Pierre Pansu
email: pierre.pansu@math.u-psud.fr

Michel Rumin
email: michel.rumin@math.u-psud.fr

ON THE ℓ q,p COHOMOLOGY OF CARNOT GROUPS

Keywords: ℓ q, p cohomology, Carnot groups, global analysis on manifolds

We study the simplicial ℓ q,p cohomology of Carnot groups G. We show vanishing and non-vanishing results depending of the range of the (p, q) gap with respect to the weight gaps in the Lie algebra cohomology of G.

1. Introduction 1.1. ℓ q,p cohomology. Let T be a countable simplicial complex. Given 1 ≤ p ≤ q ≤ +∞, the ℓ q,p cohomology of T is the quotient of the space of ℓ p simplicial cocycles by the image of ℓ q simplicial cochains by the coboundary d,

ℓ q,p H k (T ) = (ℓ p C k (T ) ∩ ker d)/d(ℓ q C k-1 (T )) ∩ ℓ p C k (T ).
It is a quasiisometry invariant of bounded geometry simplicial complexes whose usual cohomology vanishes in a uniform manner, see [START_REF] Bourdon | Cohomologie l p et espaces de Besov[END_REF][START_REF] Elek | Coarse cohomology and l p -cohomology[END_REF][START_REF] Genton | Scaled Alexander-Spanier Cohomology and L q,p Cohomology for Metric Spaces[END_REF][START_REF] Gromov | Asymptotic invariants of infinite groups[END_REF][START_REF] Pansu | Cup-products in l q,p -cohomology: discretization and quasi-isometry invariance[END_REF]. Riemannian manifolds M with bounded geometry admit quasiisometric simplicial complexes (a construction is provided below, in Section 3). Uniform vanishing of cohomology passes through. Therefore one can take the ℓ q,p cohomology of any such complex as a definition of the ℓ q,p cohomology of M.

One should think of ℓ q,p cohomology as a (large scale) topological invariant. It has been useful in several contexts, mainly for the class of hyperbolic groups where the relevant value of q is q = p, see [START_REF] Bourdon | Some applications of ℓ p -cohomology to boundaries of Gromov hyperbolic spaces[END_REF][START_REF] Bourdon | Cohomologie l p et espaces de Besov[END_REF][START_REF] Cornulier | Contracting automorphisms and L p -cohomology in degree one[END_REF][START_REF] Drutu | Random groups, random graphs and eigenvalues of p-laplacians[END_REF] for instance. It is interesting to study a class of spaces where values of q = p play a significant role. The goal of the present paper is to compute ℓ q,p cohomology, to some extent, for certain Carnot groups. Even the case of abelian groups is not straightforward. 1.2. Carnot groups. Let G be a Carnot group, i.e. a simply connected real Lie group whose Lie algebra g is equipped with a derivation whose invariant vectors generate g. The derivation defines gradations, called weight, on g and Λ • g * . The cohomology of g is graded by degree and weight,

H • (g) = k,w H k,w (g).
For k = 0, . . . , dim(g), let w min (k) (resp. w max (k)) be the smallest (resp. the largest) weight w such that H k,w (g) = 0. Let p and q be real numbers.

(i) If

1 < p, q < ∞ and 1 p - 1 q ≥ δN max (k) Q .
Then the ℓ q,p cohomology in degree k of G vanishes. (ii) If

1 ≤ p, q ≤ ∞ and 1 p - 1 q < δN min (k) Q ,
then the ℓ q,p cohomology in degree k of G does not vanish.

The non-vanishing statement has a wider scope, see Theorem 9.2. It holds in particular on more general homogeneous groups.

Theorem 1.1 is sharp when both H k-1 (g) and H k (g) are concentrated in a single weight. This happens in all degrees for abelian groups and for Heisenberg groups, for instance. This happens for all Carnot groups in degrees 1 and n: δN max (1) = δN min (1) = 1 = δN max (n) = δN min (n).

Even when not sharp, the result seems of some value as it relates large scale quasiisometric analytic invariants of G to its infinitesimal Lie structure. For instance for k = 2, the weights on H 2 (g) can be interpreted as the depth of the relations defining G with respect to a free Lie group over g 1 , see e.g. [START_REF] Rumin | Around heat decay on forms and relations of nilpotent Lie groups[END_REF]. The results yield global discrete Poincaré inequalities of type d -1 ω q ≤ C ω p on 2-cocycles, as long as 1 < p, q < +∞ and

1 p -1 q ≥ wmax(2)-1 Q
, while there exist ℓ p 2-cocycles without ℓ q primitive when 1 p -1 q < w min (2)-1 Q .

We shall also illustrate the results on the Engel group in Section 9.5, and show in particular that, apart from degrees 1 and n, the natural Carnot homogeneous structure does not give the best range for the non-vanishing result in general. 1.4. Method. We briefly describe the scheme of the proof of Theorem 1.1. The first step of the vanishing statement is a Leray type lemma which relates the discrete ℓ q,p cohomology to some Sobolev L q,p cohomology of differential forms. This is proved here in the more general setting of manifolds of bounded geometry of some high order. One has to take care of an eventual lack of uniformity in the coverings in order to be able to use local Poincaré inequalities.

A feature of this Sobolev L q,p cohomology is that its forms are a priori quite smooth, and need only be integrated into much less regular ones. This is because ℓ p -cochains at the discrete level transform into smooth forms made from a smooth partition of unity, while reversely, less regular L p forms can still be discretized into some ℓ p data.

We then focus on Carnot groups. Although they possess dilatations, the de Rham differential is not homogeneous when seen as a differential operator. Nevertheless, as observed in [START_REF] Rumin | Differential geometry on C-C spaces and application to the Novikov-Shubin numbers of nilpotent Lie groups[END_REF], it has some type of graded hypoellipticity, that can be used to produce global homotopies K. These homotopies are pseudodifferential operators as studied by Folland and Christ-Geller-Głowacki-Polin in [START_REF] Christ | Pseudodifferential operators on groups with dilations[END_REF][START_REF] Folland | Subelliptic estimates and function spaces on nilpotent Lie groups[END_REF]. They can be thought of as a kind of generalized Riesz potentials, like ∆ -1 δ on 1-forms, but adapted here to the Carnot homogeneity of the group.

One needs then to translate the graded Sobolev regularity of K into a standard one to get the L q,p Sobolev controls on d. This is here that the weight gaps of forms arise to control the (p, q) range. Actually, in order to reduce the gap to cohomology weights in H * (g) only, we work with a contracted de Rham complex d c instead, available in Carnot geometry. It shares the same graded analytic regularity as d, but uses forms with retracted components over H * (g) only. It is worthwhile noting that although the retraction of de Rham complex on d c costs a lot of derivatives, this is harmless here due to the feature of Sobolev L q,p cohomology we mentionned above. Actually, in this low energy large scale problem, loosing regularity is not an issue and the less derivatives the homotopy K c controls, the smaller is the (p, q) gap in Sobolev inequality, and the better becomes the ℓ q,p vanishing result.

The non-vanishing result (ii) in Theorem 1.1 relies on the construction of homogeneous closed differential forms of any order and controlled weights. The contracted de Rham complex is useful to this end too. Such homogeneous forms belong to L p space with explicit p, but can not be integrated in L q for q too close to p, as seen using Poincaré duality (construction of compactly supported test forms and integration by parts).

Local Poincaré inequality

In this section, differential forms of degree -1, Ω -1 , are meant to be constants. The complex is completed with the map d : Ω -1 → Ω 0 which maps a constant to a constant function with the same value. Definition 2.1. Let M be a Riemannian manifold. Say that M has C h -bounded geometry if injectivity radius is bounded below and curvature together with all its derivatives up to order h are uniformly bounded. For ℓ ≤ h-1, let W ℓ,p denote the space of smooth functions u on M which are in L p as well as all their covariant derivatives up to order ℓ. When

p = ∞, W ℓ,∞ = C ℓ ∩ L ∞ . Remark 2.2. On a C h -bounded geometry n-manifold, these Sobolev spaces are interlaced. Let q ≥ p. Then W h,p ⊂ W h-1-n/p,q .
Indeed, on a ball B of size smaller than the injectivity radius, usual Sobolev embedding holds, W ℓ,p (B) ⊂ L q (B) provided 1 p -1 q ≤ ℓ n , an inequality which is automatically satisfied if ℓ ≥ n p + 1. Pick a covering B i of M by such balls with bounded multiplicity. Let

u i = f W h,p (B) and v i = f W h-ℓ,q (B) . Then v i ≤ C u i . Furthermore f W h-ℓ,q ≤ C (v i ) ℓ q ≤ C (v i ) ℓ p ≤ C ′ (u i ) ℓ p ≤ C ′′ f W h,p .
Remark 2.3. According to Bemelmans-Min Oo-Ruh [START_REF] Bemelmans | Smoothing Riemannian metrics[END_REF], for any fixed h, any complete Riemannian metric with bounded curvature can be approximated by an other one with all derivatives of curvature up to order h uniformly bounded.

Therefore assuming bounded geometry up to a high order is not a restriction for our overall purposes. The main point is that curvature and injectivity radius be bounded. Nevertheless, it helps in technical steps like the following Proposition. Proposition 2.4. Let M be a Riemannian manifold with sectional curvature bounded by K, as well as all covariant derivatives of curvature up to order h. Let R < π 2 √ K . Assume that M has a positive injectivity radius, larger than 2R. Let y ∈ M. Let U j be balls in M containing y of radii ≤ R, and U = j U j . The Cartan homotopy is an operator P on differential forms on U which satisfies 1 = P d + dP and maps C ℓ Ω k (U) to C ℓ Ω k-1 (U) for all ℓ ≤ h -1 and all k = 0, . . . , n, with norm depending on K, h and R only.

Assume further that U contains B(y, r) for some r > 0. Then P is bounded from

W h-1,p Ω k (U) to W h-n-1,q Ω k-1 (U), provided p ≥ 1, q ≥ 1, h > n + 1.
Its norm depends on K, h, R and r only.

Proof. The assumptions on R guarantee that minimizing geodesics between points at distance < R are unique and that all balls of radii ≤ R are geodesically convex. For x, y ∈ U, let γ x,y denote the unique minimizing geodesic from x to y, parametrized on [0, 1] with constant speed d(x, y). Fix y ∈ U. Consider the vectorfield ξ y defined as follows,

ξ y (x) = γ ′
x,y (0). It is smooth, since in normal coordinates centered at y, ξ y is the radial vectorfield ξ y (u) = -u. Let φ y,t denote the diffeomorphism semi-group generated by ξ y . For t ∈ R + , φ y,t maps a point x to γ y,x (e -t d(y, x)).

Let k ≥ 1. Following H. Cartan, define an operator P y on k forms ω by

P y (ω) = - +∞ 0 φ * y,t ι ξy ω dt.
Then, on k-forms,

dP y + P y d = - +∞ 0 φ * y,t (dι ξy + ι ξy d) dt = - +∞ 0 φ * y,t L ξy dt = -φ * y,+∞ + φ * y,0 = 1, if k ≥ 1,
since φ y,0 is the identity and φ y,+∞ is the constant map to y. On 0-forms, define P y (ω) = ω(y). Then dP y = ω(y) viewed as a (constant) function on U, whereas P y dω = ω -ω(y), hence dP y + P y d = 1 also on 0-forms.

In normal coordinates with origin at y, P y has a simple expression

P y ω(x) = +∞ 0 e -kt ω e -t x (x, • • • ) dt = 1 0 s k-1 ω sx (x, • • • ) ds.
It shows that P y , read in normal coordinates, is bounded on C ℓ for all ℓ.

The domain of exponential coordinates, V = exp -1 y (U), is convex. If it contains a ball of radius r, there is a bi-Lipschitz homeomorphism of the unit ball to V with Lipschitz constants depending on R and r only, hence Sobolev embeddings

W 1,p ⊂ L q for 1 p -1 q ≤ 1 n with uniform constants (if q = ∞, p > n is required and L q is replaced with C 0 ). If p ≥ 1, this implies that W n+1,p ⊂ C 0 , hence W ℓ,p ⊂ C ℓ-n-1 . Obviously, C ℓ-n-1 ⊂ W ℓ-n-1,q .
Since curvature and its derivatives are bounded up to order h, the Riemannian exponential map and its inverse are C h-1 -bounded, hence P y is bounded on

C ℓ for ℓ ≤ h -1. If h > n + 1, the embeddings W ℓ,p ⊂ C ℓ-n-1 ⊂ W ℓ-n-1,q
hold on U with bounds depending on K, R and r only, hence P maps W ℓ,p to W ℓ-n,q , with uniform bounds.

3. ℓ q,p cohomology and Sobolev L q,p cohomology Definition 3.1. Let M be a Riemannian manifold. Let h, h ′ ∈ N. The Sobolev L q,p cohomology is

L q,p h ′ ,h H • (M) = {closed forms in W h,p }/d({forms in W h ′ ,q }) ∩ W h,p . Remark 3.2. On a bounded geometry n-manifold, this is nonincreasing in q in the fol- lowing sense. Let q ′ ≥ q. Then L q,p h ′ ,h H k (M) surjects onto L q ′ ,p h ′ -qn-1,h H k (M), see Remark 2.2. Theorem 3.3. Let 1 ≤ p ≤ q ≤ ∞.
Let M be a Riemannian manifold of C ℓ -bounded geometry, with ℓ > n n+1 + 1. Let T be a simplicial complex quasiisometric to M. For every integers h, h ′ such that n n+1 < h, h ′ ≤ ℓ -1, there exists an isomorphism between the ℓ q,p cohomology of T and the Sobolev L q,p cohomology L q,p h ′ ,h H • (M).

The proof is a careful inspection of Leray's acyclic covering theorem. First construct a simplicial complex T quasiisometric to M. Pick a left-invariant metric on M. Up to rescaling, one can assume that sectional curvature is ≤ 1/n 2 and injectivity radius is > 2n. Pick a maximal 1/2-separated subset {x i } of M. Let B i be the covering by closed unit balls centered on this set. Let T denote the nerve of this covering. Let U i = B(x i , 3). Note that if x i 0 , . . . , x i j span a j-simplex of T , then the intersection

U i 0 ...i j := j m=0 U im
is contained in a ball of radius 3 and contains a concentric ball of radius 1.

Pick once and for all a smooth cut-off function with support in [-1, 1], compose it with distance to points x i and convert the obtained collection of functions into a partition of unity χ i by dividing by the sum.

Define a bicomplex C j,k = skew-symmetric maps associating to j + 1-tuples (i 0 , . . . , i j ) differential k-forms on j + 1-fold intersections

U i 0 ∩ • • • ∩ U i j . It is convenient to extend the notation to C -1,k = Ω k (M), C j,-1 = C j (T ), C j,• = C •,j = 0 if j < -1.
The two commuting complexes are d : C j,k → C j,k+1 and the simplicial coboundary

δ : C j,k → C j+1,k defined by δ(φ) i 0 ...i j+1 = φ i 0 ...i j -φ i 0 ...i j-1 j j+1 + • • • + (-1) j+1 φ i 1 ...i j+1 , restricted to U i 0 ∩ • • • ∩ U i j+1 .
By convention, d : C j,-1 → C j,0 maps scalar j-cochains to skew-symmetric maps to functions on intersections which are constant. Also, δ : C -1,k → C 0,k maps a globally defined differential form to the collection of its restrictions to open sets U i . Other differentials vanish.

The coboundary δ is inverted by the operator

ǫ : C j,k → C j-1,k defined by ǫ(φ) i 0 ...i j-1 = m χ m φ mi 0 ...i j-1 .
By an inverse, we mean that δǫ + ǫδ = 1. This identity persists in all nonnegative bidegrees provided ǫ :

C 0,k → C -1,k is defined by ǫ(φ) = m χ m φ m and ǫ = 0 on C j,k , j < 0.
Consider the maps

Φ j = (ǫd) j+1 : C j (T ) = C j,-1 → C -1,j = Ω j (M).
By definition, given a cochain κ, Φ j (κ) is a local linear expression of the constants defining κ, multiplied by polynomials of the χ i and their differential. Therefore Φ j (κ) is C ∞ and belongs to W h,q (Ω j (M)) for all

q ≥ p ≥ 1 if κ ∈ ℓ p (C j (T )). Since (ǫd)δ = ǫδd = (1 -δǫ)d = d -δǫd, for j ≥ 0, on C j,-1 , (ǫd) j+2 δ = (ǫd) j+1 (d -δǫd) = -(ǫd) j+1 δ(ǫd) = • • • = (-1) j+1 (ǫd)δ(ǫd) j+1 = (-1) j+1 (d -δǫd)(ǫd) j+1 = (-1) j+1 d(ǫd) j+1 -(-1) j+1 δ(ǫd) j+2 = (-1) j+1 d(ǫd) j+1 .
Indeed, (ǫd) j+2 (C j,-1 ) ⊂ C -2,j+1 = {0}. In other words,

Φ j+1 • δ = (-1) j+1 d • Φ j .
We now proceed in the opposite direction to produce a cohomological inverse of Φ. Let us mark each intersection

U i 0 ...i j := U i 0 ∩ • • • ∩ U i j with the point y = x i 0 . Proposition 2.
4 provides us with an operator P i 0 ...i j on usual k-forms on U i 0 ...i j . Putting them together yields an operator P :

C j,k → C j,k-1 such that 1 = dP + P d. Furthermore, P is bounded from W •,p to W •-n-1,q .
Exchanging the formal roles of (δ, ǫ) and (d, P ), we define

Ψ k = (P δ) k+1 : Ω k (M) = C -1,k → C k,-1 = C k (T ),
As above, one checks easily that

Ψ k+1 • d = (-1) k+1 δ • Ψ k .
Observe that the maps Ψ j • Φ j , j = 0, 1, . . ., put together form a morphism of the complex C •,-1 = C • (T ) (i.e. they commute with δ). We next show that it is homotopic to the identity. Let us prove by induction on i that, on C

•,-1 , (1) (P δ) i (ǫd) i = 1 -R i δ -δR i-1 , with R 0 = 0 and R i = i-1 k=0 (-1) k (P δ) k P (ǫd) k+1
. This implies the result for Ψ j • Φ j .

Proof. For i = 1, one has on C •,-1 1) for i, one writes

(P δ)(ǫd) = P (1 -ǫδ)d = P d -P ǫdδ = 1 -(P ǫd)δ, since P d = 1 on C •,-1 . Assuming (
(P δ) i+1 (ǫd) i+1 = (P δ) i P δǫd(ǫd) i = (P δ) i P (1 -ǫδ)d(ǫd) i = (P δ) i (1 -dP -P ǫδd)(ǫd) i = (P δ) i (ǫd) i -(P δ) i dP (ǫd) i -(P δ) i P (ǫd)δ(ǫd) i . ( 2
)
About the second term in (2), one finds that δ(P δ)d = -δd(P δ), hence by induction, one can push the isolated d term to the left,

(P δ) i d = (-1) i-1 P δd(P δ) i-1 = (-1) i-1 P dδ(P δ) i-1 = (-1) i-1 δ(P δ) i-1 ,
when the image lies within C •,-1 , as it does in [START_REF] Bourdon | Some applications of ℓ p -cohomology to boundaries of Gromov hyperbolic spaces[END_REF].

For the third term in (2), one sees that (ǫd)δ(ǫd) = (ǫd)(1 -ǫδ)d = -(ǫd) 2 δ, so that one can push the isolated δ term to the right,

(P δ) i P (ǫd)δ(ǫd) i = (-1) i (P δ) i P (ǫd) i+1 δ .
Gathering in [START_REF] Bourdon | Some applications of ℓ p -cohomology to boundaries of Gromov hyperbolic spaces[END_REF] gives

(P δ) i+1 (ǫd) i+1 = 1 -(R i + (-1) i (P δ) i P (ǫd) i+1 )δ -δ(R i-1 + (-1) i-1 (P δ) i-1 P (ǫd) i ), that proves (1). Similarly, Φ • Ψ is homotopic to the identity on the complex (C -1,• , d), Φ • Ψ = 1 -dR ′ - R ′ d.
Finally, let us examine how Sobolev norms behave under the class of endomorphisms we are using. Maps from cochains to differential forms, i.e. ǫ, Φ are bounded from ℓ p to W h-1,p . Maps from differential forms, i.e. P , Ψ, loose derivatives (but this is harmless since the final outputs are scalar cochains) so are bounded from W h,p to ℓ p , h ≤ ℓ -1. One merely needs h large enough to be able to apply P n times, whence the assumption h ≥ n n+1 . Maps from cochains to cochains, e.g. R, are bounded on ℓ p , maps from differential forms to differential forms, e.g. R ′ , are bounded from W h ′ ,p to W h,p for every

h ′ ≥ n n+1 such that h ′ ≤ ℓ -1. If q ≥ p, the ℓ q -norm is controlled by the ℓ p -norm, hence R is bounded from ℓ p to ℓ q . It is also true that R ′ is bounded from W h ′ -1,p to W h-1,q
. Indeed, it is made of bricks which map differential forms to cochains or cochains to differential forms, so no loss on derivatives affects the final differentiability. For the same reason, one can gain local integrability from L p loc to L q loc without restriction on p and q but p, q ≥ 1. It follows that Φ and Ψ induce isomorphisms between the ℓ q,p cohomology of T and the Sobolev L q,p cohomology.

De Rham complex and graduation on Carnot groups

From now on, we will work on Carnot Lie groups. These are nilpotent Lie groups G such that their Lie algebra g splits into a direct sum

g = g 1 ⊕ g 2 ⊕ • • • ⊕ g r satisfying [g 1 , g i ] = g i+1 for 1 ≤ i ≤ r -1.
The weight w = i on g i induces a family of dilations δ t = t w on g ≃ G.

In turn, the tangent bundle T G splits into left invariant sub-bundles

H 1 ⊕ • • • ⊕ H r with H i = g i at the origin. Finally, differential forms decompose through their weight Ω k G = ⊕ w Ω k,w G with Ω k 1 H * w 1 ∧ • • • ∧ Ω k i H * w i of weight w = k 1 w 1 + • • • + k i w i . De Rham differential d itself splits into d = d 0 + d 1 + • • • + d r ,
with d i increasing weight by i. Indeed, this is clear on functions where d 0 = 0 and d i = d along H i . This extends to forms, using d(f α) = df ∧ α + f dα and observing that for left invariant forms α and left invariant vectors X i , Cartan's formula reads

dα(X 1 , • • • , X k+1 ) = 1≤i<j≤k+1 (-1) i+j α([X i , X j ], • • • , X i,j , • • • X k+1 ) = d 0 α(X 1 , • • • , X k+1 ).
Hence d = d 0 is a weight preserving algebraic (zero order) operator on invariant forms, and over a point, ker d 0 / Im d 0 = H * (g) is the Lie algebra cohomology of G. Note also that from these formulas, d i is a homogeneous differential operator of degree i and increases the weight by i. It is homogeneous of degree 0 trough h * λ , as d, since dh * λ = h * λ d. This algebraic d 0 allows to split and contract de Rham complex on a smaller subcomplex, as we now briefly describe. This was shown in [START_REF] Rumin | Differential geometry on C-C spaces and application to the Novikov-Shubin numbers of nilpotent Lie groups[END_REF] and [START_REF] Rumin | Around heat decay on forms and relations of nilpotent Lie groups[END_REF] in the more general setting of Carnot-Caratheodory manifolds. More details may be found there.

Pick an invariant metric so that the g i are orthogonal to each others, and let δ 0 = d * 0 and d -1 0 be the partial inverse of d 0 such that ker

d -1 0 = ker δ 0 , d -1 0 d 0 = Π ker δ 0 and d 0 d -1 0 = Π Im d 0 . Let E 0 = ker d 0 ∩ ker δ 0 ≃ Ω * H * (g). Iterating the homotopy r = 1 -d -1 0 d -dd -1
0 one can show the following results, stated here in the particular case of Carnot groups. (1) The de Rham complex on G splits as the direct sum of two sub-complexes E ⊕ F , where

E = ker d -1 0 ∩ ker dd -1 0 and F = Im d -1 0 + Im(dd -1 0 ). (2) The retractions r k stabilize to Π E the projection on E along F . Π E is a homotopy equivalence of the form Π E = 1 -Rd -dR where R is a differential operator. (3) One has Π E Π E 0 Π E = Π E and Π E 0 Π E Π E 0 = Π E 0 so that the complex (E, d) is conjugated through Π E 0 to the complex (E 0 , d c ) with d c = Π E 0 dΠ E Π E 0 .
This shows in particular that the de Rham complex, (E, d) and (E 0 , d c ) are homotopically equivalent complexes on smooth forms. For convenience in the sequel, we will refer to (E 0 , d c ) as the contracted de Rham complex (also known as Rumin complex) and sections of E 0 as contracted forms, since they have a restricted set of components with respect to usual ones.

We shall now describe its analytical properties we will use.

Inverting d c and d on G

De Rham and contracted de Rham complexes are not homogeneous as differential operators, but are indeed invariant under the dilations δ t taking into account the weight of forms. This leads to a notion of sub-ellipticity in a graded sense, called C-C ellipticity in [START_REF] Rumin | Differential geometry on C-C spaces and application to the Novikov-Shubin numbers of nilpotent Lie groups[END_REF][START_REF] Rumin | Around heat decay on forms and relations of nilpotent Lie groups[END_REF], that we now describe.

Let ∇ = d 1 the differential along H = H 1 . Extend it on all forms using ∇(f α) = (∇f )α for left invariant forms α on G. Kohn's Laplacian ∆ H = ∇ * ∇ is hypoelliptic since H is bracket generating on Carnot groups, and positive self-adjoint on L 2 . Let then

|∇| = ∆ 1/2 H
denotes its square root. Following [8, Section 3] or [START_REF] Christ | Pseudodifferential operators on groups with dilations[END_REF], it is a homogeneous first order pseudodifferential operator on G in the sense that its distributional kernel, acting by group convolution, is homogeneous and smooth away from the origin. It possesses an inverse |∇| -1 , which is also a homogeneous pseudodifferential operator of order -1 in this calculus. Actually according to [START_REF] Folland | Subelliptic estimates and function spaces on nilpotent Lie groups[END_REF]Theorem 3.15], it belongs to a whole analytic family of pseudodifferential operators |∇| α of order α ∈ C. Note that kernels of these homogeneous pseudodifferential operators may contain logarithmic terms, when the order is an integer ≤ -Q. We refer to [START_REF] Christ | Pseudodifferential operators on groups with dilations[END_REF] and [8, Section 1] for more details and properties of this calculus.

A particularly useful test function space for these operators is given by the space of Schwartz functions all of whose polynomial moments vanish,

S 0 = {f ∈ S ; f, P = 0 for every polynomial P },

where S denotes the Schwartz space of G and f, P = G f (x)P (x) dx. Unlike more usual test functions spaces as C ∞ c or S, this space S 0 is stable under the action of pseudodifferential operators of any order in the calculus, see [4, Proposition 2.2], so that they can be composed on it. In particular by [8, Theorem 3.15], for every α, β ∈ C, (4)

|∇| α |∇| β = |∇| α+β on S 0 .
We shall prove in Proposition 7.1 that S 0 is dense in all Sobolev spaces W h,p with h ∈ N and 1 < p < +∞, but we shall work mainly in S 0 in this section.

Now let N = w on forms of weight w. Consider the operator |∇| N , preserving the degree and weight of forms, and acting componentwise on S 0 in a left-invariant frame.

From the previous discussion, d ∇ = |∇| -N d|∇| N and d ∇ c = |∇| -N d c |∇| N are both homogeneous pseudodifferential operators of (differential) order 0. Indeed, as observed in Section 4, d splits into d = i d i where d i is a differential operator of horizontal order i which increases weight by i. On forms of weight w, |∇| N has differential order w, d i |∇| N has order w + i and maps to forms of weight w + i, on which |∇| -N has order -(w + i), hence |∇| -N d|∇| N has order 0. The same argument applies to d c .

Viewed in this Sobolev scale, these complexes become invertible in the pseudodifferential calculus. Let By [4, Theorem 6.2], this amounts to show that these Laplacians satisfy Rockland's injectivity criterion. This means that their symbols are injective on smooth vectors of any non trivial irreducible unitary representation of G.

∆ ∇ = d ∇ (d ∇ ) * + (d ∇ ) * d ∇ and ∆ ∇ c = d ∇ c (d ∇ c ) * + (d ∇ c ) * d ∇ c , not to be confused with the non homogeneous d ∇ c (d * c ) ∇ +(d * c ) ∇ d ∇ c = |∇| -N (d c d * c +d * c d c )|∇| N .
This leads to a global homotopy for d c on G. Indeed following [8, Proposition 1.9], homogeneous pseudodifferential operators of order zero on G, such as

d ∇ c , ∆ ∇ c and Q ∇ c , are bounded on all L p (G) spaces for 1 < p < ∞. Therefore the positive self-adjoint ∆ ∇ c on L 2 (G) is bounded from below since Q ∇ c ∆ ∇ c = 1. Hence, it is invertible in L 2 (G) and Q ∇ c = (∆ ∇ c ) -1 is the inverse of ∆ ∇ c . Since (∆ ∇ c ) -1 commute with d ∇ c
, the zero order homogeneous pseudodifferential operator

K ∇ c = (d ∇ c ) * (∆ ∇ c ) -1 is a global homotopy, 1 = d ∇ c K ∇ c + K ∇ c d ∇ c . Let us set K c = |∇| N K ∇ c |∇| -N on S 0 , in order that 1 = d c K c + K c d c . Since K ∇ c is bounded from L p to L p for 1 < p < ∞, K c is bounded on S 0 endowed with the graded Sobolev norm α |∇|,N,p := |∇| -N α p .
Actually K c stays bounded for the whole Sobolev scale with shifted weights N -m. Proposition 5.2. For every constant m ∈ R and 1 < p < +∞, the homotopy K c is also bounded on S 0 with respect to the norm |∇|,N -m,p .

Proof. Indeed for α ∈ S 0 ,

K c α |∇|,N -m,p = |∇| m-N K c α p = |∇| m |∇| -N K c |∇| N |∇| -m |∇| m-N α p by (4), = |∇| m K ∇ c |∇| -m |∇| m-N α p ≤ C |∇| m-N α p = C α |∇|,N -m,p , since |∇| m K ∇ c |∇| -m
is pseudodifferential of order 0 and homogeneous, hence bounded on

L p if 1 < p < ∞.
Remark 5.3. Note that using Theorem 5.1, one can also produce a homotopy in the same way for the full de Rham complex itself. But as we will see in Section 8.1, it leads to a weaker vanishing theorem for ℓ q,p cohomology.

Relating the |∇|-graded to standard Sobolev norms

The next step is to compare the |∇|,N,p norms, depending on the weight of forms, to usual Sobolev norms of positive order.

Fix a basis X i of g 1 , viewed as left invariant vectorfields on G. Let ∇ denote the horizontal gradient ∇f = (X 1 f, . . . , X n 1 f ). Acting componentwise on tuples of functions allows to iterate it. One also extends it on differential forms using their components in a left invariant basis. For 1

≤ p ≤ ∞ and h ∈ N, define the Sobolev W h,p c norm α W h,p c = h k=0 ∇ k α p .
According to Folland [8, Theorem 4.10, Corollary 4.13], these norms are equivalent to Sobolev norms defined using |∇|, provided 1 < p < +∞. Namely one has then (5)

W h,p c ≍ h k=0 |∇| k p .
We shall now compare these norms to the graded ones we introduced in the previous section. (ii) Let µ ∈ N, µ < Q, and

1 < p < q < ∞ satisfy 1 p -1 q = µ Q . Then for some C > 0 it holds on Ω [a,b] ∩ S that C W h,q c ≤ b+µ+h m=a+µ |∇|,N -m,p . Proof. (i) By definition a ≤ N = w ≤ b on Ω [a,b] , hence a+h m=b |∇|,N -m,p = a+h m=b |∇| m-N p ≤ N +h m=N |∇| m-N p = h k=0 |∇| k p .
One has also b+h m=a

|∇|,N -m,p = b+h m=a |∇| m-N p ≥ N +h m=N |∇| m-N p = h k=0 |∇| k p .
(ii) Since |∇| -µ is a homogeneous pseudodifferential operator of order -µ and µ < Q, its kernel is homogeneous of degree -Q + µ. According to [START_REF] Folland | Subelliptic estimates and function spaces on nilpotent Lie groups[END_REF]Proposition 1.11

], |∇| -µ is bounded from L p to L q if 1 < p < q < +∞ satisfy 1 p -1 q = µ Q , giving the Hardy-Littlewood- Sobolev inequality (6) |∇| -µ α q ≤ C α p . Then for α ∈ Ω [a,b] ∩ S b+µ+h m=a+µ α |∇|,N -m,p ≥ N +µ+h m=N +µ |∇| m-N α p = h k=0 |∇| µ+k α p ≥ 1/C h k=0 |∇| k α q ≍ α W h,

Density of S 0 and extension of K c

One knows by Proposition 5.2 that K c is continuous with respect to the whole shifted norms |∇|,N -m,p , but it is only defined and provides an homotopy for d c on the initial domain S 0 so far. Hence, we have to show that S 0 is dense in S for the standard Sobolev norms W h,p c in order to extend K c on forms coming from ℓ p cocycles in Proposition 3.3. Recall from (3) that S 0 is the space of Schwartz functions with all vanishing polynomial moments. It does not contain any non vanishing function with compact support as seen using Fourier transform or Stone-Weierstrass approximation theorem.

Proposition 7.1. For every

h ∈ N and 1 < p < +∞, S 0 is dense in W h,p c . If p = 1, S 0 is dense in {f ∈ W h,1 c ; f, 1 = 0}. If p = ∞, S 0 is dense in the space C h 0 (G) of functions of class C h that tend to 0 at infinity.
Proof. The group exponential map exp : g → G maps Schwartz space to Schwartz space, polynomials to polynomials, and Lebesgue measure to Haar measure, hence S 0 to S 0 . Pick coordinates adapted to the splitting g = g 1 ⊕ • • • ⊕ g s .

First we construct (on g) a family of functions (g α ) α∈N n which is dual to the monomial basis (x β ) β∈N n in the sense that

g α , x β =    1 if α = β, 0 otherwise.
Let F denote the Euclidean Fourier transform on g, let F -1 denote its inverse. Let n = dim(G), let α ∈ N n denote a multiindex, |α| = n i=1 α i its usual length, α! = n j=1 α j !. Fix a smooth function χ with compact support in the square {max |ξ j | ≤ 1} of g * , which is equal to 1 in a neighborhood of 0. Let

g α = F -1 (χ(ξ) (iξ) α α! ).
Then g α ∈ S. One computes

(ix) β g α (x) = F -1 ∂ |β| ∂ξ β (χ(ξ) (iξ) α α! ) .
Hence

g α , x β = (-i) |β| ∂ |β| ∂ξ β (χ(ξ) (iξ) α α! ) (0) = (-i) |β| ∂ |β| ∂ξ β ( (iξ) α α! ) (0)
which vanishes unless β = α, in which case it is equal to 1.

From

∂g α ∂x γ = (-1) |γ| (α + γ)! α! g α+γ , it follows that (ix) β ∂g α ∂x γ = (-1) |γ| (α + γ)! α! F -1 ∂ |β| ∂ξ β (χ(ξ) (iξ) α+γ (α + γ)! ) .
Hence

x β ∂g α ∂x γ ∞ ≤ 1 α! ∂ |β| ∂ξ β (χ(ξ)ξ α+γ ) 1 .
Thanks to Leibnitz' formula,

∂ |β| ∂ξ β (χ(ξ)ξ α+γ ) = η+η ′ =β β η ∂ |η| χ ∂ξ η ∂ |η ′ | ξ α+γ ∂ξ η ′ = η+η ′ =β β η ∂ |η| χ ∂ξ η (α + γ)! (α + γ -η ′ )! ξ α+γ-η ′ , where β η = n j=1 β j η j and, by convention, ξ α+γ-η ′ = 0 if α + γ -η ′ / ∈ N n .
If β and γ are fixed, this is a polynomial in α of bounded degree |β| and bounded coefficients. So is its L 1 norm. Therefore

x β ∂g α ∂x γ ∞ ≤ P β,γ (α) α! ,
where P β,γ is a polynomial on R n . Let w(j) denote the weight of the j-th basis vector. For β ∈ N n , let w(β) = n j=1 β j w(j). Let g α,t = t w(α)+Q g α • δ t . Then, for all β, x β , g α,t = t w(α)-w(β) x β , g α . Hence, for every choice of sequence (t α ), the family (g α,tα ) α∈N n is again dual to the monomial basis. Also,

x β ∂g α,t ∂x γ = t w(α)-w(β)+w(γ)+Q (x β ∂g α ∂x γ ) • δ t , hence, for every 1 ≤ p ≤ ∞ and p ′ = p p-1 , x β ∂g α,t ∂x γ p = t w(α)-w(β)+w(γ)+ Q p ′ x β ∂g α ∂x γ p .
Given an arbitrary sequence m = (m α ) α∈N n , and a positive sequence t = (t α ) α∈N n , define the series

f m,t = α∈N n m α g α,tα .
If t α and |m α |t w(α)/2 α stay bounded by 1, then for every constant C, |m α |t w(α)-C α stays bounded, the series converges in S. Indeed, for every β, γ ∈ N n , the sum of L ∞ norms x β ∂ ∂x γ m α g α,tα ∞ is bounded above by

α∈N n |m α |t w(α)-w(β)+w(γ)+Q α P β,γ (α) α! < ∞.
By construction, all functions f m,t have prescribed moments x α , f m,t = m α . Fix a finite set F of pairs (β, γ) such that w(β) ≤ w(γ). Let W F,p denote the completion of smooth compactly supported functions for the norm

f W F,p = max{ x β ∂ ∂x γ m α g α,tα p ; (β, γ) ∈ F }. Denote by N F,p α := max{ x β ∂g α ∂x γ p ; (β, γ) ∈ F }.
Pick t such that, in addition to the previous assumptions, the series |m α |t w(α)+Q/p ′ α N F,p α converges. Then for every ǫ > 0, the series f m,ǫt converges in W F,p and for ǫ ≤ 1

f m,ǫt W F,p ≤ ǫ Q/p ′ |m 0 |N F,p 0 + ǫ α =0 |m α |t w(α)+Q/p ′ α N F,p α .
Therefore, as ǫ tends to 0, f m,ǫt tends to 0 in W F,p (if p = 1, one must assume that m 0 = 0). Given f ∈ W F,p (assume furthermore that f, 1 = 0 if p = 1 or that f ∈ C h 0 (G) if p = ∞), approximate f with an element g ∈ S (resp. such that g, 1 = 0 if p = 1). Set m α = x α , g . Pick t satisfying the above smallness assumptions with respect to m. Then g -f m,ǫt ∈ S 0 and f m,ǫt tends to 0 in W F,p , thus f belongs to the closure of S 0 .

Finally, f W h,p c ≤ f W F,p for a suitable finite set F . Indeed, G admits a basis of left-invariant vectorfields X i of the form ( 7)

X i = ∂ ∂x i + j>i P ∂ ∂x j ,
where P i,j is a δ t -homogeneous of weight w(P i,j ) < w(j).

One can now extend K c from S 0 to some Sobolev spaces, depending on the weights on the source and the target. Let W h,p c denotes the completion of S, and therefore S 0 , with respect to the norm

W h,p c . Corollary 7.2. Let a ≤ b ≤ a + h and a ′ ≤ b ′ ≤ a ′ + h ′ be integers. Suppose moreover that 1 < p < q < ∞ satisfy 1 p -1 q = µ Q with 0 ≤ µ = b -a ′ < Q and h = h ′ + b ′ -a ′ + b -a. Let (K c ) [a ′ ,b ′ ] denotes the components of K c lying in Ω [a ′ ,b ′ ] . Then K c extends continuously on Ω [a,b] ∩ W h,p c so that (K c ) [a ′ ,b ′ ] (Ω [a,b] ∩ W h,p c ) ⊂ W h ′ ,q c . Proof. Let α ∈ Ω [a,b] ∩ S 0 . Apply first Proposition 6.1 (ii) to (K c ) [a ′ ,b ′ ] (α) with a ′ , b ′ , µ and h ′ as above C (K c ) [a ′ ,b ′ ] (α) W h ′ ,q c ≤ b ′ +µ+h ′ m=a ′ +µ K c α |∇|,N -m,p = a+h m=b K c α |∇|,N -m,p ≤ C ′ a+h m=b α |∇|,N -m,p by Proposition 5.2, ≤ C ′′ α W h,p c by Proposition 6.1 (i) since α ∈ Ω [a,b] .
Remark 7.3. The statement becomes simpler when forms are smooth, meaning in W +∞,p , as those coming from ℓ p -cocycles on G. Namely in that case, one can let h, h ′ go to +∞ with b ′ = Q, a = 0. Then K c integrates smooth forms in W +∞,p and of weight lower than b, into smooth forms whose components of weight larger than a ′ lies in W +∞,q . Note that the (p, q) gap is only determined by the weight gap µ = b -a ′ using Sobolev rule.

Proof of Theorem 1.1(i)

Fix a left-invariant Riemannian metric µ on G. Let ∇ µ denote its Levi-Civita connection. Let ∇ g denote the connection which makes left-invariant vectorfields parallel. Since ∇ µ -∇ g is left-invariant, hence ∇ g -parallel, higher covariant derivatives computed using either ∇ g or ∇ µ determine each other via bounded expressions. Therefore the Riemannian Sobolev space W h,p can be alternatively defined using ∇ g , i.e. using derivatives along left-invariant vectorfields. Since every left-invariant vectorfield is a combination of compositions of at most r horizontal derivatives X i , where r is the step of g,

W h,p ⊂ W h,p c ⊂ W h/r,p . ( 8
)
According to Proposition 3.3, it is sufficient to find some large integer h such that every (usual

) closed differential form α ∈ W h,p Ω k (G) writes α = dβ with β ∈ W H,q Ω k-1 (G) for H = n n+1 + 1.
The relevant value of h will arise from the proof.

We first retract α in the sub-complex (E, d) using the differential homotopy Π E . Indeed from Theorem 4.1,

α = Π E α + dRα + Rdα = Π E α + dRα ,
where Π E and R are left invariant differential operators of (horizontal) order at most Q, the homogeneous dimension of g. Then Π E α, Rα and dRα all belong to W h-Q,p c , so that α and Π E α are homotopic in L p,p , and a fortiori L q,p Sobolev cohomology for q ≥ p (see Remark 3.2).

We now deal with We can now apply Corollary 7.2, with

α E = Π E α. Its algebraic projection on E 0 , α c = Π E 0 α E is a contracted d c -closed differential form in W h-Q,p c too. Since α c ∈ E k 0 ≃ H k (g),
µ = b -a ′ = w max (k) -w min (k -1) = δN max (k) and h ′ = h -Q + a -b + a ′ -b ′ . (Observe that µ < Q except for G = R in which case L 1 one forms have bounded primitives.) We get that β c = K c α c ∈ W h ′ ,q c with d c β c = α c and 1 p -1 q = δNmax(k) Q . Finally, let β E = Π E β c . Then β E ∈ W h ′ -Q,q c
since Π E is a differential operator of order at most Q. By construction, α E = dβ E . By inclusions [START_REF] Folland | Subelliptic estimates and function spaces on nilpotent Lie groups[END_REF], β E belongs to the Riemannian Sobolev space W (h ′ -Q)/r,q . Thus let us choose h = rH + 2Q + b -a + b ′ -a ′ . We have shown that every k-form in W h,p is homotopic in L p,p Sobolev cohomology to a form that has a primitive in W H,q , where

1 p - 1 q = µ Q = δN max (k) Q . Proposition 3.3 implies that ℓ q,p H k (G) = 0. A fortiori, ℓ q ′ ,p H k (G) = 0 for all q ′ ≥ q.
8.1. Remarks on the weight gaps. Implicit in the statement of Theorem 1.1 is that for any 1 ≤ k ≤ n, the weight gap δN(k) = w max (k) -w min (k -1) is positive. Actually, one has

(9) w max (k) -w max (k -1) ≥ 1 and w min (k) -w min (k -1) ≥ 1.
Proof. Since E 0 = ker d 0 ∩ ker δ 0 is Hodge- * symmetric with w( * α) = Q -w(α) (see e.g. [START_REF] Rumin | Differential geometry on C-C spaces and application to the Novikov-Shubin numbers of nilpotent Lie groups[END_REF]), one can restrain to prove the statement about w min . Let α ∈ H k (g) be non zero with minimal weight w min (k). See it as a left invariant retracted form in E k 0 . Following Section 4.1, one has then dα = d 0 α = d c α = 0. Now (E 0 , d c ) being locally exact (homotopic to de Rham complex), one has α = d c β for some β ∈ E k-1 0 . Since d c increases the weight by 1 at least (as d 0 = 0 on it), β has a non vanishing component of weight < w min (k), whence w min (k -1) ≤ w min (k) -1.

Another remark is about the use of the contracted complex here. As observed in Remark 5.3, de Rham complex being C-C elliptic too, one could directly use a similar homotopy K for it in the previous proof. But then, it would lead to a weaker integration result of closed W p c forms in W q c for a larger gap 1 p -1 q = δN Q . Indeed this δN is the maximal weight gap between the whole Ω k (G) and Ω k-1 (G), instead of the restricted ones. This gives a weaker vanishing condition of ℓ q,p H k (G). The point here is that the first homotopy Π E to E, being differential, "costs a lot of derivatives" that would be a shame locally, but is harmless here when working with the quite smooth W h,p c forms coming from our ℓ p simplicial cocycles. Indeed, we have seen that Π E = 1 in L p,p Sobolev cohomology.

Still about these ideas of "loosing" or "gaining" derivatives, things go exactly in the opposite direction as usual here. Namely, one sees in the proof that the more derivatives the homotopy K c controls at some place, the larger becomes the (p, q) gap from Hardy-Littlewood-Sobolev inequality in Corollary 7.2. This is of course better in local problems since L q loc gets smaller, but is weaker on global smooth forms as W q c gets larger. In this large scale low frequency integration problem, the less derivatives you gain is the better. No gain, no pain.

Nonvanishing result

As we will see, in order to prove the non-vanishing of the ℓ q,p and Sobolev L q,p cohomology of G for some 1 ≤ p ≤ q ≤ +∞, we shall construct closed k-forms ω ∈ W h,p (G), h large enough, and n -k-forms ω ′ j such that dω ′ j q ′ tend to zero whereas G ω ∧ ω ′ j stays bounded away from 0. Here q ′ is the dual exponent of q : 1 q + 1 q ′ = 1. The building blocks will be differential forms which are homogeneous under dilations δ t . In this section, one can use any expanding one-parameter group s → h s of automorphisms of G. Expanding means that the derivation D generating h s has positive eigenvalues. The one-parameter group s → δ e s is an example, but others may be useful, see below. Left-invariant differential forms on G split into weights w under h s . Say a smooth differential form ω on G \ {1} is homogeneous of degree λ if h * s ω = e sλ ω for all s ∈ R. Note that homogeneity is preserved by d. A left-invariant differential form of weight w is homogeneous of degree w Let ρ denote a continuous function on G which is homogeneous of degree 1, and smooth and positive away from the origin. Let β be a nonzero continuous differential form which is homogeneous of degree λ, smooth away from the origin and has weight w, then

β = ρ λ-w a i θ i ,
where θ i 's are left-invariant of weight w and a i are smooth homogeneous functions of degree 0, hence are bounded. Therefore

β ∈ L p ({ρ ≥ 1}) ⇐⇒ +∞ 1 ρ p(λ-w) ρ T -1 dρ < ∞ ⇐⇒ λ -w + T p < 0.
It follows that if γ is a differential form which is homogeneous of degree λ and has weight ≥ w,

(10) λ -w + T p < 0 =⇒ γ ∈ L p ({ρ ≥ 1}).
Start with a closed differential k-form ω which is homogeneous of degree λ and of weight ≥ w (and no better). Pick a differential n -k-form α ′ which is homogeneous of degree λ ′ . Assume that dα ′ has weight ≥ w ′ (and no better). Set

ω ′ j = χ j α ′ ,
where χ j = χ • ρ • h -j and χ is a cut-off supported on [START_REF] Bemelmans | Smoothing Riemannian metrics[END_REF][START_REF] Bourdon | Some applications of ℓ p -cohomology to boundaries of Gromov hyperbolic spaces[END_REF]. The top degree form ω ∧ α ′ is homogeneous of degree λ + λ ′ and has weight T . It belongs to L 1 if λ + λ ′ < 0. Thus, in order that ω ∧ ω ′ j does not tend to 0, it is necessary that λ + λ ′ ≥ 0. By construction,

ω ′ j = e λ ′ j h * -j ω ′ 1 .
Let β denote a component of dω ′ 1 of weight w, and s = -j. By the change of variable formula, h * s β q ′ q ′ = |h * s β| q ′ dvol = e q ′ ws |β| q ′ • h s dvol = e (q ′ w-T )s |β| q ′ dvol = e (q ′ w-T )s β q ′ q ′ . This works as well for q ′ = ∞. Since dω ′ 1 has weight ≥ w ′ ,

ω ′ j q ′ ≤ e (λ ′ -w ′ + T q ′ )j ω ′ 1 q ′ . One concludes that ω ′ j q ′ → 0 ⇐⇒ λ ′ -w ′ + T q ′ < 0.
It holds for q ′ = ∞ as well. Remember that ω ∈ L p ⇐⇒ λ -w + T p < 0. Note that

λ -w + T p + λ ′ -w ′ + T q ′ = λ + λ ′ -w -w ′ + T + T ( 1 p - 1 q ).
Finally, one sees that one can pick λ and λ ′ such that ω ∈ L p , ω ′ j q ′ → 0 and λ + λ ′ ≥ 0 iff [START_REF] Pansu | Cup-products in l q,p -cohomology: discretization and quasi-isometry invariance[END_REF] 1 p -1 q < w + w ′ -T T .

A numerical invariant of homogeneous groups.

A lower bound for the sum w + w ′ appearing in above inequation [START_REF] Pansu | Cup-products in l q,p -cohomology: discretization and quasi-isometry invariance[END_REF] is provided by the following definitions.

Let Σ denote the level set {ρ = 1}. It is a smooth compact hypersurface, transverse to the vectorfield ξ which generates the 1-parameter group s → h e s . Differential k-forms which are homogeneous of degree λ on G \ {1} correspond to smooth sections of the pullback of the bundle Λ k T * G by the injection Σ ֒→ G. Given such a section σ, a form α is defined as follows. At a point x where ρ(x) = e s , α(x) = h * -s σ. Conversely, given a homogeneous form α, consider its values σ along Σ (not to be confused with the restriction of α, which belongs to Λ k T * Σ). A similar construction applies to contracted forms as E 0 is stable by dilations.

On spaces of homogeneous forms of complementary degrees k and n -k and complementary degrees of homogeneity λ and -λ, define a pairing as follows: if β and β ′ are homogeneous of degrees λ and -λ, set

I(β, β ′ ) = Σ ι ξ (β ∧ β ′ ).
This is a nondegenerate pairing. Indeed, pointwise, an n-form ω is determined by the restriction of ι ξ (ω) to T Σ, hence the pointwise pairing (β, β ′ ) → ι ξ (β ∧ β ′ ) |T Σ is nondegenerate. For instance, one has β ∧ * β = β 2 dvol pointwise. Note that the n-form β ∧ β ′ is homogeneous of degree 0, i.e. dilation invariant. The n -1-form ι ξ (β ∧ β ′ ) is closed, the integral I(β, β ′ ) only depends on the cohomology class of this form. The boundary of any smooth bounded domain containing the origin can be used to perform integration instead of Σ. Definition 9.1. Let G be a homogeneous Lie group of homogeneous dimension T . For k = 1, . . . , n = dim(G), define ws G (k) as the maximum of sums w + w ′ -T such that for a dense set of real numbers λ, there exist (1) a differential form α of degree k -1 on G \ {1}, homogeneous of degree λ, such that dα has weight ≥ w, (2) a differential form α ′ of degree n -k on G \ {1}, homogeneous of degree λ ′ = -λ, such that dα ′ has weight ≥ w ′ , such that I(dα, α ′ ) = 0.

Note that for all k ≥ 1, ws G (k) = ws G (n -k + 1). For instance, when Carnot dilations are used, nonzero 1-forms of weight ≥ 2 are never closed, and n-forms are always closed and of weight T , hence ws G (1) = ws G (n) = 1. 9.3. Cohomology nonvanishing. Theorem 9.2. Let G be a homogeneous Lie group of homogeneous dimension T . Then ℓ q,p H k (G) = 0 provided 1 ≤ p, q < +∞ and

1 p - 1 q < ws G (k) T .
Proof. By assumption, ǫ = w+w ′ -T T -1 p + 1 q > 0. Pick a real number λ in the dense set given in Definition 9.1 and such that [START_REF] Rumin | Differential geometry on C-C spaces and application to the Novikov-Shubin numbers of nilpotent Lie groups[END_REF] w -

T p - T 2 ǫ ≤ λ < w - T p . Then λ ′ = -λ satisfies λ ′ -w ′ + T q ′ = -λ -w ′ + T q ′ ≤ -w + T p + T ǫ 2 -w ′ + T q ′ = - T ǫ 2 < 0. ( 13
)
By definition, there exist differential k-1 and n-k-forms α and α ′ , homogeneous of degrees λ and λ ′ , such that dα and dα ′ have weights ≥ w and ≥ w ′ . Then dα ∧ α ′ is homogeneous of degree 0. Using the notations of Section 9.1, for all j, dα ∧ χ

j α ′ = h * -j (dα ∧ χ 1 α ′ ), hence G dα ∧ χ j α ′ = G dα ∧ χ 1 α ′ = I(dα, α ′ ) R χ(t) dt = 0.
Since dα is homogeneous of degree λ and has weight ≥ w, it belongs to L p (away from a neighborhood of the origin) by ( 10) and [START_REF] Rumin | Differential geometry on C-C spaces and application to the Novikov-Shubin numbers of nilpotent Lie groups[END_REF]. Furthermore, derivatives along left invariant vector fields decrease homogeneity. Hence all such derivatives of dα belong to L p . After smoothing α near the origin, we get a closed form ω on G that coincides with dα on {ρ ≥ 1}, and which belongs to W h,p for all h. Set ω ′ j = χ j α ′ . Then G ω ∧ ω ′ j does not depend on j. Assume by contradiction that ω = dφ where φ ∈ W h,q . In particular, φ ∈ L q . Since ω ′ j are compactly supported, Stokes theorem applies and

| G ω ∧ ω ′ j | = | G dφ ∧ ω ′ j | = | G φ ∧ dω ′ j | ≤ φ q dω ′ j q ′
which tends to 0, as α ′ and dα ′ ∈ L q ({ρ ≥ 1}) by ( 10) and ( 13), contradiction. We conclude that [ω] = 0 in the Sobolev L q,p cohomology of G. According to Proposition 3.3, this implies that the ℓ q,p cohomology of G does not vanish. 9.4. Lower bounds on ws G . We give here two lower bounds on ws G . Combined with Theorem 9.2, they complete the proof of Theorem 1.1(ii) in the wider setting of homogeneous groups. We start with a lemma on the contracted complex. We see that Theorem 1.1 is sharp in degrees 1 and 4. However, there are gaps in degrees 2 and 3. In particular, H 2,q,p (E 4 ) vanishes when 1 p -1 q ≥ 3 7 and does not when 1 p -1 q < 2 7 , provided 1 < p, q < +∞. implies that ℓ q,p H 2 (E 4 ) = {0}. We see that a non-Carnot homogeneous structure may yield a better interval for cohomology nonvanishing, which is intriguing for a large scale geometric invariant.

1. 3 .Theorem 1 . 1 .

 311 Main result. Let G be a Carnot group of dimension n and of homogeneous dimension Q. Let k = 1, . . . , n. Denote by δN max (k) = w max (k) -w min (k -1), δN min (k) = max{1, w min (k) -w max (k -1)}.

Theorem 4 . 1 .

 41 [START_REF] Rumin | Differential geometry on C-C spaces and application to the Novikov-Shubin numbers of nilpotent Lie groups[END_REF] Theorem 1]

Theorem 5 . 1 .

 51 [START_REF] Rumin | Differential geometry on C-C spaces and application to the Novikov-Shubin numbers of nilpotent Lie groups[END_REF] Theorem 3],[START_REF] Rumin | Around heat decay on forms and relations of nilpotent Lie groups[END_REF] Theorem 5.2] The Laplacians ∆ ∇ and ∆ ∇ c have left inverses Q ∇ and Q ∇ c , which are zero order homogeneous pseudodifferential operators.

Proposition 6 . 1 .

 61 Let 1 < p < ∞ be fixed, and a, b, h ∈ N be such that a ≤ b ≤ a + h. Let Ω [a,b] denote the space of differential forms whose components have weights a ≤ w ≤ b.

  (i) It holds on Ω [a,b] that

p c using ( 6 )

 6 , (5) and |∇| µ+k = |∇| µ |∇| k on S for µ, k ∈ N as comes from [8, Theorem 3.15].

1 0,

 1 the weights of its components belong to the interval [a, b] where a = w min (k) and b = w max (k). Moreover, since K c α c ∈ E k-it belongs to Ω [a ′ ,b ′ ] with a ′ = w min (k -1) and b ′ = w max (k -1).

9. 1 .

 1 Homogeneous differential forms. Fix an expanding one-parameter group s → h s of automorphisms of G, generated by a derivation D. The data (G, (h s )) is called a homogeneous Lie group. Denote by T = trace(D) its homogeneous dimension.

Lemma 9 . 3 .Proposition 9 . 4 . 9 . 5 .

 939495 Let G be a homogeneous Lie group of dimension n, let k = 1, . . . , n. Then for an open dense set of real numbers λ, there exist smooth non d c -closed contracted k-1-forms on G \ {1} which are homogeneous of degree λ.Proof. The differential d c ≡ 0 on E k-1 0 , since the complex (d c , E * 0 ) is a resolution on G. Then, by the Stone-Weierstrass approximation theorem, their exist non d c -closed contracted forms with homogeneous polynomial components in an invariant basis. Pick one term P α 0 with α 0 ∈ E k-1 0 invariant, and a non constant homogeneous polynomial P such that d c (P α 0 ) = 0. Up to changing P into -P , pick x 0 ∈ G such that d c (P α 0 )(x 0 ) = 0 and P (x 0 ) > 0. Consider the mapF : λ ∈ C → d c (P λ α 0 )(x 0 ).Since d c is a differential operator, F is analytic. Since F (1) = 0, one has F (λ) = 0 except for a set of isolated values of λ. Let χ be a smooth homogeneous function on G \ {1} of degree 0 with support in {P > 0} and χ = 1 around x 0 . Then α = χP λ α 0 is a smooth non d c -closed homogeneous contracted form on G \ {1} of degree w(α) = λw(P ) + w(α 0 ). Let G be a homogeneous Lie group of dimension n. For all k = 1, . . . , n,ws G (k) ≥ max{1, w min (k) -w max (k -1)}.Proof. By Lemma 9.3, pick a non d c -closed contracted k -1-form α, homogeneous of degree λ. Assume that d c α has weight ≥ w and no better (i.e. its weight w component (d c α) w does not vanish identically). Pick a smooth contracted n -k-form α ′ of weight T -w, homogeneous of degree -λ and such that I(d c α, α ′ ) = 0. For instance α ′ = ρ -2λ+2w-T * (d c α) w will do. Set α E = Π E α and α ′ E = Π E α ′ . By construction (see Theorem 4.1), Π E = Π E 0 + D where D strictly increases the weight. Hence dα E -d c α = Π E d c α -d c α has weight ≥ w + 1, and α ′ E -α ′ has weight ≥ T -w + 1. Therefore dα E ∧ α ′ E -d c α ∧ α ′ has weight ≥ T + 1, thus vanishes. Then it holds thatI(dα E , α ′ E ) = I(d c α, α ′ ) = 0 .Consider now the weight of dα ′ E . By construction, E ⊂ ker d 0 , so that d strictly increases the weight on E, see Section 4. Thereforew(dα ′ E ) ≥ w(α ′ E ) + 1 = w(α ′ ) + 1 = T -w(d c α) + 1 = T -w(dα E ) + 1,hence ws G (k) ≥ w(dα E ) + w(dα ′ E ) -T ≥ 1 as needed. One has also thatw(dα ′ E ) = w(d c α ′ ) ≥ w min (n -k + 1) = T -w max (k -1),by Hodge * -duality, see proof of (9), whilew(dα E ) = w(d c α) ≥ w min (k) .This gives ws G (k) ≥ w(dα E ) + w(dα ′ E ) -T ≥ w min (k) -w max (k -1). An example: Engel's group. We illustrate the non-vanishing results on the Engel group E4 .It has a 4-dimensional Lie algebra with basis X, Y, Z, T and nonzero brackets [X, Y ] = Z and [X, Z] = T . One finds, see e.g. [13, Section 2.3], that H 1 (g) ≃ span(θ X , θ Y ) and H 2 (g) ≃ span(θ X ∧ θ Z , θ Y ∧ θ Z ) .The following table gives the values of δN max and δN min for E 4 with respect to its standard Carnot weight : w(X) = w(Y ) = 1, w(Z) = 2 and w(T ) = 3. One has Q =

Following [ 13 ,

 13 Section 4.2], let us also use the expanding one-parameter group of automorphisms of E 4 generated by the derivation D defined byD(X) = X, D(Y ) = 2Y, D(Z) = 3Z, D(T ) = 4T.Then trace(D) = 10, and with this choice of derivation, the table of weights becomes k 1 2 3 4 w max (k) 2 5 9 10 w min (k) 1 5 8 10 δN max (k) 2 4 4 2 δN min (k) 1 3 3 1 According to Proposition 9.4, with respect to this homogeneous structure, ws E 4 (2) ≥ δN min (2) = 3. Then with Theorem 9.2, 1 ≤ p, q < +∞ and 1

Both authors supported in part by MAnET Marie Curie Initial Training Network. P.P. supported by Agence Nationale de la Recherche, ANR-15-CE40-0018.


Justin Dirrenberger

Samuel Forest

Dominique Jeulin

Christophe Colin

Jean-Dominique D Bartout

Mathieu Faessel

François Willot

Effective properties of architectured materials

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

The MANSART project, for "architectured sandwich materials", aims at developing innovative architectured materials, exhibiting improved functional properties in comparison with traditional composites. Our contribution is focused on modeling and simulation of architectured microstructures, prediction of their effective properties using numerical homogenization implemented with finite elements (Z-set software). Rapid prototyping using selective laser melting allows us to fabricate samples and validate the numerical prediction. Two types of morphologies have been identified: randomly distributed entangled Poisson's fibres and auxetic periodic lattices. Effective properties of Poisson's fibres is investigated although estimation is challenging since statistical representative volume element (RVE) size is a priori unknown, due to an infinite integral range as shown in [Jeulin, 1989]. Scaling laws for variances of properties are computed. Auxetics are architectured materials exhibiting a negative Poisson's ratio. Mechanical behavior of such materials is investigated for both elastic and plastic domains. 2D and 3D lattices are proposed for industrial applications.

Keywords: Homogenization, architectured materials, auxetics, Poisson's fibres, RVE, finite elements, mathematical morphology, selective laser melting.

Determination of statistical RVE and effective properties for Poisson's fibres

Conclusions and prospects

-Modeling and homogenization tools, in association with the set of goals for the MANSART project, resulted in the development of architectured materials likely to replace aluminium honeycombs currently serving as core for sandwich composites intensively used in the aerospace industry. The non-linear static and dynamic mechanical behavior of such materials has yet to be characterized using FEM and experiments; indentation and acoustic damping tests will be conducted in order to bring out interesting functional properties for auxetics, due to their high shear modulus. Quasi-isotropic 3D auxetic lattices shall be developed. -The study of Poisson's fibres highlighted a scaling law for the volume fraction variance proportional to V 2 3 . Efforts are being made to obtain scaling laws concerning elastic moduli, simulations with more fibres are needed. Statistical convergence is slower; for the same precision, Poisson's fibres' RVE is larger than for a classical Boolean variety. Future work on these microstructures will include: static uniform boundary conditions (SUBC) calculations, larger simulation size, mesh rationalization, domain decomposition, thermal homogenization and Fast Fourier Transform computations.

-

  Modeling straight infinite randomly entangled interpenetrating Poisson's fibres with intensity λ such as, λ = ln(1-V V ) π 2 R 2 with R fibres radius and V V volume fraction of fibres.-Optimization of surface (vtkSim, Yams) and volume meshes (GHS3D) for finite element computation.-Estimation of RVE size based on method established in[Kanit et al., 2003].-Simulation of samples with an increasing quantity of fibres; scaling laws are computed.-Homogenization using mixed, kinetic (KUBC) and static uniform boundary conditions (SUBC). N = 8 N = 32 N = 71 N = 127 Homogenization and mechanical investigation of auxetic lattice structures -Modeling of original auxetic periodic cells and examples from the literature. -Numerical homogenization (FE) using periodic boundary conditions with imposed macroscopic stress or strain. -Analytical representation of elastic moduli and Poisson's ratio in 3D. -Design and manufacture of 3D quasi-isotropic auxetic lattice structures. Auxetic lattice from the literature (hexachiral) ν = -0.83, V V = 7% [Prall and Lakes, 1997] Original transverse isotropic chiral structure ν = -0.25, V V = 7% [Dirrenberger et al., 2011] Cubic anti-chiral structure from the literature ν = f (φ), V V = 7% [Alderson et al., 2010] Elastic moduli as function of φ for the anti-chiral cell Original 3D periodic auxetic cubic cell (hexatruss) Elastic moduli as function of φ for the hexatruss cell ε p ν 12 Plastic Poisson's ratio ν 12 vs. cumulative plastic strain ε p for a hexachiral cell for an imposed macroscopic stress. ε p map for an imposed macroscopic strain E 11 = 0.1 of an auxetic microstructure from [Milton, 1991] Direct fabrication using selective laser melting and experimental validation -Direct fabrication from alloy or polymer powders -Compressive, 3 and 4-points bending tests -Image correlation measurements -Comparison between FEM and experiments -Available process for production of microarchitectured materials in the industry (aerospace, automotive, biomedical, etc.) ⇒ 316L steel hexachiral lattice sample made with SLM ⇒ 4-points bending test of PA 12 re-entrant honeycombs

[

  Alderson et al., 2010] Alderson A. et al. (2010). Composites Science and Technology, vol. 70 (7), pp 1042-1048. [Dirrenberger et al., 2011] Dirrenberger J., Forest S., Jeulin D., Colin C. (2011). ICM11, Procedia Engineering, in press. [Jeulin, 1989] Jeulin, D. (1989). Modèles de Fonctions Aléatoires Multivariables, Ecole des Mines de Paris, Centre de Géostatistique. [Kanit et al., 2003] Kanit T., Forest S., Galliet I., Mounoury V. and Jeulin D. (2003). International Journal of Solids and Structures, vol. 40, pp 3647-3679. [Milton, 1992] Milton G.W. (1992). Journal of the Mechanics and Physics of Solids, vol. 40 (5), pp 1105-1137. [Prall and Lakes, 1997] Prall D. and Lakes R. S. (1997). International Journal of Mechanical Sciences, vol. 39 n • 3, pp 305-314. Contact : justin.dirrenberger@mines-paristech.fr CENTRE DES MATERIAUX P.M FOURT


François Willot

Sofiane S Guessasma

Dominique Jeulin

Guy G Della Valle

Anne-Laure A.-L Reguerre

Fast solution to determine the elastic behavior of 3D reconstructed biopolymer cellular structures

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.


Tania Dukic Willstrand
email: tania.willstrand@vti.se

Thierry Bellet
email: thierry.bellet@ifsttar.fr

Thomas Broberg
email: thomas.broberg@volvocars.com

Christina Stave
email: christina.stave@vti.se

Jean-Christophe Paris
email: jean-christophe.paris@ifsttar.fr

Björn Peters
email: bjorn.peters@vti.se

Claude Marin-Lamellet
email: claude.marin-lamellet@ifsttar.fr

What ADAS are the most promising for our future older drivers? Evidences reported from France and Sweden

Keywords: older drivers, focus group, driving support, navigation, intersections, speed, ADAS. 

Focus groups were conducted in both France and Sweden as part of the SAFEMOVE project. The aim of the study was to identify and asses difficulties experienced by older drivers (+70) due to agerelated declines in sensory, physical and cognitive abilities and potential consequences in terms of both traffic safety and mobility. Furthermore, the aim was to identify Advanced Driving Assistance Systems (ADAS) liable to improve safe mobility for the target group and to compare the situation between France and Sweden. Three main topics investigated were trip planning and navigation task, speed control and regulation, and intersection crossing (more particularly when turning on the left). For each one, data collected focused on both older drivers' experienced difficulties and their interests or expectations towards driver support like ADAS. There was in general a positive attitude to driver support systems but participants were also concerned about costs. Furthermore, several differences between French and Swedish older drivers were found.

INTRODUCTION

During the last decades, more and more attention has been devoted to the older drivers' situation. The older drivers are usually experienced and careful drivers, without taking overestimated risks [START_REF] Langford | Older drivers do not have a high crash risk--A replication of low mileage bias[END_REF]. However, aging is frequently associated with some declines in sensory, physical and cognitive capacities. For instance, [START_REF] Vrkljan | Advancements in vehicular technology: potential implications for the older driver[END_REF] indicated that changes in the visual and cognitive systems may reduce the ability to drive a car safely. Physical abilities like neck flexibility can also affect various aspect of driving performance. For [START_REF] Baldwin | Designing in-vehicle technologies for older drivers: application of sensory-cognitive interaction theory[END_REF] age effect on information processing could be linked to an interaction between cognitive and sensory changes: if vision is affected by aging in all its components, it is higher order visual processes including dynamic visual acuity and the simultaneous use of central and peripheral vision which have been shown to have some correlation with accident rates. Audition is also affected by aging, notably pure tone detection and speech comprehension. As a consequence, a minimum of 10 Db increasing is requested by older people for an auditory presentation if they are involved in a multiple task in comparison to younger drivers. From Koppel et al (2010), it appears that when cognitive load increases, processing speed of older people decreases. This could be the result of slowed integration of relevant information, limited memory and attentional capacities as well as a decline in executive functions, high order cognitive processes like planning, inhibition of irrelevant information, mental flexibility, judgement and self-adaptation. Then older drivers could be more subject to distraction than other drivers because they have less spare attentional capacity, and multitask can be problematic because of the reduction of time-sharing skills. Another set of studies [START_REF] Bolstad | Situation awareness: does it change with age?[END_REF][START_REF] Bailly | Conscience de la Situation des conducteurs: aspects fondamentaux, méthodes et applications pour la formation des conducteurs[END_REF][START_REF] Kaber | Prithima Mosaly[END_REF] also showed negative age effects on drivers' Situation Awareness (SA). Situation Awareness was defined by Endsely (1995) as 'the perception of elements in an environment within a volume of time and space, the comprehension of their meaning, and projection of their status in the near future'. In the frame of car driving activity, SA corresponds to mental representation of the road environment, as perceived and understood by the driver [START_REF] Bellet | A theoretical and methodological framework for studying and modelling drivers' mental representations[END_REF]. Through a simulator experiment, [START_REF] Bolstad | Situation awareness: does it change with age?[END_REF] showed that, due to age-related declines in perceptive and cognitive functions, older drivers' mental models of road environments are less complete of SA elaborated by youngest experienced drivers. Similar results were also found by [START_REF] Bailly | Conscience de la Situation des conducteurs: aspects fondamentaux, méthodes et applications pour la formation des conducteurs[END_REF] in normal driving conditions and, more recently, by [START_REF] Kaber | Prithima Mosaly[END_REF] for hazardous situations: age negatively impacts the content and the adequacy of SA, more particularly for complex driving situations requiring high attentional demands. Facing the complexity of some traffic situations, older drivers may have difficulties to perceive and integrate all the relevant pieces of information required to make the right decision, resulting in higher involvement in certain types of accidents. This is typically the case for accidents occurring at intersections with crossing traffic [START_REF] Larsen | Multidisciplinary in-depth investigations of head-on and left-turn road collisions[END_REF][START_REF] Preusser | Fatal crash risk for older drivers at intersections[END_REF]. [START_REF] Braitman | Factors leading to older drivers' intersection crashes[END_REF] reported that failure to yield was found in more than half of the crashes involving drivers +80 in a US-study. These crashes occur more often when drivers are turning left and at stop signs. [START_REF] Braitman | Factors leading to older drivers' intersection crashes[END_REF] attribute this fact to misjudgements of the time needed to proceed through the turning manouver; moreover, they also reported that "look but failed to see" errors were predominant among drivers +80. Based on these characteristics, the development of driving assistance systems might be an option to support older drivers and prolong their time as active safe drivers [START_REF] Simoes | Road users who are elderly: drivers and pedestrians[END_REF], [START_REF] Davidse | Older drivers and ADAS-which systems improve road safety?[END_REF]). To support older drivers, the driving aids have to be adapted to their characteristics, needs and expectations in order to be efficient and well accepted.

The overall aim of this focus group study conducted within the SAFEMOVE project was to acquire knowledge about the difficulties older drivers have when they drive their car in everyday life both in France and in Sweden to compare findings. A second aim was to identify existing or future Advanced Driving Assistance Systems (ADAS) capable in supporting older drivers and liable to help improve their mobility in a safe way. A first step to address safe mobility in the SAFEMOVE project was through a large survey reaching a total of 4200 persons in France and Sweden together. Onroad observations with sub-sets of the survey respondents were performed in both countries as a next step. During the on-road experiments, participants drove a car in real traffic conditions on a specific route. A driving instructor was in the car to ensure safety and an occupational therapist was responsible to identify problematic situations and behaviours observed for the drivers. Based on the experience gained from the survey and the on-road experiments, complemented with a literature review (Chin et al, 2014), three common areas of interest were identified to be further investigated among older drivers through focus groups in Sweden and France [START_REF] Stave | Older drivers' needs for safety and comfort systems in their cars -A focus group study in Sweden VTI Notat 31A-2014[END_REF][START_REF] Bellet | Older drivers' needs and priorities for DVIS functionality (in french)[END_REF].

METHOD

Even if the approaches applied in France and Sweden were similar, there were some methodological specificities in each country concerning the way the focus group sessions were carried out, the type of data that was collected and the analysis performed by each team.

Focus groups

France: Focus group sessions as implemented in France, were organised around a set of main topics, briefly introduced by the experimenters, and then sub-divided in a sub-set of questions to be successively discussed by the participants. Data collected were both individual opinions per question and then common discussions between the participants dedicated to each main topic.

Sweden: For the Swedish focus group sessions, using an explorative approach, quality rather than quantity was the lead word striving for consensus where counting frequencies or categories was not of interest. Focus group interviews support interaction amongst participants, which can enrich data in a way individual interviews cannot [START_REF] Kitzinger | The methodology of focus groups: the importance of interactions between research participants[END_REF].

Recruitment of participants

France: 30 older drivers participated in the focus group sessions. All the French participants were recruited among a group of 76 older drivers who previously participated in the SAFEMOVE on-road driving experiment. These 76 participants were initially selected from a face-to-face survey (including interview, questionnaires and cognitive tests) implemented among 800 older drivers living in the Rhône area. Six sessions of five participants (from 4 to 6) were organized.

Sweden: The focus groups were conducted at two occasions, 2011 and 2013. The participants in the first round were recruited through two different national pensioners' organisations. The second round of participants was recruited among the 40 participants in the on-road assessment conducted in SAFEMOVE [START_REF] Henriksson | Challenging situations, self-reported driving habits and capacity among older drivers (70+) in Sweden. A questionnaire study[END_REF]. In total 63 drivers living in the Gothenburg area participated. They were 70 years or older, had a valid drivers' licence and were still active drivers.

Procedure and themes

France: focus group sessions were organised in 7 "Blocks", each one dedicated to a specific topic, jointly defined with the Swedish team. The themes derived from the previous studies of [START_REF] Broberg | Safe mobility for elderly drivers-Considerations based on expert and self-assessment[END_REF] and [START_REF] Peters | On Road Assessment in Sweden -First Results[END_REF] where difficulties were observed during the on-road assessment performed in the SAFE MOVE project. The first one was dedicated to trip planning issues and the navigation task. The second topic concerned speed control and speed adaptation strategies of older drivers according to the traffic condition and the situational context, including their awareness of speed limit when driving. The last topic focused on intersection crossing issues, more specifically when turning to the left, involving complex interaction with other roads users. Each theme was organised in two parts: one focused on the difficulties met by the participants when driving and the second one was dedicated to their use and/or expectations regarding ADAS and future driving aids (Fig 1). Finally, each of these two parts were supported by a set of questions to be successively answered by the participants (around 12 questions per block). For each question, participants had first to provide an individual judgement (by using a Likert scale). Then, after each part (i.e. half-block), a common discussion between the participants was implemented under the moderation of the experimenters. The total duration of a session was half a day (4 hours), including two 15 minute breaks. Sweden: A structured interview guide was used where the three pre-defined themes were introduced to get the discussion going (see the French description earlier). As an introductory "warm up" topic, participants were asked what kind of driver assistance or safety technology they have in their cars today. The moderator introduced themes and facilitated the discussion among the participants. The discussion last for two hours including a short break.

Analysis

France: Concerning the individual opinions collected among the participants based on Likert scales, they had to provide numerical values (from 0 % to 100 %) on an individual paper grid (one numerical value was collected for each question, among each participant). Concerning the common discussion phases (coming after each part of 6-10 questions), they were video recorded and notes were taken by the experimenter. Analysis implemented were primarily focused on individual opinions, and collective discussion were mainly used to assess if common consensus exist or not between the participants or if several contradictory point of views appeared.

Sweden: All focus groups were audio recorded and notes were taken. After each focus group, a debriefing was performed in order to see if the researchers had a common understanding of what they had just heard. Then, the researchers who performed the focus groups listened to the recordings to extract answers, first individually and secondly together. For the three pre-defined themes covered as well as the introductory question, a simple form of content analysis was performed.

RESULTS

Trip planning and navigation task

Difficulties for Navigation in France: The majority of older drivers participating did not experience navigation as a very difficult task. Indeed, for a large part of them, they primarily drive their car on familiar routes during their everyday life, and they can generally adapt their trip in order to avoid unfavourable driving conditions (like bad weather, night driving, or high congestion time). In addition, they generally have a good confidence in their own ability to navigate when driving. If they have to perform the driving task on unfamiliar routes, the navigation task is assessed as significantly more difficult. In this case, they spend time to carefully plan their trips. More than half of them know and frequently use web services like Mappy or ViaMichelin, and they print the trip information to support them during the travel. All of them also have and use classical paper maps. However, despite this they declared a high level of confidence in their navigation skills, it should be noted that 4 participants tracked the trip for going to IFSTTAR, the day before the experiment.

Difficulties for Navigation in Sweden: The majority of the participants agreed that every activity take more time nowadays. They spend more time to plan their trips. They expressed a need to plan even for shorter trips, specifically if it concerned new destinations. However, generally it is not a big problem as they do not visit new locations very often and they tend to use the well-known roads as they have done for many years. They actually have more need to reroute when a sudden change occurs such as traffic accidents or a roadwork. They experience trouble when rerouting spontaneously in such situations. Planning a trip is about creating large margins to stay inside their comfort zones. They often take into account possible information about weather, distance, the traffic situation, parking, time of the day, specific roads or places to avoid, different kinds of breaks (rest, sleep, food, refuel). They try to avoid driving in darkness, especially if there is a risk of being blinded by headlights and if it is winter conditions. Aid in France: Two thirds of the French participants declared they have a Navigation system. Moreover, a large part of them frequently use it, also for familiar routes. This high rate of equipment and use is partially due to the French law on speed limit. Indeed, the most popular Navigation systems in France are also equipped with warning functions informing the driver of the current speed limits (including radar detection until 2011), and several drivers primarily activate their GPS to better control their speed and to avoid speed tickets. In a general way, a majority of the participants have a positive opinion about GPS, in terms of both perceived utility and acceptability, even if they also sometimes detect some dis-functioning or sub-optimal route suggestions provided by their GPS. However, advanced HMI, like head up display or augmented reality technologies (presented to the participants from illustrative pictures during the focus groups), were assessed by seniors as very promising and liable to better support their own needs against the existing systems.

Aid in Sweden: Participants agreed that GPS is a solution for the future that will be good for route planning, travel time and refuel information and more. However, even if some participants tell they had a navigation system it was not clear how often they use them. However, to increase their use, they requested to have better interfaces and perhaps voice guided navigation. Traffic information through radio was also considered a good source of information to plan or reroute. Traffic information regarding accidents on the road is automatic and sent to your radio automatically. Regardless if the radio is on or not it will stop broadcasting and send warnings regarding accidents, their location and measures to avoid the black spot if possible. Older drivers using cell phones stated that "the cell phone gives a sense of security and you can call for help and communicate if you have any problems". Some participants would like to see better visible road signs that can be seen at a longer distance in complex traffic locations.

Speed control, regulation and adaptation

Difficulties in France: The main topic investigated during the Focus Groups in France regarding speed control was the difficulties the participants have (or not) to respect the legal speed limits. First at all, it must be noted that this issue appeared as one of the highest difficulty, and probably the main concern of older drivers in France, when they have to drive their car in their everyday life. This fact is surely due to strict French laws applied since 2005 regarding speed limit violations, and to the massive deployment -during the last ten years -of automatic speed-controls radars (speed ticket of 90 Euros and the lost on 1 point /12 since the first km/h exceeding the legal speed limit). As a consequence, the majority of the French older drivers are fully aware of the high probability to be controlled for their speed during driving (several of them received a speed ticket during the last 3 years), and they pay high attention to control and regulate their speed under the legal limit. In addition, the more the route they use is unfamiliar (i.e. difficulty to know the current speed limit) and/or includes frequent changes of speed limit (like short section of 70-km/h on rural road or urban highway, generally limited to 90-km/h, or 30-km/h sections in urban areas), the more they have to pay attention to regulate their own speed, and the more they have difficulties to adequately perform the driving task (however, proportionally, the 30-km/h sections generates more difficulties among men, against 70-km/h sections among women).

Difficulties in Sweden: There was a general high acceptance for driving too fast and the participants seldom discussed that their own high speed would be a problem. Quite the contrary, many considered people driving slowly as more dangerous, possibly causing dangerous situations. The participants felt that they have to follow the traffic flow even if it means driving too fast or driving over their own ability. However, they mentioned that it was important to respect 30-km/h signs posted at schools. Driving slowly wasn't perceived as good driving practice. The ideal driving was driving fast in a more offensive way, which indicates that you are a skilled driver with great control of the car and with an ability to quickly and accurately scan and perceive all road users. It creates a problem for those who need more time since the worst thing that could happen, according to the participants, is to obstruct the traffic flow. Some difficulty in speed adaptation was noticed during the on-road driving in SAFE MOVE, i.e. driving too fast in certain scenarios such as down town where many pedestrians and bicyclists are present. You do not necessarily exceed speed limit but you do not drive with a safe speed in this particular situation. Moreover with age you need more time to scan the traffic, to perceive other road users, especially bicyclists which suddenly appears, and at last to react to sudden event. Therefore recommendation are to decrease speed in order to give more time to react. There were many different explanations as to why they were driving too fast. Difficulties in knowing the actual speed limit was one. Another reason often mentioned was the traffic rhythm, i.e. you have to follow the flow to keep your position in traffic, and it is perceived as more important than to respect the speed limit. The participants mentioned that if you do not follow the traffic flow, dangerous overtaking situations could appear. In addition, it was specifically mentioned that it was important to respect 30-km/h signs posted at schools. Aid in France: Regarding driving assistance, one of the main results collected during the focus groups is that half of the participants have, and frequently use, a driving aid to support them in their speed regulation. The main systems used are the automatic cruise control, information or warning systems provided by a GPS, and the speed limiter. Regarding the perceived utility and the acceptance level of such driving aids, it must be noted that devices providing information (i.e. current speed limit) and warning (in case of over-speed) received highest values among the participants as a whole, against active ADAS based on partial automation (like speed limiter, cruise control, or adaptive cruise control).

Aid in Sweden: Aid mentioned were more signs to frequently remind drivers about the speed limit; an alternative considered was having a display in the car to inform the driver about the current speed limit. Many participants were positive to signs in the infrastructure that give feedback on actual speed. Cruise control and navigation system were also mentioned as means. They also mentioned 2+1 roads or highways where it is possible for other drivers to overtake safely so that they could keep driving at their own pace "without being in the way" for other road users. Infrastructure solutions such as speed bumps were suggested to help control speed but were also noted as negative if not correctly designed or easy to detect.

Intersection crossing and left turn

Difficulties in France: Intersection crossing, more particularly when they have to leave the priority to other road users, was assessed by the French participants as a potential difficult task during driving. Nevertheless, this task was assessed as less difficult than overtaking a vehicle, to respect the speed limits, or to merge on to an urban highway. The main difficulties they have are about identifying the priority rules in unfamiliar intersections, and to make decision for turning to the left at urban crossroadsmore particularly to adequately assess in this last case the speed of incoming vehicles. In addition, difficulties for crossing intersections were assessed as significantly higher by women, compared to men. This was the case for all the types of crossroads and manoeuvres investigated during the focus groups.

Difficulties in Sweden: Most of the participants agreed that intersections could be difficult, especially when turning left on rural roads. "I feel stressed when the traffic is pushing from behind, trying to find a gap ahead while determining the speed of the oncoming cars". "There are also risks with those who are close behind, are there any driver assistance systems for that?" one driver asked reflecting on several comments regarding the problem with those who don't keep a safe distance. They also blamed other road users (drivers, pedestrians and cyclists) for not giving way, which make them confused and create difficulties "It is mostly young people who drive very fast into the intersections and brake hard; you cannot trust that people will stop at the intersection".

Aid in France: For the French focus group sessions, driving aids to assist intersection crossing were primarily focused on the Left Turn (LT) manoeuvre. To support the discussions, three type of LT aids were first presented and explained indepth to the participants: (1) a perception support system, providing information on approaching cars (having the priority), specifically regarding their distance and their speed, (2) a decision support system, assessing gaps and giving the time drivers have to safely cross the opposite lane, and (3) a warning support system, designed to detect potential collision risks and then to alert the driver in case of an erroneous decision or a critical behaviour. Data collected among the participants (having to give their opinion on each system and then to establish a hierarchy between the 3 devices) clearly showed the highest levels of interest, acceptance and perceived utility for the last warning support system. Then, the perception support system was assessed as very useful by women (close to the Warning system), but more poorly accepted by men. Finally, Decision support system (LT gap assessment) received the lowest judgement values from our participants as a whole.

Aid in Sweden: Automatic road signs actually showing that a cyclist is approaching were given as an example of a good solution. Another suggested solution was to drive through an intersection or roundabout with more care, looking more often to the sides and also using the rear mirrors, being more attentive and careful or trying to avoid those locations in the first place. One person suggested "having good foresight and automatic transmission so that no attention is spent on changing gears". From the infrastructure side, one could implement better and more signs to help inform about the way so that it becomes easier to plan the driving. One could also replace dangerous intersections with roundabouts which are highly appreciated and perceived as safer compared to other crossings. To build separate lanes to be able to turn left in a safe way without being in the way for other road users or to introduce traffic islands are other ways of improving infrastructure. Some participants stated that the visibility in new cars has been reduced and better mirrors could be a possible improvement. A slower pace in traffic as a whole was requested by many.

DISCUSSION

The aim of the present study was two folded: first, to identify potential areas were driver support system could contribute to an improved safe mobility for older (+70y) drivers and to identify possible communalities and differences among French and Swedish older drivers. On one side, similarities may indicate more universal difficulties when driving, or similar expectations towards driving aids liable to be shared by all older drivers living in different European countries. By contrast, differences may reveal some cross-cultural specificities between Swedish and French drivers that could be considered to adequately assist them or to support their respective needs for driving aids.

Concerning the demographic issues of older drivers, there has been a similar change in the two countries during the last decades. In Sweden, the proportion of license holders among older citizens and particularly among women has increased sharply in recent decades. Today, 83 percent of all those who are 67 years or older have a driving license, which can be compared with the share of 30 percent in 1980. Among women 67 and older, the proportion of the license increased from ten per cent in 1980 to 73 per cent in 2013 (facts from BilSweden). In a similar way, the number of older drivers also considerably increased in France during the last decades [START_REF] Marin-Lamellet | Les aînés : quels enjeux pour les décideurs publics? Contribution au débat national[END_REF]. This evolution is due to the increasing number of French households having a car (from 64 % of households in 1994 against 85 % in 2008), but also to the highest number of women who have a driving licence and who drive a car (42 % in 1982, 64 % in 1994 and 76 % en 2008). Despite these demographic similarities, there are differences in special driving conditions for Sweden and France which should be mentioned. For instance, the traffic situation is dramatically different in Sweden compared to France during winter. The poor weather conditions during winter with a long period of darkness, presence of snow and black ice in many places, significantly reduces driving for Swedish older drivers during wintertime. In France, and more particularly in the Lyon area, weather conditions are generally better and older drivers may drive to a larger extent all the year round. Other differences are related to specific traffic laws in each country. An example, recent changes in Sweden and in France regarding speed limits have influenced older drivers differently and their concerns towards speed limit respect for each country (see 4.2.).

Trip planning and navigation task

For older drivers in both countries planning the trip and performing the navigation task, especially on unfamiliar roads, is a situation of concern where the participants welcome support systems. However, it is generally not a major problem for them because they primarily drive in familiar places using well-known roads. In other cases, they spend more time carefully planning their trips before travelling (from map or web services). They often plan their trip according to weather, distance, or traffic conditions. Furthermore, they also consider places to avoid. Both Swedish older drivers and a large part of French older drivers (specifically women) avoid driving during night, especially if there is a risk of glare by headlights of other vehicles and if there is bad weather conditions.

Experience of navigation aid systems for support in this situation seems more positive in France than in Sweden, probably because French drivers have more experience in using GPS-navigation systems. Partly, because they have a specific concern to avoid speeding tickets. Moreover, both French and Swedish drivers pointed out the need to design a better interface for the GPS navigation system (to avoid visual distraction or to facilitate GPS information matching with the road environment) and other support to really use them and avoid irritation. This comment is also common for all support systems where they do not want to be disturbed or overloaded too often by a visual or an auditive interface. In a recent study, tactile-based navigation system was tested with hearing impaired drivers (common impairment among older drivers) and normal hearing drivers, with very positive results for both groups [START_REF] Thorslund | Hearing loss and a supportive tactile signal in a navigation system: Effects on driving behavior and eye movements[END_REF].

Speed control, regulation and adaptation

For older drivers in both countries, there is a perceived difficulty to respect speed limits although the reasons are not completely similar. To be aware of the speed limit is a common explanation in both countries where they do not feel to have sufficient support today. Results in Sweden also show that the traffic rhythm is contributing to excessive speed. Regarding aid for speed control, drivers in both countries wants to get feedback and information on their own speed and the speed limits, from the vehicle. Despite these similarities, inter-country differences also appear clearly in the focus group data. These differences seem directly related to specific traffic laws in each country. For instance, in 2011, the Swedish authorities made a reform regarding speed limits where several new speed limits were implemented giving that signs posting 30, 40, 50, 60, 70, 80, 90, 100, 110 and 120 km/h are present on the roads. As a consequence, driving in Sweden requires frequent updating of speed limit from a large set of possible values, and it is not always easy for older drivers to be fully aware of the speed limit on the road section where they are currently driving. For France, the recent development and the massive deployment of automatic speed-control radars since 2005 on the French roads, have considerably impacted drivers' behaviour and their concern towards speed limits. Around ten percent of the older French drivers interviewed during the focus group had received a speed ticket during the past 5 years, and a large part of the others feel highly concerned by this potential risk. This concern is clearly due to the automatic "contrôle-sanction" policy in France that occurred during the last decade. There is no tolerance with the new radars that are able to automatically generate a speed ticket of 90 Eurosand the fact that you loose one point on a 12 point-scale-from the first km/h exceeding the legal speed limit. As a consequence, knowing the legal speed limit on the road section they drive is a major concern to older French drivers, and they allocate a lot of their capacity to the speed control task. In addition, several of the French drivers decided to equip their car with a GPS system including warning in case of speed-limit violation. This fact may explain the higher level of experience and the more frequent use of GPS navigation systems by French older drivers compared to Swedish.

Intersection crossing and left turn

The common difficulties at an intersection was to leave priority, especially when turning left, to other vehicles and at the same time find a gap to turn, assess speed of in-coming vehicle and manoeuvre the own vehicle. Both French and Swedish drivers were interested to have support in this task.

They were both open to have systems inside their car and also proposed road infrastructure solutions such as to remove "left turn intersections and introduce roundabouts instead. The results in Sweden and in France showed that no participants knew of any driver assistance system that could help them in intersections but confronted with such a possibility during the focus group, they were very positive. In France, intersection crossing in general, and left turn manoeuvre in particular, are assessed as more difficult by women. Moreover, by considering the left turn manoeuvre successfully implemented by the French participants during the on-road experiment, it seems that some of the older drivers underestimate the difficulties they have to adequately perform the task. They indicated during the focus group that is not an easy task to perform, but they don't feel "highly" concerned about this situation (compared with speed limit respect, for instance). In contrast, several critical situations occurred during the on-road experiment when they had to turn left at urban crossroads with incoming traffic, and speed limit was generally well respected.

Method

Even though extracted from a larger cohort of older drivers in both countries all participants in this study were volunteers. Generalizing the application of the findings to older drivers in general is therefore to some extent restricted due to possible bias (i.e. those with most severe problems are less likely to attend). More so, it showed difficult within the focus groups to find ideas of specific solutions for driver aid. However, the focus groups made it possible to identify problems and needs for support to older drivers. The next step would therefore be to provide representatives of the target group with examples of technical solution they might not have any idea existed. One such next step within the SAFEMOVE project will be to create demonstrators of driver assistance concepts in vehicle and driving simulator environment for further studies using the results from the focus group as input.

CONCLUSION

Amongst both the French and Swedish drivers, there was a great variation in experience of own assistance technology but generally a positive attitude despite a fear of costs. Car manufacturers have to take the economic situation of older drivers into account when pricing the systems and designing the safety package as an option. When the French & Swedish participants were asked about future ADAS, they were positive if it increased safety. Information regarding ADAS effect has to reach the public and future users. By knowing the existence and understanding their effect, it would increase the penetration rate, which is still very low, and consequently has an effect on safety. ADAS systems have the potential to increase mobility for older drivers. The use of systems can delay driving cessation by maintaining the task difficulty within their ability. Future ADAS development has to include all travels with an holistic approach, from planning the travel to returning back home. New planning tools have to be designed to plan the travel before sitting in the vehicle. Information has to be available during driving to increase situation awareness and increase perception when needed such as during turning left manoeuvre. Feedback during and after the drive has to be available as a training tool for the driver to learn from mistakes. This holistic approach is a key factor to increase safe mobility not only for older but for all drivers.

Fig 1 :

 1 Fig 1: Example of illustration presented to the French participants to collect their opinions and their expectations on future navigation systems based on augmented reality (to the left) or on head-up display (to the right).

The authors would like to acknowledge the SAFEMOVE project. For the financial support, we would like to thank VINNOVA, SAFER and Region Västra Götaland in Sweden and MOVEO, ANR in France.