebraic step

As in [START_REF] Baldi | Sharp a priori estimates for div-curl systems in Heisenberg groups[END_REF], our proof of Theorem 1.3 relies on the fact (precisely stated in Theorem 5.1 below) that the components with respect to a given basis of closed forms in E h 0 can be viewed as the components of a horizontal vector field with vanishing horizontal divergence if h = n, n + 1 or vanishing "generalized horizontal divergence" if h = n, n + 1. More precisely, we have:

Theorem 5.1. Let α = J α J ξ h J ∈ Γ(E h 0 ), 1 ≤ h ≤ 2n, be such that d c α = 0.

Then

• if h = n then each component α J of α, J = 1, . . . , dim E h 0 , can be written as

α J = dim E h+1 0 I=1 2n i=1 b J i,I G I,i ,
where the b J i,I 's are real constants and for any I = 1, . . . , dim E h+1 0 the G I,i 's are the components of a horizontal vector field

G I = i G I,i W i with i W i G I,i = 0, I = 1, . . . , dim E h+1 0 .
Moreover there exist a geometric constant C > 0 such that for I = 1, . . . , dim E h+1

0 and 1 ≤ p ≤ ∞ (13) G I L p (H n , 1 h 1 ) ≤ C α L p (H n ,E h 0 ) . • If h = n, then each component α J of α, J = 1, . . . , dim E n
0 , can be written as

α J = dim E n+1 0 I=1 i,j b J i,j,I (G Sym I ) i,j .
Here the b J i,j,I 's are real constants and for any I = 1, . . . , dim E n+1 0 the (G Sym I ) i,j 's are the components of the symmetric part (see Proposition 3.1) of the 2-tensor

G I = i G I,i,j W i ⊗ W j that satisfies i,j W i W j G I,i,j = 0, I = 1, . . . , dim E n+1 0 .
Moreover there exist a geometric constant C > 0 such that for I = 1, . . . , dim E n+1

0 and 1 ≤ p ≤ ∞ (14) G I L p (H n ,⊗ 2 h 1 ) ≤ C α L p (H n ,E n 0 ) .
The proof of this theorem requires several preliminary steps. First of all, we want to prove that the exterior differential d c is invariant (i.e. "natural") under the action of a class of intrinsic transformation of H n . Theorem 5.2. If A belongs to the symplectic group Sp 2n (R), we associate with A the real (2n + 1) × (2n + 1) matrix [START_REF] Michel Rumin | An introduction to spectral and differential geometry in Carnot-Carathéodory spaces[END_REF] or [START_REF] Franchi | Wave and Maxwell's equations in Carnot groups[END_REF].

(15) f A : h → h, f A = A 2n×2n 0 2n×1 0 1×2n 1 . Then i) f A (h 1 ) = h 1 ; ii) f A induces a homogeneous group isomorphism exp •f A • exp -1 still denoted by f A such that f A : H n → H n ; iii) f * A : Γ(E * 0 ) → Γ(E * 0 ); iv) for any h-form α ∈ Γ(E h 0 ) d c (f * A α) = f * A (d c α); Proof. See
The next step consists in proving that the symbols σ(d c ) and Σ(d c ) are injective.

First of all, we remind that, by [START_REF] Weil | Introduction à l'étude des variétés kählériennes[END_REF], Chapter I, Theorem 3 and Corollary at p. 28), the following proposition holds: Proposition 5.3. Let P h be space of primitive forms defined in [START_REF] Bourgain | On the equation div Y = f and application to control of phases[END_REF]. Then i) if 1 ≤ h ≤ 2n, then the following orthogonal decomposition holds:

h h 1 = i≥(n-h) + L i (P h-2i ); ii) P h = {0} if h > n; iii) the map L n-h : h h 1 → 2n-h h 1 is a linear isomorphism; iv) if h ≤ n, then P h = ker L n-h+1 ; v) the map L n-h : P h → 2n-h h 1 ∩ ker L is a linear isomorphism; vi) a symplectic map A ∈ Sp 2n (R) commutes with L, i.e. if 1 ≤ h ≤ 2n -2, then [⊗ h A, L] = 0.
The injectivity of the symbols will follow from the following result.

Proposition 5.4. The symplectic group Sp 2n (R) acts irreducibly on P h for h = 1, . . . , n and on ker L for h = n + 1, . . . , 2n -1.

Proof. If 1 ≤ h ≤ n, then the statement is proved in [START_REF] Bourbaki | Éléments de mathématique[END_REF], p. 203. Suppose now h > n. If A ∈ Sp 2n (R) and α ∈ ker L, then by Proposition 5.3, vi), ⊗ h AL ∈ ker L, so that A acts on ker A. On the other hand, if V ⊂ ker L ∩ h h 1 is invariant under the action of Sp 2n (R), then, by Proposition 5.3, v) and vi), (L h-r ) -1 V is also invariant under the action of Sp 2n (R), and the assertion follows by the first part of the proof.

Using Definition 2.2, we have:

Corollary 5.5. If A ∈ Sp 2n (R) and f A is defined as in [START_REF] Dieudonné | Éléments d'analyse. Tome III: Chapitres XVI et XVII. Cahiers Scientifiques[END_REF], then f A acts irreducibly on E h 0 for h = 1, . . . , 2n. Since d c is equivariant under all smooth contact transformations, it is in particular Sp 2n (R)-equivariant. It follows that the kernels ker σ(d c ) and ker Σ(d c ) are invariant subspaces for the action of Sp 2n (R), so that the injectivity will follow. In fact, we have:

Proposition 5.6. Keeping in mind Definition 3.6, if 1 ≤ h ≤ 2n , h = n, then ker σ(d c )(e) is invariant under the action of Sp 2n (R), i.e. if A ∈ Sp 2n (R), then we have: if ᾱ ∈ E h 0 and σ(d c )(e)(ᾱ) = 0 then σ(d c )(e)((⊗ h A) ᾱ) = 0. If h = n, then ker Σ(d c )(e) is invariant under the action of Sp 2n (R), i.e., A ∈ Sp 2n R), then we have: if ᾱ ∈ E h 0 and Σ(d c )(e)(ᾱ) = 0 then Σ(d c )(e)((⊗ h A) ᾱ) = 0. Proof. Suppose first h < n and let α ∈ Γ(E h
0 ) be a differential form such that α(e) = ᾱ. Let f A be the matrix associated with A as in [START_REF] Dieudonné | Éléments d'analyse. Tome III: Chapitres XVI et XVII. Cahiers Scientifiques[END_REF]. We notice first that ( 16)

f * A (α)(e) = (⊗ h A)ᾱ, since f A (e) = e. Let now u be a smooth function such that u(e) = 0. We set v := u • f -1
A . Keeping again in mind that ( 16), we have also that v(e) = 0. By Theorem 5.2 and Remark 3.4, we have:

d c (uf * A α)(e) = d c (f * A (vα))(e) = f * A (d c (vα))(e) = (⊗ h+1 f A )(d c (vα)(e)) = (⊗ h+1 A)(d c (vα)(e)) = I,k (A -1 P I,k u)(e)ᾱ k (⊗ h+1 A)(ξ h+1 I (e)). (17)
Hence, by ( 16)

σ(d c )(e)((⊗ h A)ᾱ) = σ(d c )(e)((f * A α)(e)) = I A -1 k ᾱk P I,k (e) ⊗ (⊗ h+1 A)(ξ h+1 I (e)). (18)
On the other hand, by assumption,

σ(d c )(e)((⊗ h A) ᾱ) = 0.
Consider now the case h = n, when d c is a second order operator in the horizontal derivatives. We stress that E n+1 0 contains only vertical forms, i.e. forms that are multiple of the contact form θ. Suppose Σ(d c )(e)ᾱ = 0. Then, by [START_REF] Bourgain | New estimates for the Laplacian, the div-curl, and related Hodge systems[END_REF], [START_REF] Franchi | Regular submanifolds, graphs and area formula in Heisenberg groups[END_REF]

i,j,k ᾱk FI,k,i,j (W i ⊗ W j + W j ⊗ W i ) = 0, I = 1, . . . , dim E h+1 0 .
We take now u ∈ E(H n ) satisfies u(e) = 0 and W i u(e) = 0, i = 1, . . . , 2n. As above, we set v := u • f -1

A . Keeping in mind that f A (e) = e, we have also that v(e) = 0 and (W i v)(e) = 0, i = 1, . . . , 2n. Then equations [START_REF] Folland | Hardy spaces on homogeneous groups[END_REF] become

d c (uf * A α)(e) = d c (f * A (vα))(e) = f * A (d c (vα))(e) = (⊗ h+1 f A )(d c (vα)(e)) = I,k i,j F I,k,i,j (A -1 W i )(A -1 W j )u(e)ᾱ k (⊗ n+1 f A )(ξ n+1 I (e)).
Therefore, keeping in mind Remark 3.4 and ( 16), we have:

Σ(d c )(e)((⊗ h A)ᾱ) = Σ(d c )(e)((f * A α)(e)) = I i,j,k ᾱk FI,k,i,j (A -1 W i )(e) ⊗ (A -1 W j )(e) +(A -1 W j )(e) ⊗ (A -1 W i )(e) ⊗ (⊗ n+1 f A )(ξ n+1 I (e)) = I A -1 i,j,k ᾱk FI,k,i,j W i (e) ⊗ W j (e) + W j (e) ⊗ W i (e) ⊗ (⊗ n+1 f A )(ξ n+1
I (e)) = 0, by [START_REF] Franchi | Regular submanifolds, graphs and area formula in Heisenberg groups[END_REF].

Finally, the proof for h > n can be carried out precisely as in the case h < n, with only minor changes. In particular, ( 16) must be replaced taking into account that a form α ∈ E h 0 has also a vertical component of the form β ∧ θ and that f * A (β ∧ θ)(e) = (⊗ h A)β(e) ∧ θ. This completes the proof of the proposition. ). We deal first with the case h = n and we set

B h ∈ Hom (h 1 ⊗ E h+1 0 , E h 0 ), B n ∈ Hom (Sym (⊗ 2 h 1 ) ⊗ E n+1 0 , E n 0 ) such that (20) α = B h (σ(d c )α) for all α ∈ Γ(E h 0 ), h = n,
B h (W i ⊗ ξ h+1 I ) := J b J i,I ξ h J .
Then, if we write α = J α J ξ h J ∈ Γ(E h 0 ) and

P I,k = i F I,k,i W i ,
(where the F I,k,i 's are real constants) identity [START_REF] Franchi | Wave and Maxwell's equations in Carnot groups[END_REF] becomes

α = B h ( I,k α k P I,k ⊗ ξ h+1 I ) = J   I,k,i b J i,I F I,k,i α k   ξ h J , (22)
so that

α J = I,k,i b J i,I F I,k,i α k , for J = 1, . . . , dim E h 0 . ( 23
G I = i G I,i W i := i k F I,k,i α k W i then i W i G I,i = 0, I = 1, . . . , dim E h+1 0 .
Thus [START_REF] Maheux | Analyse sur les boules d'un opérateur souselliptique[END_REF] reads as

α J = I,i b J i,I G I,i ,
achieving the proof in the case h < n.

We deal now with the case h = n and we set

B n ((W i ⊗ W j + W j ⊗ W i ) ⊗ ξ n+1 I ) := J b J i,j,I ξ n J .
Thus, if we write α = J α J ξ h J ∈ Γ(E n 0 ) by [START_REF] Bourgain | New estimates for the Laplacian, the div-curl, and related Hodge systems[END_REF], identity (21) becomes α = B n (

I,k,i,j α k FI,k,i,j (W i ⊗ W j + W j ⊗ W i ) ⊗ ξ n+1 I ) = J   I,k,i,j b J i,j,I FI,k,i,j α k   ξ n J , (24)
so that

α J = I,k,i,j b J i,j,I FI,k,i,j α k , for J = 1, . . . , dim E n 0 . ( 25
)
Denote by G I the horizontal tensor field

G I = i,j G I,i,j W i ⊗ W j := i,j k F I,k,i,j α k W i ⊗ W j . (26)
By Proposition 3.1 we can write

G I = G Sym I + G Skew I , where
G Sym I = i,j (G Sym I ) i,j (W i ⊗ W j + W j ⊗ W i ) := i,j k FI,k,i,j α k (W i ⊗ W j + W j ⊗ W i ).
We suppose now that d c α = 0, that, by [START_REF] Pansu | Differential forms and connections adapted to a contact structure, after M. Rumin[END_REF], in coordinates is i,j

W i W j G I,i,j = 0, I = 1, . . . , dim E n+1 0 .
Thus (25) reads as

α J = I,i,j b J i,j,I (G Sym I ) i,j for J = 1, . . . , dim E n 0 ,
achieving the proof in the case h = n.

Proof of Theorem 1.3

The proof follows the lines of [START_REF] Baldi | Sharp a priori estimates for div-curl systems in Heisenberg groups[END_REF]. Let us reming few facts of harmonic analysis in homogeneous groups.

A differential operator P : Γ(E h 0 ) → Γ(E k 0 ) is said left-invariant if for all q ∈ H n P (τ q ) * α = (τ q ) * (P α) for all α ∈ Γ(E h 0 ). If f is a real function defined in H n , we denote by v f the function defined by v f (p) := f (p -1 ), and, if T ∈ D ′ (H n ), then v T is the distribution defined by v T |φ := T | v φ for any test function φ.

Following e.g. [START_REF] Folland | Hardy spaces on homogeneous groups[END_REF], we can define a group convolution in H n : if, for instance, f ∈ D(H n ) and g ∈ L 1 loc (H n ), we set [START_REF] Michel Rumin | Formes différentielles sur les variétés de contact[END_REF] f * g(p) := f (q)g(q -1 • p) dq for q ∈ H n .

We remind that, if (say) g is a smooth function and P is a left invariant differential operator, then

P (f * g) = f * P g.
We remind also that the convolution is again well defined when f, g ∈ D ′ (H n ), provided at least one of them has compact support. In this case the following identities hold As in [START_REF] Folland | Hardy spaces on homogeneous groups[END_REF], we also adopt the following multi-index notation for higherorder derivatives. If I = (i 1 , . . . , i 2n+1 ) is a multi-index, we set

W I = W i 1 1 • • • W i 2n 2n T i 2n+1
. By the Poincaré-Birkhoff-Witt theorem, the differential operators W I form a basis for the algebra of invariant differential operators in H n . Furthermore, we set |I| := i 1 + • • • + i 2n + i 2n+1 the order of the differential operator W I , and d(I)

:= i 1 + • • • + i 2n + 2i 2n+1 its degree of homogeneity with respect to group dilations. Suppose now f ∈ E ′ (H n ) and g ∈ D ′ (H n ). Then, if ψ ∈ D(H n ), we have (W I f ) * g|ψ = W I f |ψ * v g = (-1) |I| f |ψ * (W I v g) = (-1) |I| f * v W I v g|ψ . (29)
Following [START_REF] Folland | Subelliptic estimates and function spaces on nilpotent Lie groups[END_REF], we remind now the notion of kernel of order a, as well as some basic properties. Definition 6.1. A kernel of order a is a homogeneous distribution of degree a -Q (with respect to group dilations), that is smooth outside of the origin. Proposition 6.2. Let K ∈ D ′ (H n ) be a kernel of order a.

i) v K is again a kernel of order a; ii) W ℓ K is a a kernel of order a -1 for any horizontal derivative W ℓ K, ℓ = 1, . . . , 2n; iii) If a > 0, then K ∈ L 1 loc (H n ). Definition 6.3. In H n , following [START_REF] Michel Rumin | Formes différentielles sur les variétés de contact[END_REF], we define the operator ∆ H,h on E h 0 by setting

∆ H,h =    d c δ c + δ c d c if h = n, n + 1; (d c δ c ) 2 + δ c d c if h = n; d c δ c + (δ c d c ) 2 if h = n + 1.
Notice that -∆ H,0 = 2n j=1 (W 2 j ) is the usual sub-Laplacian of H n . For sake of simplicity, once a basis of E h 0 is fixed, the operator ∆ H,h can be identified with a matrix-valued map, still denoted by ∆ H,h

∆ H,h = (∆ ij H,h ) i,j=1,...,N h : D ′ (H n , R N h ) → D ′ (H n , R N h ) (30)
. This identification makes possible to avoid the notion of currents: we refer to [START_REF] Baldi | Compensated compactness for differential forms in Carnot groups and applications[END_REF] for this more elegant presentation.

Combining [START_REF] Michel Rumin | Formes différentielles sur les variétés de contact[END_REF], Section 3, and [START_REF] Baldi | Hypoellipticity, fundamental solution and Liouville type theorem for matrix-valued differential operators in Carnot groups[END_REF], Theorems 3.1 and 4.1, we obtain the following result. Theorem 6.4. If 0 ≤ h ≤ 2n + 1, then the differential operator ∆ H,h is hypoelliptic of order a, where a = 2 if h = n, n + 1 and a = 4 if h = n, n + 1 with respect to group dilations. Then i) for j = 1, . . . , N h there exists

(31) K j = K 1j , . . . , K N h j , j = 1, . . . N h with K ij ∈ D ′ (H n ) ∩ E(H n \ {0}), i, j = 1, . . . , N ; ii) if a < Q, then the K ij 's are kernels of type a for i, j = 1, . . . , N h If a = Q, then the K ij 's satisfy the logarithmic estimate |K ij (p)| ≤ C(1 + | ln ρ(p)|) and hence belong to L 1 loc (H n ). Moreover, their hor- izontal derivatives W ℓ K ij , ℓ = 1, . . . , 2n, are kernels of type Q -1; iii) when α ∈ D(H n , R N h ), if we set (32) Kα := j α j * K 1j , . . . , j α j * K N h j , then ∆ H,h Kα = α. Moreover, if a < Q, also K∆ H,h α = α. iv) if a = Q, then for any α ∈ D(H n , R N h ) there exists β α := (β 1 , . . . , β N h ) ∈ R N h , such that K∆ H,h α -α = β α .
Remark 6.5. Coherently with formula (30), the operator K can be identified with an operator (still denoted by K) acting on smooth compactly supported differential forms in D(H n , E h 0 ). Proof of Theorem 1.3. The case h = 0 is well known ( [START_REF] Franchi | Sobolev and isoperimetric inequalities for degenerate metrics[END_REF], [START_REF] Capogna | The geometric Sobolev embedding for vector fields and the isoperimetric inequality[END_REF], [START_REF] Maheux | Analyse sur les boules d'un opérateur souselliptique[END_REF]). Suppose now h = n -1. In order to estimate the terms of (34), we have to estimate terms of the form (37)

G I,i , (d c Kφ) ℓ L 2 (H n ) = G I , Φ L 2 (H n ,h 1 )
, where Φ = (d c Kφ) ℓ W i . We can apply Theorem 4.2. Keeping in mind [START_REF] Capogna | The geometric Sobolev embedding for vector fields and the isoperimetric inequality[END_REF], we obtain

(38) f ℓ , (d c Kφ) ℓ L 2 (H n ) ≤ C f L 1 (H n ,E h+1 0 ) ∇ H d c Kφ L Q (H n ,E h+1 0
) . On the other hand, ∇ H d c Kφ can be expressed as a sum of terms with components of the form φ j * W I Kij with d(I) = 2.

By Theorem 6.4, iv) and Proposition 6.2, ii) W I Kij are kernels of type 0, so that, by [START_REF] Folland | Subelliptic estimates and function spaces on nilpotent Lie groups[END_REF], Proposition 1.9 we have

(39) | f ℓ , (d c Kφ) ℓ L 2 (H n ) | ≤ C f L 1 (H n ,E h+1 0 ) φ L Q (H n ,E h 0 )
. The same argument can be carried out for all the components of f , yielding

(40) | f, d c Kφ L 2 (H n ,E h+1 0 ) | ≤ C f L 1 (H n ,E h+1 0 ) φ L Q (H n ,E h 0 )
. Suppose now h = n -1 then we have to estimate terms of the form

(41) (G Sym I ) i,j , (d c Kφ) ℓ L 2 (H n ) = G I , Φ L 2 (H n ,⊗ 2 h 1 ) , Φ = (d c Kφ) ℓ W i ⊗ W j + W j ⊗ W i ) ∈ Γ(H n , Sym (⊗ 2 h 1 )
). We can apply Theorem 4.3 and we obtain

(42) | f, d c Kφ L 2 (H n ,E n 0 ) | ≤ C f L 1 (H n ,E n 0 ) φ L Q (H n ,E n-1 0
) . This achieve the estimate of the first term of (33) for all 1 < h < 2n, h = n, n + 1.

  then we have d c (uα)(p) = I,k (P I,k u)(p)α k (p)ξ h+1 I (p). Hence we set σ(d c )(p)ᾱ := I,k ᾱj P I,k (p) ⊗ ξ h+1 I (p).

  2n, then we have d c (uα)(p) = I,k (P I,k u)(p)α k (p) ξ n+1 I (p). Hence we set σ(d c )(p)ᾱ := I,k ᾱk P I,k (p) ⊗ ξ n+1 I (p).

,

  ,k (e) = 0 I = 1, . . . , dim E h+1 0 since the ξ h+1 I 's are linearly independent. Thus eventually from[START_REF] Franchi | Sobolev and isoperimetric inequalities for degenerate metrics[END_REF]

Proof of Theorem 5 . 1 .

 51 First of all, we notice that, by Corollary 5.5 and Proposition 5.6, both ker σ(d c )(e) (if h = n) and ker Σ(d c )(e) (if h = n) are the null space {0}, and hence both σ(d c )(e) and Σ(d c )(e) have a left inverseB h ∈ Hom (h 1 ⊗ (E h+1 0 ) e , (E h 0 ) e ) if h = n. and B n ∈ Hom (Sym (⊗ 2 h 1 ) ⊗ (E n+1 0 ) e , (E n 0 ) e ) if h = n.By the commutativity of the diagram[START_REF] Bourgain | New estimates for elliptic equations and Hodge type systems[END_REF], B h and B n can be identified with constant coefficient maps

and ( 21 )

 21 α = B n (Σ(d c )α) for all α ∈ Γ(E n 0

)F

  Suppose now d c α = 0. Then, writing the identity in coordinates, if I = 1, . . . , dim E h+1 0 I,k,i α k = 0, so that, if we denote by G I the horizontal vector field

( 28 )

 28 f * g|φ = g| v f * φ and f * g|φ = f |φ * v g for any test function φ.

Case 1 < 33 ),.By Theorem 5 . 1 ,.

 13351 h < 2n and h = n, n + 1. If u, φ ∈ D(H n , E h 0 ), we can write u, φ L 2 (H n ,E h 0 ) = u, ∆ H,h Kφ L 2 (H n ,E h 0 ) = u, (δ c d c + d c δ c )Kφ L 2 (H n ,E h 0 ) .(Consider now the first term in the previous sum,u, δ c d c Kφ L 2 (H n ,E h 0 ) = d c u, d c Kφ L 2 (H n ,E h+10) . If we write f := d c u, then d c f = 0. From now on, without loss of generality, for 1 ≤ h ≤ 2n + 1 we take an orthonormal basis of E h 0 , still denoted by {ξ hℓ ; ℓ = 1, . . . , dim E h 0 }. Thus, since f, d c Kφ ∈ E h+1 0 Kφ) ℓ ξ h+1 ℓ, and hence we can reduce ourselves to estimate[START_REF] Weil | Introduction à l'étude des variétés kählériennes[END_REF] f ℓ ,(d c Kφ) ℓ L 2 (H n ) for ℓ = 1, . . . , dim E h+1 0 if h = n -1 , each component f ℓ of f can be written as G I,i ,where the b ℓ i,I 's are real constants and for any I = 1, . . . , dim E h+2 0 the G I,i 's are the components of an horizontal vector fieldG I = i G I,i W i with (35) i W i G I,i = 0, I = 1, . . . , dim E h+20 On the other hand, for h = n-1, each component f ℓ of f , ℓ = 1, . . . , dim E n Here the b ℓ i,j,I 's are real constants and for any I = 1, . . . , dim E n+1 0 the (G Sym I ) i,j 's are the components of the symmetric part (see Proposition 3.1) of the 2-tensor G I = i G I,i,j W i ⊗ W j that satisfies (36) i,j W i W j G I,i,j = 0, I = 1, . . . , dim E n+1 0 .

1 0)

 1 (43) | u, δ c d c Kφ L 2 (H n ,E h 0 ) | ≤ C f L 1 (H n ,E h+1 0 ) φ L Q (H n ,E h 0 ) . Consider now the second term in (33) u, d c δ c Kφ L 2 (H n ,E h 0 ) = δ c u, δ c Kφ L 2 (H n ,E h-1 0 ) = g, δ c Kφ L 2 (H n ,E h-= * g, * δ c Kφ L 2 (H n ,E 2n+2-h 0 ) ,

  . . , X n and Y 1 , . . . , Y n . Coherently, from now on, we refer to X 1 , . . . , X n , Y 1 , . . . , Y n (identified with first order differential operators) as to the horizontal derivatives. Denoting by h 2 the linear span of T , the 2-step stratification of h is expressed by

A.B. and B.F. are supported by University of Bologna, funds for selected research topics, by GNAMPA of INdAM, Italy and by MAnET Marie Curie Initial Training Network. P.P. is supported by Agence Nationale de la Recherche, ANR-10-BLAN 0116.

where * denotes the Hodge duality. We notice now that from δ c u = g, by Hodge duality we have d c * u = * g. Hence d c ( * g) = 0, and thus, arguing precisely as above, we get

Combining ( 44) with (43), we get eventually

, and hence

) . This completes the proof of statement iii) of the theorem. Case h = 1, 2n. By Hodge duality we may restrict ourselves to the case h = 1. Again we write

In order to estimate the first term f, d c Kφ L 2 (H n ,E 1 0 ) , we repeat verbatim the arguments above for the corresponding term in the case h = n, n + 1. As for the second term, by Theorem 6.4, formula [START_REF] Van Schaftingen | Estimates for L 1 vector fields under higher-order differential conditions[END_REF], and keeping in mind that δ c is an operator of order 1 in the horizontal derivatives when acting on E 1 0 the quantity δ c Kφ can be written as a sum of terms such as φ j * W ℓ Kij , with ℓ = 1, . . . , 2n.

On the other hand,

Notice the W ℓ Kij 's and hence the v (W ℓ Kij )'s are kernels of type 1. Thus, by Theorem 6.10 in [START_REF] Folland | Hardy spaces on homogeneous groups[END_REF],

. Combining this estimate with the one in (43), we get eventually

This completes the proof of statement ii) of the theorem. Case h = n, n + 1. By Hodge duality we may restrict ourselves to the case h = n.

If u, φ ∈ E n 0 are smooth compactly supported forms, then we can write

Consider now the term

Let us write f := d c u. Again, d c f = 0, and hence, as above, if

, and thus we can reduce ourselves to estimate (47)

By Theorem 5.1, each component f ℓ of f can be written as

where the b ℓ i,I 's are real constants and for any I = 1, . . . , dim E n+2 0 the G I,i 's are the components of an horizontal vector field

As in the previous cases, in order to estimate the terms of (47), we have to deal terms of the form

,

We can apply Theorem 4.2. Again keeping in mind [START_REF] Capogna | The geometric Sobolev embedding for vector fields and the isoperimetric inequality[END_REF], we obtain

) . On the other hand, ∇ H d c Kφ can be expressed as a sum of terms with components of the form

since the differential d c on n-forms has order 2 in the horizontal derivatives. By Theorem 6.4, iv) and Proposition 6.2, ii) W I Kij are kernels of type 1, so that, by [START_REF] Folland | Subelliptic estimates and function spaces on nilpotent Lie groups[END_REF], Proposition 1.11 we have

. The same argument can be carried out for all the components of f , yielding

. Consider now the second term in (46). We have

0 , and then we can repeat the arguments leading to (42) for f in the case h = n -1, obtaining

As above, ∇ H d c δ c Kφ can be expressed as a sum of terms with components of the form

0 is an operator of order 1 in the horizontal derivatives, as well as d c : E n-1 0 → E n 0 . By Theorem 6.4, iv) and Proposition 6.2, ii) W I Kij are kernels of type 1, so that, by [START_REF] Folland | Subelliptic estimates and function spaces on nilpotent Lie groups[END_REF], Proposition 1.11 we have

. Combining this estimate with the one in (52), we get eventually

, and hence

To achieve the proof of statement iv) of the theorem we have to consider separately the cases f = 0 and g = 0. Suppose h = n + 1 (i.e. g = 0). The proof for h = n (i.e. f = 0) follows by Hodge duality. In the case h = n + 1 identity (46) read as

Since d c f = 0, by Theorem 5.1 we can apply Theorem 4.2, and we get

by [START_REF] Folland | Subelliptic estimates and function spaces on nilpotent Lie groups[END_REF], Proposition 1.9, since ∇ H d c δ c d c K is a kernel of type 0. Then we can conclude by duality as of the proof of case iii), achieving the proof of statement iv) of the theorem that now is completely proved.

Final remarks

The estimates in Theorem 1.3 for n-forms and (n + 1)-forms can be reformulated in the spirit of the estimates proved in [START_REF] Baldi | Gagliardo-Nirenberg inequalities for horizontal vector fields in the Engel group and in the 7-dimensional quaternionic Heisenberg group[END_REF]. To state our result, we must recall preliminarily few definitions of the function spaces we need for our results.

If p, q ∈ [1, ∞], we define the space

We have:

Again if p, q ∈ [1, ∞], we can endow the vector space L p (H n ) + L q (H n ) with the norm

Analogous spaces of forms can be defined in the usual way.

The following characterization of (L p,q (H n )) * can be proved by standard arguments of functional analysis. Proposition 7.1. If p, q ∈ (1, ∞) and p ′ , q ′ are their conjugate exponents, then i)

for some C > 0 and for all φ

Using the function spaces defined above, we can reformulate Theorem 1.3 in the critical cases h = n as follows. Since a similar formulations lacks in [START_REF] Baldi | Sharp a priori estimates for div-curl systems in Heisenberg groups[END_REF], we state and prove the theorem also of n = 1. Theorem 7.2. Denote by (E * 0 , d c ) the Rumin's complex in H n , n ≥ 1.

Consider the system

) and for any u ∈ D(H 1 , E 2 0 ), respectively. Proof. By Hodge duality we may restrict ourselves to the case h = n.

If u, φ ∈ E n 0 are smooth compactly supported forms, then we can write

The estimate of the first term of the last line of ( 54) is already given in (52) and reads

Consider now the second term in the last line of (54). If n ≥ 2, we have

We notice now that * g is a d c -closed form in E n+2 0 . Then we can repeat the arguments of the proof of Theorem 1.3 and we get * g, * δ

As in the proof of Theorem 1.3, ∇ H δ c d c δ c Kφ can be expressed as a sum of terms with components of the form

0 is an operator of order 1 in the horizontal derivatives, as well as d c : E n-1 0 → E n 0 . By Theorem 6.4, iv) and Proposition 6.2, ii) W I Kij are kernels of type 0, so that, keeping in mind (57), by Proposition 1.9 we have g

If n = 1, we write instead

, where K is a kernel of type 1. By Hölder inequality and [START_REF] Folland | Hardy spaces on homogeneous groups[END_REF], Theorem 6.10,

To conclude the proof, if n > 1, combining (59) with ( 56) or (60), we get eventually

If n = 1 the same estimate holds with g L 1 replaced with g H 1 .

Indeed, if n > 1 and we replace (56) and ( 61) in (54), we obtain by duality (Proposition 7.1 -iv) )

) . Again, g L 1 must be replaced by g H 1 if n = 1. This completes the proof of the theorem.


Hélène Barral
email: helene.barral@lgge.obs.ujf-grenoble.fr

E Vignon

E Bazile

O Traullé

H Gallée

C Genthon

C Brun

F Couvreux

P Le Moigne

SUMMER DIURNAL CYCLE AT DOME C

ON THE ANTARTIC PLATEAU

INTRODUCTION

Antarctic boundary layers experience persistent and strong temperature inversions [START_REF] Phillpot | The surface temperature inversion over the Antarctic continent[END_REF][START_REF] King | Antarctic meteorology and climatology[END_REF]. A better understanding of the physical processes and their coupling involved in stable boundary layers is necessary to model the Antarctic meteorology and for future predictions of the regional climate and sea level. The study of the Antarctic atmosphere may also help to investigate the stable boundary layers. Indeed, the ice-sheet offers us "laboratory cases" with endless snow covered, relatively homogeneous, flat or sloped areas with persistent and strong stable stratifications resulting in low level jets. Dome C (75°06 S, 123°E) on the East Antarctic Plateau is one of them.

Dome C is a place of particular interest partly because of the numerous observations performed there, all year long. The nearby permanent scientific station Concordia, jointly operated by the French and Italian polar institutes (IPEV and PNRA) allows routinely operating permanent observations despite the remoteness of the site, the very low temperatures and the frost deposition.

Figure 1 shows the topography of the dome culminating at 3233 m above sea level ; the local slope is no greater than 1 ‰ so that the place does not feed the famous strong and persistent katabatic winds of Antarctica [START_REF] Parish | Reexamination of the near-surface airflow over the Antarctic continent and implications on atmospheric circulations at high southern latitudes[END_REF]. No clear annual cycle in wind speed has been noticed, the annual speed averages 4.5 ms -1 at 10 m according to In summer, this high latitude area is permanently insolated, still the boundary layer experiences a clear diurnal cycle in temperature and wind speed. This summertime boundary layer has been investigated through several measurement methods : in-situ meteorological sensors [START_REF] Gallée | Meteorological atmospheric boundary layer measurements and ECMWF, analyses during summer at Dome C, Antarctica[END_REF][START_REF] Genthon | Two years of atmospheric boundary layer observations on a 45-m tower at Dome C on the Antarctic plateau[END_REF], groundbased sodar [START_REF] Argentini | Summer boundary-layer height at the plateau site of Dome C, Antarctica[END_REF][START_REF] Argentini | The surface layer observed by a high-resolution sodar at Dome C, Antarctica[END_REF], and ground-based microwave radiometer [START_REF] Ricaud | Summer to winter diurnal variabilities of temperature and water vapour in the lowermost troposphere as observed by HAMSTRAD over Dome C, Antarctica[END_REF]. At "night" that is when the sun zenith angle is the largest, the albedo of the surface is also the highest (around 0.85), the net long-wave deficit LW ↑ -LW ↓ exceeds the net short-wave gain SW ↓ -SW ↑ leading to surface cooling. Clear sky and weak wind conditions favour the development of a strongly stably stratified boundary layer. During the day, high downward solar radiative fluxes and dryness associated to low temperatures (∼ -30 °C) favour the initiation of convection [START_REF] King | Contrasts between the summertime surface energy balance and boundary layer structure at Dome C and Halley stations, Antarctica[END_REF]. A well mixed layer grows up to 200 or 300 m above the surface [START_REF] Argentini | Summer boundary-layer height at the plateau site of Dome C, Antarctica[END_REF][START_REF] Argentini | The surface layer observed by a high-resolution sodar at Dome C, Antarctica[END_REF][START_REF] Ricaud | Summer to winter diurnal variabilities of temperature and water vapour in the lowermost troposphere as observed by HAMSTRAD over Dome C, Antarctica[END_REF]. Such a diurnal cycle, with a boundary layer experiencing such a broad range of turbulence regime is not common even in Antarctica, the land of uncommons. [START_REF] King | Contrasts between the summertime surface energy balance and boundary layer structure at Dome C and Halley stations, Antarctica[END_REF] compare the diurnal responses of the boundary layer of Halley and Dome C, both around 75°S. At Halley which surface receives approximately the same amount of daily radiation, the measured diurnal temperature amplitude (at 2 m) typically reaches 3 °C compared to 10 °C at Dome C. The difference is attributed to 1) a lower daytime heating due to different partitioning of available surface energy into fluxes of latent or sensible heat ; 2) a lower night-time cooling due to difference in cloud cover. [START_REF] King | Contrasts between the summertime surface energy balance and boundary layer structure at Dome C and Halley stations, Antarctica[END_REF] conclude that in order to experience a summertime convective boundary layer in Antarctica, one should be northerly enough for sufficient daytime solar radiation and also elevated enough for low temperatures (so that the energy absorbed by the surface is transferred to the boundary layer through sensible rather than latent heat, and thus is able to initiate convection). However, on the ice-sheet altitude is roughly correlated with latitude. Moreover, going northward, cloud cover and mechanical mixing due to katabatic wind prevent the formation of a strongly stably stratified boundary layer. Dome C appears to be a good synergy between latitude and altitude.

Low level jets are commonly observed in stable boundary layers. They may be downslope flows [START_REF] Manins | A model of katabatic winds[END_REF] or "short-lived" jet, induced by an inertial oscillation during the "evening transition" (van de [START_REF] Van De Wiel | A conceptual view on inertial oscillations and nocturnal low-level jets[END_REF]. This study focuses on a particular diurnal cycle : the 11th and 12th December 2009. Single column model simulations are performed and compared to in-situ observations. Then we focus on night-time turbulent mixing and search for possible underlying mechanisms for a nocturnal low level jet. Wind speed and direction are measured with Young 05103 aerovanes, air temperature and relative humidity are sampled with Vaisala HMP155, in aspirated shields. Up to now, a 5-year quasi-continuous data series has been recorded. [START_REF] Gallée | Meteorological atmospheric boundary layer measurements and ECMWF, analyses during summer at Dome C, Antarctica[END_REF][START_REF] Genthon | Atmospheric temperature measurement biases on the Antarctic plateau[END_REF][START_REF] Genthon | Two years of atmospheric boundary layer observations on a 45-m tower at Dome C on the Antarctic plateau[END_REF] give further details on this tower measurements, the obtained mean statistics and the deduced main characteristics of the boundary layer. In addition to these "standard" meteorological sensors, six sonic thermo-anemometers were deployed between 7 and 42 m. In December 2009, five of them were running.

Everyday, insights of the state of the atmosphere up to 20-25 km over Dome C are provided by launching Vaisala RS92 radiosound around 12 UTC (20 LT) as part as the Rou-tine Meteorological Observation program † . In December 2009, during the Concordiasi campaign detailed by [START_REF] Rabier | The Concordiasi project in Antarctica[END_REF], an additional radiosound was launched around 0 UTC (8 LT).

The Meso-NH numerical model

Meso-NH is a non-hydrostatic, anelastic model designed to simulate a broad range of atmospheric motions from mesoscale to turbulent eddies [START_REF] Lafore | The Meso-NH atmospheric simulation system. Part I : adiabatic formulation and control simulations[END_REF]. Here, it was used in single column mode coupled with the SURFEX surface scheme [START_REF] Masson | The SURFEXv7.2 land and ocean surface platform for coupled or offline simulation of earth surface variables and fluxes[END_REF]. A 1.5-order turbulent scheme as described in [START_REF] Cuxart | A turbulence scheme allowing for mesoscale and large-eddy simulations[END_REF], based on a prognostic equation for T KE and involving a Richardson-number-dependent formulation for the Prandtl number [START_REF] Redelsperger | Méthode de representation de la turbulence associee aux precipitations dans un modèle tri-dimensionnel de convection nuageuse[END_REF] is implemented. It is used here in its one dimensional formulation with a Bougeault and Lacarrère (1989) mixing length. The long-waves and short-waves radiative transfers are computed separately using the ECMWF operational radiation code [START_REF] Gregory | Revision of convection, radiation and cloud schemes in the ECMWF integrated forecasting system[END_REF]. Cloud dynamics is simulated using a mixed-phase microphysical scheme [START_REF] Pinty | A mixedphase cloud parameterization for use in mesoscale non-hydrostatic model : simulations of a squall line and of orographic precipitations[END_REF] ; shallow convection is parameterized with a combined eddydiffusity / mass-flux approach [START_REF] Pergaud | A parameterization of dry thermals and shallow cumuli for mesoscale numerical weather prediction[END_REF].

Initialization and large-scale forcing

The case starts at 0 UTC that is 8 LT in the morning. A radiosounding is available at that time. Actually, this observational data are not directly used. Instead, temperature, humidity and geostrophic wind initial fields have been deduced and simplified [START_REF] Bazile | GABLS4 : An intercomparison case to study the stable boundary layer with surface interactions on the Antarctic plateau[END_REF] from a 4D-var re-analysis with the ARPEGE global circulation model. The re-analysis was performed using the ARPEGE configurations used during the Concordiasi field campaign : a stretched grid centred on Dome C with horizontal resolution of 10 km on the Antarctic plateau. Each radiosounding of the period was taken into account with its whole set of data levels. †. RMO, http://www.climantartide.it/.

For meaningful comparison between simulations and observations, we must prescribe a realistic large scale atmospheric forcing corresponding to the studied period. The needed large-scale fields are the geostrophic wind and the dynamical tendencies of temperature and humidity induced by the synoptic-scale weather conditions. These fields and their temporal evolutions are not observed but deduced by comparison of modelling and radiosounding data [START_REF] Bazile | GA-BLS4 : an intercomparison case for 1D models to study the stable boundary layer at Dome C on the Antartic plateau[END_REF]. An ensemble of simulations with the three dimensional limited area model AROME [START_REF] Seity | The AROME-France convective-scale operational model[END_REF] forced by the ARPEGE reanalysis and coupled with different physical packages were performed.

The AROME horizontal pressure gradient is used as an estimator of the geostrophic wind. The dynamic contributions to tendencies ∂T ∂t and ∂q ∂t were isolated by subtracting the mean spatially averaged physical tendencies only obtained with the AROME ensemble simulations from the tendencies observed between two consecutive radiosounding measurements.

Four profiles for +0 h, +12 h, +24 h and +36 h have been designed. In between, the model computes a linear interpolation of the two consecutive forcings. The advection tendencies of temperature and humidity are kept constant in each 12 h period. These initial and forcing fields were preliminary in-test versions of the GABLS4 intercomparison project [START_REF] Bazile | GABLS4 : An intercomparison case to study the stable boundary layer with surface interactions on the Antarctic plateau[END_REF].

Model setup

A reference simulation was performed with 10 m resolution in the low troposphere and a first level at 5 m. The stretched vertical grid extends up to 9000 m above the surface, with a damping layer above 8 km. The time step is set to 30 s. Microphysics, radiation, turbulence, shallow convection and surface schemes are activated. The chosen surface scheme consists in a 3 layers soil with the surface characteristics of fresh snow. The following surface parameters are fixed and do not evolve : the albedo (α = 0.8), the surface roughness (z 0 = 0.01 m) and the emissivity ( = 0.98). The simulation starts on 11 December at 00 UTC (8 LT) and runs 36 hours.

THE DIURNAL CYCLE

on DECEMBER 11-12th, 2009.

Comparison with December climatology

To assess whether the study period (11 and 12 December 2009) is representative of the summertime at Dome C, observed data from five December months (2009 to 2013) have been gathered and compared to the data collected during the study period [START_REF] Vignon | Couches limites atmosphériques extrêmes en Antarctique : le cas GABLS4[END_REF]. For a saturated air parcel at -30 °C and 650 hPa, specific humidity is of about 0.35 g • kg -1 . This is a very low value : the atmosphere may be considered as dry. Consequently, the present study does not discuss moisture aspects of the boundary layer but focuses on the temperature and wind climatology. In fact, no moisture climatology has been drawn due to the lack of accurate humidity measurements. In such dry conditions, standard humidity measurements are questionable [START_REF] Genthon | Two years of atmospheric boundary layer observations on a 45-m tower at Dome C on the Antarctic plateau[END_REF].

December climatology

Located 1000 km away from the Coast, cloud cover is relatively thin or absent [START_REF] King | Antarctic meteorology and climatology[END_REF]. For the last 5 summers the measured LW ↓ flux is under 125 Wm -2 more than 75 % of the time (Figure 3). Temperature. Temperature usually averages around -30 °C in summer but experiences diurnal variations with an amplitude of about 10 °C at 3 m, damping to 4 °C at 42 m as shown in Figure 4. Daytime profiles are quasi-uniform along the tower. This turbulent mixed layer fed by convection deepens from about 20-30 m at 9 am up to 100-300 m, as estimated by sodar measurements [START_REF] Argentini | Summer boundary-layer height at the plateau site of Dome C, Antarctica[END_REF][START_REF] Argentini | The surface layer observed by a high-resolution sodar at Dome C, Antarctica[END_REF]. From 4-5 pm, as soon as the heat turbulent fluxes decrease below zero, the mixed layer collapses to 50 m within only 2 hours. A stable stratification starts to set up in the surface layer. Observed night-time temperature profiles obtained from the tower and radiosoundings may be divided in three layers. The inversion layer extends in the lowest 20 meters. The inversion layer characterised by a temperature gradient of 0.7 °C • m -1 thickens regularly from 8 pm to 7 am, before shrinking suddenly. An overlying residual layer, with uniform temperature follows and overpasses the top of the tower. Above and up to the tropopause, the atmosphere is weakly stable with a quasi-uniform gradient, and a Brunt Väisälä frequency about N ∼ 0.014s -1 . The maximum daytime temperature is reached 1 hour after midday for the surface temperature, 2 hours later (3 pm) at 3 m and around 4-4.5 pm at 40 m above the surface. The minimum night-time temperature is reached at about 2 am for the surface temperature, 2 hours later (4 am) at 3 m and at about 6 am at 25 m above the surface.

Wind. More than 80 % of the time, wind blows from South, South-West, bringing dry and cold air from the high plateau (Figure 5). Northerly winds occur sometimes and are correlated with so-called warm events [START_REF] Gallée | Meteorological atmospheric boundary layer measurements and ECMWF, analyses during summer at Dome C, Antarctica[END_REF]. Daytime vertical wind profiles are quasi-homogeneous in the mixed layer. Conversely, night-time wind profiles are characterized by a strong wind shear dW dz ∼ 0.075 s -1 . The near surface wind, weaker by night, plateaus at 4.5 ms -1 by day, while active turbulent mixing may bring momentum from the upper troposphere. On the tower top, the midday plateau consists in a minimum, the maximum is reached by night. Temperature and velocity profiles in the boundary layer are typical of a summer day. In the lowest 15 m, the temperature is a bit colder than the climatological mean as other clear sky days. LW ↓ varies between 90 and 100 Wm -2 . Wind is South-Westerly. Along the tower it is slightly weaker (∼ 1 ms -1 ) than the climatological mean but still in the ± one standard deviation interval. Figure 6 shows that the near-surface stratification in the early morning is one of the strongest observed (T 3.3m -T s > 10 K), as expected for clear-sky and weak wind conditions.

Diurnal cycle in temperature

Isocontours of the observed and simulated potential temperature profiles are presented in Figure 8. The simulated temperature field experiences a clear diurnal cycle in phase with the observed cycle. Neither the surface temperature nor the fluxes are prescribed, so that, the cycle expresses the response of the boundary layer to solar forcing. The simulated boundary layer is stably stratified at night (with a temperature gradient reaching 0.11 Km -1 between 3 and 60 m) and relatively well mixed during the day. Evening and morning transitions seem on the whole well reproduced and occur in time.

However, the vertical distribution of temperature is not satisfactory. In order to be com- parable, simulated and observed temperature profiles are shown in Figure 8 using two different vertical scales. The simulated temperature field is plotted up to 180 m above the surface, whereas observed temperatures are plotted only along the 45 m of the tower. The simulated inversion layer follows similar thickening and shrinking phases than the observed one but extends up to 60 m. In addition, there is no clear residual layer above.

The order of magnitude of the daytime near-surface temperature is well reproduced with a temperature maximum around -27.5 °C (θ ∼ 278 K). But, the surface layer does not cool enough at night, leading to a warm bias in the simulation. The near surface diurnal amplitude is thus underestimated, scaling around 10 °C at the surface and 9 °C at 3 m whereas, in the observational data, the diurnal amplitude reaches 17 °C at the surface and 12 °C at 3 m. The amplitude is damped to 8 °C at 20 m ; the simulated damping is underestimated and only reaches 1 °C, consistent with a too deep inversion layer.

Diurnal cycle in wind speed

The simulated 3.5 m mean wind speed is quite close to the observed one (∼ 2.5 ms -1 ), however, the 10 m mean wind speed is un-derestimated by about 0.25 ms -1 . Figure 9 shows the velocity profile (1 h averaged) during the whole simulation. During the "day" (between 11 am et 4 pm) the velocity vertical profile is uniform along the tower. Whereas, at "night", a strong shear is observed in the first 20 meters. Despite an overestimation of daytime shear and an underestimation of nighttime shear, the diurnal cycle in wind speed is in overall well reproduced by the model.

Figure 10 exhibits the wind speed isocontours as observed (top) and simulated (bottom). Two velocity maxima can be seen by night. From tower observations, a first maximum is visible at 10 pm with a 6.1 ms -1 speed. The maximum speed is measured by the highest anemometer, we suppose this is the signature of a jet since the speed exceeds the geostrophic speed as estimated by AROME (section 2.3, and used to force the model). Nevertheless, we do not know exactly the jet nose position. It could be higher, or a bit lower since the difference in the two speed measured from the two highest anemometers is far lower than the factory stated accuracy (∆W ∼ ±0.03 ms -1 < σ W = 0.3 ms -1 ). The jet nose looks going down below 10 m at 2 h30. Around 9 am, a second maximum is measured at the top.

The model also produces two distinct wind speed maxima. A jet is simulated around 1 am with a nose located at 58 m and a speed of 7.1 ms -1 . The speed exceeds the geostrophic wind speed G by 2.3 ms -1 . A second supergeostrophic maximum of 6.5 ms -1 = G+1.3 ms -1 is reached at about 9 am and is located near 200 m. Figure 11 compares the temporal series of observed and simulated wind speed. Despite the too high location of the wind maxima, the timing is quite satisfactory.

In the simulation, the geostrophic wind is forced and evolves in time. However, from 0 to 400 m the geostrophic wind speed is vertically uniform and varies only slightly temporally, G = 4.7 ms -1 with a standard deviation of 0.33 ms -1 . Nevertheless, in order to be sure that the maximum speed described here is not correlated with temporal extrema in geostrophic wind, a simulation with a constant geostrophic forcing has been performed. The results reported in Table 1 do not significantly change. Two supergeostrophic maxima are again simulated. The wind maxima are weaker consistent with a weaker geostrophic wind, W J1 = G+2.1 ms -1 , W J2 = G + 1.7 ms -1 . A dynamical explanation of the wind maxima as low level jet is sketched in section 5.

TURBULENT MIXING

The model performs quite well during the day but fails in reproducing the stable stratification at night (section 3.2). Possible reasons could be an unrealistic turbulent mixing or a misrepresentation of snow-surface processes. This section investigates the first scenario and focuses on night-time turbulent mixing. First the simulation is compared to turbulent quantities deduced from in-situ observations. Then, the turbulent scheme of the Meso-NH model is analyzed.

Figure 12 shows normalised vertical profiles of potential temperature, wind speed and TKE at midnight. Figure 13 shows the contributions of different terms in the TKE budget equations. Turbulent kinetic energy is mainly produced by shear in the 60 lowest meters. The top of the turbulent layer corresponds to the top of the inversion layer and to the height of the jet. Studying a case of nocturnal low level jets in the Duero Basin, Spain, [START_REF] Conangla | On the turbulence in the upper part of the low-level jet : An experimental and numerical study[END_REF] and [START_REF] Cuxart | Mixing processes in a nocturnal low-level jet : An LES study[END_REF] have found with Meso-NH simulations two distinct turbulent layers. Vertical profiles of TKE is maximum near the surface and above the jet nose due to shear production. These two layers of elevated turbulence are separated by a minimum around the jet nose. The two-layer structure is not simulated here (Figure 13). Maybe this is because our jet is less pronounced and "short lived". The jet studied by [START_REF] Cuxart | Mixing processes in a nocturnal low-level jet : An LES study[END_REF] is quasi-stationary between 0000 and 0200 UTC with values around 9 ms -1 .

Turbulent quantities as turbulent kinetic energy, heat and momentum fluxes and temperature variances have been computed from sonic-anemometers data and eddy correlation (EC) methods. Surface fluxes have been computed through profile method from classical meteorological measurements and similarity laws [START_REF] Vignon | Couches limites atmosphériques extrêmes en Antarctique : le cas GABLS4[END_REF]. Some of these quantities are compared here to the simulated one (Figure 14). Surface sensible heat flux is in a good agreement with observations whereas the surface friction velocity is significantly overestimated (by 40 %) (not shown). Correct values of surface sensible heat fluxes are not expected since snow-surface temperature does not cool enough at night leading to a warm bias of about 6 K. Changing the roughness length to a more realistic value z 0 ) Simu2 = 0.001 m instead of z 0 ) Simu1 = 0.01 m leads to a lower surface friction velocity, which is still overestimated, but by 30 %. The surface sensible heat flux is improved whereas TKE at 7 m deteriorates (not show). Overall the simulated boundary layer is slightly more stable and thinner (Table 1), but improvements are limited.

Figure 14 compares temporal series of turbulent quantities at 7 meters computed by the Meso-NH model and by eddy correlation methods. The height of 7 m falls between the first and second model atmospheric levels (3.7 m and 11.2 m). Turbulent kinetic energy (e), potential temperature variance (θ 2 ), sensible heat (w θ ) and momentum fluxes (w u ) are shown in Figure 14. Simulated and observed diurnal trends are similar, whereas their order of magnitude differs. At 7 m above the surface, while values of TKE (e) are comparable, and θ 2 is overestimated, w θ , w u and w v are overestimated by Meso-NH. As one goes upper, for example at 30 m, the agreement tends to be better (not shown). This overestimation of turbulent mixing intensity in the inversion layer may explain the difficulty of the model to represent the observed strength of the stable stratification and its shallowness.

In fact, misrepresentation of the second order moments : θ 2 , w θ , w u and w v , by night is expected since vertical gradients of potential temperature θ and, to a lesser extent, of wind speed are largely underestimated by the model. The model turbulent-scheme computes . the second order moments using a K-gradient approach.

First wind maximum Second wind maximum

z 0 G dT dz ) J1 t J1 z J1 W J1 t J2 z J2 W J2 m ms -1 Km -1 LT m ms -1 LT m
Figure 15 shows the vertical profile of the eddy diffusivity coefficient K h for sensible heat at midnight. The simulated K h decreases with height. The eddy-diffusivity coefficient deduced from eddy correlation appears quite noisy in this instantaneous picture, but the lowest value at 7 m is relatively constant along the night. At 7 m K h ) obs ∼ 0.005 m 2 s -1 whereas K h ) simu ∼ 0.1 m 2 s -1 . K h ) simu is computed by the Meso-NH closure scheme according to, [START_REF] Cuxart | A turbulence scheme allowing for mesoscale and large-eddy simulations[END_REF] :

K h = C h • l • √ e • φ (1)
where e denotes the turbulent kinetic energy. l is the Bougeault and Lacarrère (1989) mixing length representing the size of the most energetic subgrid eddies. φ is a stability function, it accounts for the Prandtl number dependence on Richardson number, and C h = 0.14 is a fixed parameter. In one-dimensional cases, φ gets a simple formulation, [START_REF] Cuxart | A turbulence scheme allowing for mesoscale and large-eddy simulations[END_REF] : In a layer of constant stable stratification ∂ θ ∂z , a simple analytical formulation for l can be obtained from a second order development of θ(z)-θ(z ), [START_REF] Bougeault | Parametrization of orography-induced turbulence in a mesoscale model[END_REF] :

φ(z) = 1 1 + C 1 β l 2 e ∂ θ ∂z
l = 2e β ∂ θ ∂z (3)
Combining equations ( 3) and (2) in equation ( 1), K h becomes :

K h ∼ C h √ 2 1 + 2C 1 e β ∂ θ ∂z (4)
The equation implies that K h is a decreasing function of the stratification parameter ∂z and e, it gives : K h ) analytic1 ∼ 0.13 m 2 s -1 . This is larger than K h ) simu ∼ 0.10 m 2 s -1 . However it is the right order of magnitude, and is far larger than the value deduced from observations : K h ) obs ∼ 0.001 m 2 s -1 . Suppose that the simulation performs better in reproducing the stable stratification so that at midnight the simulated vertical gradient of temperature approaches the observed one. Then, according to equation (4) : K h ) analytic2 = 0.06 m 2 s -1 , this value is lower but still one order of magnitude too large.

The values of vertical gradient of temperature and turbulent kinetic energy, at midnight, 7 m above the surface are reported in Table 2. The corresponding observed or computed eddy-diffusivity coefficients K h and fluxes w θ are reported in the same table and in Figure 16. The analytic study detailed in the case of sensible heat flux and summarized in Figure 16 shows that the model overestimates turbulent mixing. Even though the stable stratification was well reproduced, it could not be sustained due to a too large mixing length l or a too large coefficient C h . A simulation initialized with the observed temperature profile at 2 am instead of 8 am has been performed. It did not give better results : the steep initial temperature gradient is rapidly smoothed, supporting the conclusion drawn from the analytic study. Dividing C h by 30 in equation ( 1) would resolved the problem at the beginning of the night, but it may alter good results during day time. The adjustment of the parameter C h is part of ongoing work.

THE LOW LEVEL JET

Climatological study

Both in the observations and simulations, the two low level jets occur at the top of the inversion layer. The simulated jet is too high compared with the observed one but this is consistent with the weaker stability and deeper inversion layer. This kind of nocturnal jet is not specific to our case study as a jet is observed below 41 m (the highest anemometer) more than 90 % of the 5 last years December nights [START_REF] Vignon | Couches limites atmosphériques extrêmes en Antarctique : le cas GABLS4[END_REF]. Figure 4 shows that contrary to the near-surface wind, the 20 to 40-m winds reach their maximum by night. This averaged maximum speed is probably associated to a low level jet. Climatological wind standard deviation increases with height by night (reaching 1.5 ms -1 ), so that the clear maximum wind speed seen in Figure 4 at midnight, hides disparities in the low level jet nose and maximum. The jet characteristics vary from day to day probably depending on inversion strength and on geostrophic wind. Inversion strength itself depends on external forcing like cloud cover. Figure 7 shows the height of the jet nose is lower when the inversion is stronger.

Inversion related low level jets are usually associated to katabatic flows. The surface is flat at Dome C, so that katabatic flows can not be locally generated. However, map 1 shows a 300 km-long slope, down from the South-Western high plateau to a pass 200 m lower. The slope direction corroborates with the prevailing wind. From the pass the terrain height increases by 40 m over 100 km and reaches Dome C. Nocturnal radiative cooling during

December, 11-12th, 2009

The present simulations are performed on a flat single-grid domain, so that other mechanisms than katabatic flows, are at stake to generate the simulated low level jet. Hodographs from observed and simulated wind vectors are drawn in Figure 17. At each level the wind is turning anti-clockwise. A turn is completed in approximately 11-12 h. As we go up along the tower or in the model, the oscillation starts sooner and stops later, so that there is time to initiate a second turn. Going back to Figure 11, we see that above 20 m in the observations, and above 100 m in the simulation, two maxima are observed shifted from 11.5 h ± 0.15 in simulation and 11.6 h ± 0.5 in the model. This duration corresponds to the duration of one turn and to the inertial period T io = 1/f c , where f c is the Coriolis parameter equal to 1.510 -4 s -1 at Dome C.

Inertial oscillation

These observations support the inertial oscillation (IO) mechanisms for low level jets, as theorized by [START_REF] Blackadar | Boundary layer wind maxima and their significance for the growth of nocturnal inversions[END_REF] and revisited by van de [START_REF] Van De Wiel | A conceptual view on inertial oscillations and nocturnal low-level jets[END_REF] to account for frictional effects. The theory predicts that while at evening transition, the equilibrium between the gradient pressure force, the Coriolis force, and friction is broken, the wind starts to rotate around an equilibrium profile with the period T io , (van de Wiel et al., 2010) :

u -u e = (v 0 -v e ) sin( 2π
T io t) + (u 0 -u e ) cos( 2π
T io t) v -v e = (v 0 -v e ) cos( 2π
T io t) + (u 0 -u e ) sin( 2π
T io t)
where (u 0 , v 0 ) and (u e , v e ) represent the initial and equilibrium velocity components ‡ . At each height the IO is independent of the IO at other heights (van de [START_REF] Van De Wiel | A conceptual view on inertial oscillations and nocturnal low-level jets[END_REF]. Friction shuts down sooner for higher level, so that oscillation at each level are out of phase. Oscillations amplitude depend also on the height through the departure between initial and equilibrium profiles. The departure is zero near the surface where both wind speeds are zero and in the free troposphere where both wind speeds are geostrophic. In between it reaches a maximum, generally around the top of the nocturnal shear layer. This explains why we observe jet-like profile. van de [START_REF] Van De Wiel | A conceptual view on inertial oscillations and nocturnal low-level jets[END_REF] found that a "reverse ‡. ue corresponds the geostrophic wind in [START_REF] Blackadar | Boundary layer wind maxima and their significance for the growth of nocturnal inversions[END_REF].

oscillation" may take place at low level during an inertial oscillation. This may contribute to the weakening of near-surface wind. Such a "clockwise oscillation" has not been observed in our case, neither on the observations nor in the simulation.

According to the theory, the amplitude of the oscillation is larger when the contrast in turbulence intensity between day and night is larger. Such contrasts are reinforced by the weaker role of the turbulent latent heat flux at Dome C, leading to a stronger diurnal cycle of the sensible heat flux. This also explain why at Dome C, jets are observed during clear sky and weak wind conditions : this conditions leads to a stronger inversion. This fact also supports that Dome C, in the summer, is an ideal site to study inertial oscillations thanks to the occurrences of both a convective boundary layer in afternoon and a strongly stable stratification at "night". This kind of jets are thus not expected on typical winter days.

CONCLUSION

Although the sun never disappears below the horizon, the summertime boundary layer at Dome C experiences a clear diurnal cycle. A rapid transition between a very stable boundary layer and its associated jet and a diurnal convective boundary layer is observed. This offers an ideal case in the real world to test turbulent schemes, and learn about stable regimes. This is the goal of the freshly launched GABLS4 intercomparison project (http://www.cnrm.meteo.fr/ aladin/meshtml/GABLS4/GABLS4.html) organized by CNRM/GAME and LGGE. The present simulation does not perform as well as it should to fit observed data. The nocturnal cooling is underestimated leading to a far too deep and not stable enough inversion layer. Some more work is required in order to understand the simulation failures and improve the results. In particular, a simulation with a prescribed surface temperature may be helpful to discriminate the failures due to the turbulent scheme from those due to the surface scheme or to the coupling. Nevertheless, the comparison between the simulated night-time wind and temperature profiles with the obser-vations do show some similarities : on both, a short live low level jet occurs just below the top of the inversion layer.

These similarities provided clues for an interpretation of the wind speed maxima seen sometimes in the middle of the observed vertical profiles. Indeed, an insight of inertial induced jets requires fine vertical and temporal resolutions in both model and observation system. On the tower, almost ten meters separates the two highest anemometers so that, the short lived jet produces some unfamiliar maxima on one or two measurement samples of the observed vertical profile. These maxima had before often been interpreted as measurement errors.

Some evidence of oscillation shows that the inertial oscillation mechanism contributes to the formation of this jet. Of course, there may also be other underlying processes feeding the jet. Studies with three dimensional models constrained by good quality large scale atmospheric forcing and able to reproduce the fine scale turbulence as done in Gallée et al. (2014b) are necessary for a comprehensive understanding of the underlying mechanisms responsible for the low level jet.

Inertial induced low level jets at Dome C, may appear anecdotal. Yet, in other latitudes, low level jets may interfere with other processes like the dispersion of polluants in stable cases or birds migrations. Inertial induced jet and the induced shear-driven turbulence may usefully be considered to adjust the height of wind farms, design their turbines, program the flight of a drone and train carrier-pigeon. The good behaviour of the Gallée et al. (2014b) model MAR in simulating boundary layer processes at Dome C is not stranger to its good behaviour over Belgium [START_REF] Doutreloup | Comparaison entre le profil vertical de la vitesse du vent observé dans les basses couches de la troposphère et celui simulé par le modèle WRF en Belgique[END_REF]. We argue that the dataset of the fullscale laboratory Dome C could be useful for every one interested in low level jets or other stable boundary layers related features. Besides, anyone trying to design a simulation for low level jet by inertial oscillations, whether he is from the meteorological or climate communities or not, is encouraged to take part in the GABLS4 intercomparison.

Figure 1 -

 1 Figure 1 -Topography around Dome C -(Gallée et al., 2014a)

Figure 2 -

 2 Figure 2 -The 45 m tower at Dome C (Photo C. Brun)

Figure 3 -

 3 Figure 3 -Histrogram -longwave diffusive radiation down : LW ↓. BSRN data, December only from 2009 to 2013 -(Vignon, 2014).

Figure 4 -

 4 Figure 4 -Mean diurnal cycle in temperature and wind speed. Night-time December data from 2009 to 2011.

Figure 5 -

 5 Figure 5 -Wind rose -9-m wind, December (only) data from 2009 to 2011

Figure 6 -

 6 Figure 6 -Downward long-waves radiation LW ↓ versus Inversion strength in the surface layer (T 3.3m -T s ), by night. Strong winds samples are plotted in red, whereas weak winds samples in blue. Plotted samples are from night-time December (only) data from 2009 to 2013. -Adapted from[START_REF] Vignon | Couches limites atmosphériques extrêmes en Antarctique : le cas GABLS4[END_REF].

  16 shows the shape of K h as a function of ∂ θ ∂z for different values of e. Computing K h from equation (4) with modelled values of ∂ θ

Figure 7 -

 7 Figure 7 -Number of samples (30-min average) for which a jet is detected below 40 m, depending on the height (expressed in tower level) and the inversion strength (measured by T tower top -T tower bottom in °C). All the available measurements recorded during December nights from 2009 to 2013 have been plotted. -Adapted from[START_REF] Vignon | Couches limites atmosphériques extrêmes en Antarctique : le cas GABLS4[END_REF].

  Values of the vertical gradient of temperature, turbulent kinetic energy and eddy diffusivity coefficients at 0 LT, at 7 m. The gradients are computed from the two closest measurement or model points above and below 7 m. Line 1 : simulated values ; line 2 : values deduced from observation and EC methods ; line 3 : K h is computed from equation (4) and simulated values ; line 4 : K h is computed from equation (4) and observed values.

Figure 8 -

 8 Figure 8 -Potential temperature and wind speed isocontours from tower observations (top) and in the 180 first meters of the simulation (bottom). Potential temperature, calculated with a reference level at 1013 hPa, ranges so "high" above 0 °C due to the elevated altitude of Dome C and the associated low pressure (∼ 650 hPa). Corresponding absolute temperature is of about -30 °C.

Figure 9 -

 9 Figure 9 -Wind velocity profiles at 1 pm and 10 pm. Tower observations in black are compared with the control simulation (in green) and a simulation Simu2 with a lower value of the rugosity length.

Figure 10 -Figure 11 -Figure 12 -

 101112 Figure 10 -Wind speed isocontours from tower observations (top) and in the 180 first meters of the simulation (bottom).

Figure 13 -Figure 14 -Figure 15 -Figure 16

 13141516 Figure 13 -Vertical profile of TKE budget computed from 1-hour statistics : a. at 0 LT, b. at 12 LT.

Figure 17 -

 17 Figure 17 -Hodographs for different vertical levels in observation (left) and in the model (right). Dashed and coloured arrays stand for the evening velocity vectors (7 pm). Grey dashed vectors represent the maximum wind speed for each level. The observed wind is plotted from 7 pm to 6 am ; the simulated wind is plotted from 7 pm to 9 am.