Hajer Bahouri

AND Clotilde Fermanian-Kammerer

Isabelle Gallagher
email: gallagher@math.univ-paris-diderot.fr

DISPERSIVE ESTIMATES FOR THE SCHR ÖDINGER OPERATOR ON STEP 2 STRATIFIED LIE GROUPS

The present paper is dedicated to the proof of dispersive estimates on stratified Lie groups of step 2, for the linear Schrödinger equation involving a sublaplacian. It turns out that the propagator behaves like a wave operator on a space of the same dimension p as the center of the group, and like a Schrödinger operator on a space of the same dimension k as the radical of the canonical skew-symmetric form, which suggests a decay rate |t| -k+p-1 2 . In this article, we identify a property of the canonical skew-symmetric form under which we establish optimal dispersive estimates with this rate. The relevance of this property is discussed through several examples.

1. Introduction 1.1. Dispersive inequalities. Dispersive inequalities for evolution equations (such as Schrödinger and wave equations) play a decisive role in the study of semilinear and quasilinear problems which appear in numerous physical applications. Dispersion phenomena amount to establishing a decay estimate for the L ∞ norm of the solutions of these equations at time t in terms of some negative power of t and the L 1 norm of the data. In many cases, the main step in the proof of this decay in time relies on the application of a stationary phase theorem on an (approximate) representation of the solution.

Combined with an abstract functional analysis argument known as the T T * -argument, dispersion phenomena yield a range of estimates involving space-time Lebesgue norms. Those inequalities, called Strichartz estimates, have proved to be powerful in the study of nonlinear equations (for instance one can consult [START_REF] Bahouri | Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren der mathematischen Wissenschaften[END_REF] and the references therein).

In the R d framework, dispersive inequalities have a long history beginning with the articles of Brenner [START_REF] Brenner | On L p -L p estimates for the wave-equation[END_REF], Pecher [START_REF] Pecher | L p -Abschätzungen und klassische Lösungen für nichtlineare Wellengleichungen[END_REF], Segal [START_REF] Segal | Space-time decay for solutions of wave equations[END_REF] and Strichartz [START_REF] Strichartz | Restriction Fourier transform of quadratic surfaces and decay of solutions of the wave equations[END_REF]. They were subsequently developed by various authors, starting with the paper of Ginibre and Velo [START_REF] Ginibre | Generalized Strichartz inequalities for the wave equations[END_REF] (for a detailed bibliography, we refer to [START_REF] Keel | Endpoint Strichartz estimates[END_REF][START_REF] Tao | Nonlinear dispersive equations. Local and global analysis[END_REF] and the references therein). In [START_REF] Bahouri | Espaces de Besov et estimations de Strichartz généralisées sur le groupe de Heisenberg[END_REF], the authors generalize the dispersive estimates for the wave equation to the Heisenberg group H d with an optimal rate of decay of order |t| -1/2 (regardless of the dimension d) and prove that no dispersion phenomenon occurs for the Schrödinger equation. In [START_REF] Hiero | Dispersive and Strichartz estimates on H-type groups[END_REF], optimal results are proved for the time behavior of the Schrödinger and wave equations on H-type groups: if p is the dimension of the center of the H-type group, the author establishes sharp dispersive inequalities for the wave equation solution (with a decay rate of |t| -p/2 ) as well as for the Schrödinger equation solution (with a |t| -(p-1)/2 decay). Compared with the R d framework, there is an exchange in the rates of decay between the wave and the Schrödinger equations.

Strichartz estimates in other settings have been obtained in a number of works. One can first cite various results dealing with variable coefficient operators(see for instance [START_REF] Kapitanski | Some generalization of the Strichartz-Brenner inequality[END_REF][START_REF] Smith | A parametrix construction for wave equation with C 1,1 coefficients[END_REF]) or studies concerning domains such as [START_REF] Burq | Global existence for energy critical waves in 3-D domains[END_REF][START_REF] Ivanovici | Dispersion for the wave equation inside strictly convex domains I: the Friedlander model case[END_REF][START_REF] Smith | On the critical semilinear wave equation outside convex obstacles[END_REF]. One can also refer to the result concerning the full Laplacian on the Heisenberg group in [START_REF] Furioli | Strichartz inequalities for the wave equation with the full Laplacian on the Heisenberg group[END_REF], works in the framework of the real hyperbolic spaces in [START_REF] Anker | Nonlinear Schrdinger equation on real hyperbolic spaces[END_REF][START_REF] Banica | The nonlinear Schrödinger equation on hyperbolic space[END_REF][START_REF] Tataru | Strichartz estimates in the hyperbolic space and global existence for the semilinear wave equation[END_REF], or in the framework of compact and noncompact manifolds in [START_REF] Anton | Strichartz inequalities for Lipschitz metrics on manifolds and nonlinear Schrdinger equation on domains[END_REF][START_REF] Banica | Weighted Strichartz estimates for radial Schrödinger equation on noncompact manifolds[END_REF][START_REF] Burq | Strichartz inequalities and the nonlinear Schrdinger equation on compact manifolds[END_REF]; finally one can mention the quasilinear framework studied in [START_REF] Bahouri | Équations d'ondes quasilinéaires et inégalités de Strichartz[END_REF][START_REF] Bahouri | Microlocal analysis, bilinear estimates and cubic quasilinear wave equation[END_REF][START_REF] Klainerman | Rough solutions of the Einstein-vacuum equations[END_REF][START_REF] Smith | Sharp local well-posedness results for the nonlinear wave equation[END_REF], and the references therein.

In this paper our goal is to establish optimal dispersive estimates for the solutions of the Schrödinger equation on 2-step stratified Lie groups. We shall emphasize in particular the key role played by the canonical skew-symmetric form in determining the rate of decay of the solutions. It turns out that the Schrödinger propagator on G behaves like a wave operator on a space of the same dimension as the center of G, and like a Schrödinger operator on a space of the same dimension as the radical of the canonical skew-symmetric form associated with the dual of the center. This unusual behavior of the Schrödinger propagator in the case of Lie algebras whose canonical skew-symmetric form is degenerate (known as Lie algebras which are not MW, see [START_REF] Moore | Square integrable representations of nilpotent Lie groups[END_REF], [START_REF] Müller | Solvability for a class of doubly characteristic differential operators on two-step nilpotent groups[END_REF] for example) makes the analysis of the explicit representations of the solutions tricky and gives rise to uncommon dispersive estimates. It will also appear from our analysis that the optimal rate of decay is not always in accordance with the dimension of the center: we shall exhibit examples of 2-step stratified Lie groups with center of any dimension and for which no dispersion occurs for the Schrödinger equation. We shall actually highlight that the optimal rate of decay in the dispersive estimates for solutions to the Schrödinger equation is rather related to the properties of the canonical skew-symmetric form.

1.2. Stratified Lie groups. Let us recall here some basic facts about stratified Lie groups (see [START_REF] Corwin | Representations of nilpotent Lie groups and their applications, Part 1: Basic theory and examples[END_REF][START_REF] Folland | Harmonic Analysis in Phase Space Annals of Mathematics Studies[END_REF][START_REF] Folland | Hardy Spaces on Homogeneous Groups[END_REF][START_REF] Stein | Introduction to Fourier analysis on euclidean spaces[END_REF] and the references therein for further details). A connected, simply connected nilpotent Lie group G is said stratified if its left-invariant Lie algebra g (assumed real-valued and of finite dimension n) is endowed with a vector space decomposition g = ⊕ 1≤k≤∞ g k , where all but finitely many of the g k s are {0}, such that [g 1 , g k ] = g k+1 . Via the exponential map exp : g → G which is in that case a diffeomorphism from g to G, one identifies G and g. It turns out that under this identification, the group law on G (which is generally not commutative) provided by the Campbell-Baker-Hausdorff formula, (x, y) → x • y is a polynomial map. In the following we shall denote by z the center of G which is simply the last non zero g k and write

(1.1) G = v ⊕ z ,
where v is any subspace of G complementary to z.

The group G is endowed with a smooth left invariant measure µ(x), the Haar measure, induced by the Lebesgue measure on g and which satisfies the fundamental translation invariance property:

∀f ∈ L 1 (G, dµ) , ∀x ∈ G , G f (y)dµ(y) = G f (x • y)dµ(y) .
Note that the convolution of two functions f and g on G is given by

(1.2) f * g(x) := G f (x • y -1 )g(y)dµ(y) = G f (y)g(y -1 • x)dµ(y)
and as in the euclidean case we define Lebesgue spaces by

f L p (G) := G |f (y)| p dµ(y) 1 p , for p ∈ [1, ∞[, with the standard modification when p = ∞.
Since G is stratified, there is a natural family of dilations on g defined for t > 0 as follows: if X belongs to g, we can decompose X as X = X k with X k ∈ g k , and then

δ t X := t k X k .
This allows to define the dilation δ t on the Lie group G via the identification by the exponential map:

g δt → g exp ↓ ↓ exp G → exp • δt • exp -1

G

To avoid heaviness, we shall still denote by δ t the map exp

• δ t • exp -1 .
Observe that the action of the left invariant vector fields X k , for X k belonging to g k , changes the homogeneity in the following way:

X k (f • δ t ) = t k X k (f ) • δ t ,
where by definition

X k (f )(y) := d ds f y • exp(sX k ) |s=0 and the Jacobian of the dilation δ t is t Q where Q := 1≤k≤∞ k dim g k is called the homogeneous dimension of G: (1.3) G f (δ t y) dµ(y) = t -Q G f (y) dµ(y) .
Let us also point out that there is a natural norm ρ on G which is homogeneous in the sense that it respects dilations: G x → ρ(x) satisfies ∀x ∈ G, ρ(x -1 ) = ρ(x) , ρ(δ t x) = tρ(x) , and ρ(x) = 0 ⇐⇒ x = 0 .

We can define the Schwartz space S(G) as the set of smooth functions on G such that for all α in N d , for all p in N, x → ρ(x) p X α f (x) belongs to L ∞ (G), where X α denotes a product of |α| left invariant vector fields. The Schwartz space S(G) has properties very similar to those of the Schwartz space S(R d ), particularly density in Lebesgue spaces.

1.3. The Fourier transform. The group G being non commutative, its Fourier transform is defined by means of irreducible unitary representations. We devote this section to the introduction of the basic concepts that will be needed in the sequel. From now on, we assume that G is a step 2 stratified Lie group and we choose v = g 1 in (1.1).

1.3.1. Irreducible unitary representations. Let us fix some notation, borrowed from [START_REF] Ciatti | Uncertainty inequalities on stratified nilpotent groups[END_REF] (see also [START_REF] Corwin | Representations of nilpotent Lie groups and their applications, Part 1: Basic theory and examples[END_REF] or [START_REF] Müller | Solvability for a class of doubly characteristic differential operators on two-step nilpotent groups[END_REF]). For any λ ∈ z (the dual of the center z) we define a skew-symmetric bilinear form on v by

(1.4) ∀ U, V ∈ v , B(λ)(U, V ) := λ([U, V ]) .
One can find a Zariski-open subset Λ of z such that the number of distinct eigenvalues of B(λ) is maximum. We denote by k the dimension of the radical r λ of B(λ). Since B(λ) is skew-symmetric, the dimension of the orthogonal complement of r λ in v is an even number which we shall denote by 2d. Therefore, there exists an orthonormal basis

P 1 (λ), . . . , P d (λ), Q 1 (λ), . . . , Q d (λ), R 1 (λ), . . . , R k (λ)
such that the matrix of B(λ) takes the following form

                 0 . . . 0 η 1 (λ) . . . 0 0 • • • 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . 0 . . . 0 0 . . . η d (λ) 0 • • • 0 -η 1 (λ) . . . 0 0 . . . 0 0 • • • 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . 0 . . . -η d (λ) 0 . . . 0 0 • • • 0 0 . . . 0 0 . . . 0 0 • • • 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . 0 . . . 0 0 . . . 0 0 • • • 0                 
, where each η j (λ) > 0 is smooth and homogeneous of degree one in λ = (λ 1 , . . . , λ p ) and the basis vectors are chosen to depend smoothly on λ in Λ. Decomposing v as

v = p λ + q λ + r λ with p λ := Span P 1 (λ), . . . , P d (λ) , q λ := Span Q 1 (λ), . . . , Q d (λ) , r λ := Span R 1 (λ), . . . , R k (λ)
any element V ∈ v will be written in the following as

P + Q + R with P ∈ p λ , Q ∈ q λ and R ∈ r λ .
We then introduce irreducible unitary representations of G on L 2 (p λ ):

(1.5) u λ,ν X φ(ξ) := e -iν(R)-iλ(Z+[ξ+ 1 2 P,Q]) φ(P + ξ) , λ ∈ z * , ν ∈ r * λ , for any x = exp(X) ∈ G with X = X(λ, x) := P (λ, x), Q(λ, x), R(λ, x), Z(x) and φ ∈ L 2 (p λ ).
In order to shorten notation, we shall omit the dependence on (λ, x) whenever there is no risk of confusion.

1.3.2. The Fourier transform. In contrast with the euclidean case, the Fourier transform is defined on the manifold r(Λ) whose fibre above λ ∈ Λ is r * λ ∼ R k and is valued in the space of bounded operators on L 2 (p λ ). More precisely, the Fourier transform of a function f in L 1 (G) is defined as follows: for any (λ, ν) ∈ r(Λ)

F(f )(λ, ν) := G f (x)u λ,ν X(λ,x) dµ(x) .
Note that for any (λ, ν), the map u λ,ν X(λ,x) is a group homomorphism from G into the group U (L 2 (p λ )) of unitary operators of L 2 (p λ ), so functions f of L 1 (G) have a Fourier transform (F(f )(λ, ν)) λ,ν which is a bounded family of bounded operators on L 2 (p λ ). One may check that the Fourier transform exchanges convolution, whose definition is recalled in (1.2), and composition:

(1.6) F(f g)(λ, ν) = F(f )(λ, ν) • F(g)(λ, ν) .
Besides, the Fourier transform can be extended to an isometry from L 2 (G) onto the Hilbert space of two-parameter families A = {A(λ, ν)} (λ,ν)∈r(Λ) of operators on L 2 (p λ ) which are Hilbert-Schmidt for almost every (λ, ν) ∈ r(Λ), with A(λ, ν) HS(L 2 (p λ )) measurable and with norm

A := λ∈Λ ν∈r * λ A(λ, ν) 2 HS(L 2 (p λ )) |Pf(λ)|dν dλ 1 2 < ∞ ,
where |Pf(λ)| := d j=1 η j (λ) is the Pfaffian of B(λ). We have the following Fourier-Plancherel formula: there exists a constant κ > 0 such that

(1.7) G |f (x)| 2 dx = κ λ∈Λ ν∈r * λ F(f )(λ, ν) 2 HS(L 2 (p λ )) |Pf(λ)| dν dλ .
Finally, we have an inversion formula as stated in the following proposition which is proved in the Appendix page 21.

Proposition 1.1. There exists κ > 0 such that for f ∈ S(G) and for almost all x ∈ G the following inversion formula holds:

(1.8) f (x) = κ λ∈Λ ν∈r * λ tr (u λ,ν X(λ,x) ) Ff (λ, ν) |Pf(λ)| dν dλ .
1.3.3. The sublaplacian. Let (V 1 , . . . , V m ) be an orthonormal basis of g 1 , then the sublaplacian on G is defined by

(1.9) ∆ G := m j=1 V 2 j .
It is a self-adjoint operator which is independent of the orthonormal basis (V 1 , . . . , V m ), and homogeneous of degree 2 with respect to the dilations in the sense that :

δ -1 t ∆ G δ t = t 2 ∆ G .
To write its expression in Fourier space, we consider the basis of Hermite functions (h n ) n∈N , normalized in L 2 (R) and satisfying for all real numbers ξ:

h n (ξ) -ξ 2 h n (ξ) = -(2n + 1)h n (ξ) .
Then, for any multi-index α ∈ N d , we define the functions h α,η(λ) by (1.10)

∀ Ξ = (ξ 1 , . . . , ξ d ) ∈ R d , h α,η(λ) (Ξ) := d j=1
h αj ,ηj (λ) (ξ j ) and

∀(n, β) ∈ N × R + , ∀ξ ∈ R , h n,β (ξ) := β 1 4 h n β 1 2 ξ .
The sublaplacian ∆ G defined in (1.9) satisfies

(1.11) F(-∆ G f )(λ, ν) = F(f )(λ, ν) H(λ) + |ν| 2 ,
where |ν| denotes the euclidean norm of the vector ν in R k and H(λ) is the diagonal operator defined on L 2 (R d ) by

H(λ)h α,η(λ) = d j=1 (2α j + 1)η j (λ) h α,η(λ) .
In the following we shall denote the "frequencies" associated with P 2 j (λ) + Q 2 j (λ) by (1.12)

ζ j (α, λ) := (2α j + 1)η j (λ) , (α, λ) ∈ N d × Λ ,
and those associated with H(λ) by

(1.13) ζ(α, λ) := d j=1 ζ j (α, λ) , (α, λ) ∈ N d × Λ .
Note that ∆ G is directly related to the harmonic oscillator via H(λ) since eigenfunctions associated with the eigenvalues ζ(α, λ) are the products of 1-dimensional Hermite functions. Also observe that ζ(α, λ) is smooth and homogeneous of degree one in λ = (λ 1 , . . . , λ p ). Moreover, ζ(α, λ) = 0 if and only if B(λ) = 0, or equivalently by (1.4), λ = 0.

Notice also that there is a difference in homogeneity in the variables λ and ν. Namely, in the variable ν, the sublaplacian acts as in the euclidean case (homogeneity 2) while in λ, it has the homogeneity 1 of a wave operator.

Finally, for any smooth function Φ, we define the operator Φ (-∆ G ) by the formula

(1.14) F Φ(-∆ G )f (λ, ν) := Φ(H(λ) + |ν| 2 )F(f )(λ, ν) ,
which also reads

F Φ(-∆ G )f (λ, ν)h α,η(λ) := Φ |ν| 2 + ζ(α, λ) F(f )(λ, ν)h α,η(λ) ,
for all (λ, ν) ∈ r(Λ) and α ∈ N d .

F(f )(λ, ν) = F(f )(λ, ν) θ (P 2 j + Q 2 j )(λ) , ∀(λ, ν) ∈ r(Λ) . Remark 1.3.
One could expect the notion of spectral localization to relate to the Laplacian instead of each individual vector field P 2 j + Q 2 j , assuming rather the less restrictive condition

F(f )(λ, ν) = F(f )(λ, ν) θ H(λ) , ∀(λ, ν) ∈ r(Λ) .
The choice we make here is more restrictive due to the anisotropic context (namely the fact that η j (λ) depends on j): in the case of the Heisenberg group or more generally H-type groups, the notion of "strict spectral localization" in a ring C of R p actually coincides with the more usual definition of "spectral localization" since as recalled in the next paragraph η j (λ) = 4|λ| (for a complete presentation and more details on spectrally localized functions, we refer the reader to [START_REF] Bahouri | Paraproduit sur le groupe de Heisenberg et applications[END_REF][START_REF] Bahouri | Phase-space analysis and pseudodifferential calculus on the Heisenberg group[END_REF][START_REF] Bahouri | Refined Inequalities on Graded Lie Groups[END_REF]). Assumption (1.15) guarantees a lower bound, which roughly states that for F(f )(λ, ν)h α,λ to be non zero, then (1.16) ∀j ∈ {1, . . . , d}, (2α j + 1)η j (λ) ≥ c > 0 , hence each η j must be bounded away from zero, rather than the sum over j. These lower bounds are important ingredients of the proof (see Section 3.3). where (• | •) denotes the euclidean scalar product on R d . In that case the center consists in the points of the form (0, 0, s) and is of dimension 1. The Lie algebra of left invariant vector fields is generated by

X j := ∂ xj + 2y j ∂ s , Y j := ∂ yj -2x j ∂ s with 1 ≤ j ≤ d , and
S := ∂ s = 1 4 [Y j , X j ] .
The canonical skew-symmetric form associated with the frequencies λ ∈ R * writes

B(λ)(U, V ) = λ ([U, V ]) , ∀ U, V ∈ Vect (X j , Y j , 1 ≤ j ≤ d)
and its radical reduces to {0} with Λ = R * and |η j (λ)| = 4 |λ| for all j ∈ {1, . . . , d}. Note in particular that strict spectral localization and spectral localization are equivalent.

1.4.2. H-type groups. These groups are canonically isomorphic to R m+p , and are a multidimensional version of the Heisenberg group. The group law is of the form (x (1) , x (2) ) • (y (1) , y (2) ) := x

(1) 1) , U (j) y (1) , j = 1, ..., p where U (j) are m×m linearly independent orthogonal skew-symmetric matrices satisfying the property

j + y (1) j , j = 1, ..., m x (2) j + y (2) j + 1 2 x (
U (r) U (s) + U (s) U (r) = 0
for every r, s ∈ {1, ..., p} with r = s. In that case the center is of dimension p and may be identified with R p and the radical of the canonical skew-symmetric form associated with the frequencies λ is again {0}. For example the Iwasawa subgroup of semi-simple Lie groups of split rank one (see [START_REF] Korányi | Geometric properties of Heisenberg-type groups[END_REF]) is of this type. On H-type groups, m is an even number which we denote by 2 and the Lie algebra of left invariant vector fields is spanned by the following vector fields, where we have written z = (x, y) in R × R : for j running from 1 to and k from 1 to p,

X j := ∂ xj + 1 2 p k=1 2 l=1 z l U (k) l,j ∂ s k , Y j := ∂ yj + 1 2 p k=1 2 l=1 z l U (k) l,j+ ∂ s k and ∂ s k .
In that case, we have

Λ = R p \{0} with η j (λ) = λ 2 1 + • • • + λ 2
p for all j ∈ {1, . . . , } (here again, strict spectral localization and spectral localization are equivalent). 1.4.3. Diamond groups. These groups which occur in crystal theory (for more details, consult [START_REF] Ludwig | Dual topology of diamond groups[END_REF][START_REF] Poguntke | Unitary representations of diamond groups[END_REF]), are of the type Σ H d where Σ is a connected Lie group acting smoothly on H d . One can find examples for which the radical of the canonical skew-symmetric is of any dimension k, 0 ≤ k ≤ d. For example, one can take for Σ the k-dimensional torus, acting on

H d by θ(w) := (θ • z, s) := e iθ1 z 1 , . . . , e iθ k z k , z k+1 , . . . , z d , s , w = (z, s)
where the element θ = (θ 1 , . . . , θ k ) corresponds to the element e iθ1 , . . . , e iθ k of T k . Then, the product law on

G = T k H d is given by (θ, w) • (θ , w ) = θ + θ , w.(θ(w ) ,
where w.(θ(w )) denotes the Heisenberg product of w by θ(w ). As a consequence, the center of G is of dimension 1 since it consists of the points of the form (0, 0, s) for s ∈ R. Let us choose for simplicity k = d = 1, the algebra of left-invariant vector fields is generated by the vector fields ∂ θ , ∂ s , Γ θ,x and Γ θ,y where

Γ θ,x = cos θ∂ x + sin θ∂ y + 2(ycos θ -xsinθ)∂ s , Γ θ,y = -sin θ∂ x + cos θ∂ y -2(ysin θ + xcosθ)∂ s .
It is not difficult to check that the the radical of B(λ) is of dimension 1. In the general case, where k ≤ d, the algebra of left-invariant vector fields is generated by the vector fields ∂ s , the d -k -1 vectors

X = ∂ x + 2y ∂ s , Y = ∂ y -2x ∂ s ,
and the k vectors defined for 1 ≤ j ≤ k by : ∂ θj ,, Γ θj ,xj and Γ θj ,yj where

Γ θj ,xj = cos θ j ∂ xj + sin θ j ∂ yj + 2(y j cos θ j -x j sinθ j )∂ s , Γ θj ,yj = -sin θ j ∂ xj + cos θ j ∂ yj -2(y j sin θ j + x j cosθ j )∂ s .
and this provides an example with a radical of dimension k.

1.4.4. The tensor product of Heisenberg groups. Consider H d1 ⊗ H d2 the set of elements (w

1 , w 2 ) in H d1 ⊗ H d2 , which can be written (w 1 , w 2 ) = (x 1 , y 1 , s 1 , x 2 , y 2 , s 2 ) in R 2d1+1 × R 2d2+1 , equipped with the law of product: (w 1 , w 2 ) • (w 1 , w 2 ) = (w 1 • w 1 , w 2 • w 2 ),
where w 1 •w 1 and w 2 •w 2 denote respectively the product in H d1 and H d2 . Clearly H d1 ⊗ H d2 is a step 2 stratified Lie group with center of dimension 2 and radical index null. Moreover, for λ = (λ 1 , λ 2 ) in the dual of the center, the canonical skew bilinear form B(λ) has radical {0} with Λ = R * × R * , and one has η

1 (λ) = 4 |λ 1 | and η 2 (λ) = 4 |λ 2 |.
In that case, strict spectral localization is a more restrictive condition than spectral localization. Indeed, if f is spectrally localized, one has λ 1 = 0 or λ 2 = 0 on the support of F(f )(λ), while one has λ 1 = 0 and λ 2 = 0 on the support of F(f )(λ) if f is strictly spectrally localized.

1.4.5. Tensor product of H-type groups. The group R m1+p1 ⊗ R m2+p2 is easily verified to be a step 2 stratified Lie group with center of dimension p 1 +p 2 , a radical index null and a skew bilinear form B(λ)

defined on R m1+m2 with m 1 = 2 1 and m 2 = 2 2 . The Zariski open set associated with B is given by Λ = (R p1 \{0}) × (R p2 \{0}) and for λ = (λ 1 , • • • , λ p1+p2 ), we have (1.17) η j (λ) = λ 2 1 + • • • + λ 2 p1
, for all j ∈ {1, . . . , 1 } and

η j (λ) = λ 2 p1+1 + • • • + λ 2 p1+p2
for all j ∈ { 1 + 1, . . . , 1 + 2 }.

1.5. Main results. The purpose of this paper is to establish optimal dispersive inequalities for the linear Schrödinger equation on step 2 stratified Lie groups associated with the sublaplacian. In view of (1.11) and the fact that the "frequencies" ζ(α, λ) associated with H(λ) given by (1.13) are homogeneous of degree one in λ, the Schrödinger operator on G behaves like a wave operator on a space of the same dimension p as the center of G, and like a Schrödinger operator on a space of the same dimension k as the radical of the canonical skew-symmetric form. By comparison with the classical dispersive estimates, the expected result would be a dispersion phenomenon with an optimal rate of decay of order |t| -k+p-1

2

. However, as will be seen through various examples, this anticipated rate is not always achieved. To reach this maximum rate of dispersion, we require a condition on ζ(α, λ). Let us now present the dispersive inequality for the Schrödinger equation. Recall that the linear Schrödinger equation writes as follows on G:

(1.18) (i∂ t -∆ G ) f = 0 f |t=0 = f 0 ,
where the function f with complex values depends on (t, x) ∈ R ×G.

Theorem 1. Let G be a step 2 stratified Lie group with center of dimension p with 1 ≤ p < n and radical index k. Assume that Assumption 1.4 holds. A constant C exists such that if f 0 belongs to L 1 (G) and is strictly spectrally localized in a ring of R in the sense of Definition 1.2, then the associate solution f to the Schrödinger equation (1.18) satisfies

(1.19) f (t, •) L ∞ (G) ≤ C |t| k 2 (1 + |t| p-1 2 ) f 0 L 1 (G) ,
for all t = 0 and the result is sharp in time.

The fact that a spectral localization is required in order to obtain the dispersive estimates is not surprising. Indeed, recall that in the R d case for instance, the dispersive estimate for the Schrödinger equation derives immediately (without any spectral localization assumption) from the fact that the solution u(t) to the free Schrödinger equation on R d with Cauchy data u 0 writes for t = 0

u(t, •) = u 0 * 1 (-2iπt) d 2 e -i |•| 2 4t ,
where * denotes the convolution product in R d (for a detailed proof of this fact, see for instance Proposition 8.3 in [START_REF] Bahouri | Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren der mathematischen Wissenschaften[END_REF]). However proving dispersive estimates for the wave equation in R d requires more elaborate techniques (including oscillating integrals) which involve an assumption of spectral localization in a ring. In the case of a step 2 stratified Lie group G, the main difficulty arises from the complexity of the expression of Schrödinger propagator that mixes a wave operator behavior with that of a Schrödinger operator. This explains on the one hand the decay rate in Estimate (1. [START_REF] Folland | Hardy Spaces on Homogeneous Groups[END_REF]) and on the other hand the hypothesis of strict spectral localization.

Let us now discuss Assumption 1.4. As mentioned above, there is no dispersion phenomenon for the Schrödinger equation on the Heisenberg group H d (see [START_REF] Bahouri | Espaces de Besov et estimations de Strichartz généralisées sur le groupe de Heisenberg[END_REF]). Actually the same holds for the tensor product of Heisenberg groups H d1 ⊗ H d2 whose center is of dimension p = 2 and radical index null, and more generally to the case of 2 step stratified Lie groups, decomposable on non trivial 2 step stratified Lie groups : we derive indeed from Theorem 1 the following corollary.

q := 1 2 1≤m≤r (k m + p m -1) = 1 2 (k + p -r) ,
where p is the dimension of the center of G and k its radical index. Besides, this rate is optimal.

Corollary 1.6 is a direct consequence of Theorem 1 and the simple observation that ∆ G = ⊗ 1≤m≤r ∆ Gm . This result applies for example to the tensor product of Heisenberg groups, for which there is no dispersion, and to the tensor product of H-type groups R m1+p1 ⊗ R m2+p2 for which the dispersion rate is t -(p1+p2-2)/2 (see [START_REF] Hiero | Dispersive and Strichartz estimates on H-type groups[END_REF]). Corollary 1.6 therefore shows that it can happen that the "best" rate of decay |t| -(k+p-1)/2 is not reached, in particular for decomposable Lie groups. This suggests that Assumption 1.4 could be related with decomposability.

More generally, a large class of groups which do not satisfy the Assumption 1.

f 0 ∈ S(G), x ∈ G and c 0 > 0 such that ∀t ∈ R + , |e -it∆ G f 0 (x)| ≥ c 0 .
Finally we point out that the dispersive estimate given in Theorem 1 can be regarded as a first step towards space-time estimates of the Strichartz type. However due to the strict spectral localization assumption, the Besov spaces which should appear in the study (after summation over frequency bands) are naturally anisotropic; thus proving such estimates is likely to be very technical, and is postponed to future works.

1.6. Strategy of the proof of Theorem 1. In the statement of Theorem 1, there are two different results: the dispersive estimate in itself on the one hand, and its optimality on the other. Our strategy or proof is closely related to the method developed in [START_REF] Bahouri | Espaces de Besov et estimations de Strichartz généralisées sur le groupe de Heisenberg[END_REF] and [START_REF] Hiero | Dispersive and Strichartz estimates on H-type groups[END_REF] with additional non negligible technicalities.

In the situation of [START_REF] Bahouri | Espaces de Besov et estimations de Strichartz généralisées sur le groupe de Heisenberg[END_REF] where the Heisenberg group H d is considered, the authors prove that there is no dispersion by exhibiting explicitly a Cauchy data f 0 for which the solution f (t, •) to the Schrödinger equation (1.18) satisfies

(1.20) ∀ q ∈ [1, ∞] , f (t, •) L q (H d ) = f 0 L q (H d ) .
More precisely, they take advantage of the fact that the Laplacian-Kohn operator ∆ H d can be recast under the form

(1.21) ∆ H d = 4 d j=1 (Z j Z j + i∂ s ) ,
where Z 1 , Z 1 , ..., Z d , Z d , ∂ s is the canonical basis of Lie algebra of left invariant vector fields on H d (see [START_REF] Bahouri | Phase-space analysis and pseudodifferential calculus on the Heisenberg group[END_REF] and the references therein for more details). This implies that for a non zero function f 0 belonging to Ker d j=1 Z j Z j , the solution of the Schrödinger equation on the Heisenberg group f (t) = e -it∆ H d f 0 actually solves a transport equation:

f (z, s, t) = e 4dt∂s f 0 (z, s) = f 0 (z, s + 4dt)
and hence satisfies (1.20). The arguments used in [START_REF] Hiero | Dispersive and Strichartz estimates on H-type groups[END_REF] for general H-type groups are similar to the ones developed in [START_REF] Bahouri | Espaces de Besov et estimations de Strichartz généralisées sur le groupe de Heisenberg[END_REF]: the dispersive estimate is obtained using an explicit formula for the solution, coming from Fourier analysis, combined with a stationary phase theorem. The Cauchy data used to prove the optimality is again in the kernel of an adequate operator, by a decomposition similar to (1.21).

As in [START_REF] Bahouri | Espaces de Besov et estimations de Strichartz généralisées sur le groupe de Heisenberg[END_REF] and [START_REF] Hiero | Dispersive and Strichartz estimates on H-type groups[END_REF], the first step of the proof of Theorem 1 consists in writing an explicit formula for the solution of the equation by use of the Fourier transform. Let us point out that in the setting of [START_REF] Bahouri | Espaces de Besov et estimations de Strichartz généralisées sur le groupe de Heisenberg[END_REF] and [START_REF] Hiero | Dispersive and Strichartz estimates on H-type groups[END_REF], irreducible representations are isotropic with respect to the dual of the center of the group; this isotropy allows to reduce to a one-dimensional framework and deduce the dispersive effect from a careful use of a stationary phase argument of [START_REF] Stein | Oscillatory integrals in Fourier analysis[END_REF]. As we have already seen in Paragraph 1.3.1, the irreducible representations are no longer isotropic in the general case of stratified Lie groups, and thus we adopt a more technical approach making use of Schrödinger representation and taking advantage of some properties of Hermite functions appearing in the explicit representation of the solutions derived by Fourier analysis (see Section 3.3). The optimality of the inequality is obtained as in [START_REF] Bahouri | Espaces de Besov et estimations de Strichartz généralisées sur le groupe de Heisenberg[END_REF] and [START_REF] Hiero | Dispersive and Strichartz estimates on H-type groups[END_REF], by an adequate choice of the initial data. 1.7. Organization of the paper. The article is organized as follows. In Section 2, we write an explicit formulation of the solutions of the Schrödinger equation. Then, Section 3 is devoted to the proof of Theorem 1 and in Section 4, we discuss the optimality of the result and prove Proposition 1.7.

Finally, we mention that the letter C will be used to denote a universal constant which may vary from line to line. We also use A B to denote an estimate of the form A ≤ CB for some constant C.

Explicit representation of the solutions

2.1. The convolution kernel. Let f 0 belong to S(G) and let us consider f (t, •) the solution to the free Schrödinger equation (1.18). In view of (1.11), we have

F(f (t, •))(λ, ν) = F(f 0 )(λ, ν) e it|ν| 2 +itH(λ) ,
which implies easily (arguing as in the Appendix) that f (t, •) belongs to S(G). Assuming that f 0 is strictly spectrally localized in the sense of Definition 1.2, there exists a smooth function θ compactly supported in a ring C of R such that, defining (λ) . Therefore by the inverse Fourier transform (1.8), we deduce that the function f (t, •) may be decomposed in the following way:

Θ(λ) := d j=1 θ (P 2 j + Q 2 j )(λ) , then F(f (t, •))(λ, ν) = F(f 0 )(λ, ν) Θ(λ) e it|ν| 2 +itH
(2.1) f (t, x) = κ λ∈Λ ν∈r * λ tr (u λ,x X(λ,ν) ) F(f 0 )(λ, ν) Θ(λ) e it|ν| 2 +itH(λ) |Pf(λ)| dν dλ .
We set for

X ∈ R n , (2.2) k t (X) := κ λ∈Λ ν∈r * λ tr (u λ,ν X ) Θ(λ) e it|ν| 2 +itH(λ) |Pf(λ)| dνdλ .
The function k t plays the role of a convolution kernel in the variables of the Lie algebra and we have the following result.

Proposition 2.1. If the function k t defined in (2.2) satisfies

(2.3) ∀t ∈ R , k t L ∞ (R n ) ≤ C |t| k 2 (1 + |t| p- 1 2 )
, then Theorem 1 holds.

Proof. We write, according to (2. We use the exponential law y → Y = (P (λ, y), Q(λ, y), Z, R(λ, y)) and the fact that dµ(y) = dY the Lebesgue measure, then we perform a linear orthonormal change of variables

(P (λ, y), Q(λ, y), R(λ, y)) → ( P , Q, R) so that dµ(y) = dY = d P d Q dZ d R and we write f (t, x) = κ λ∈Λ ν∈r * λ ( P , Q,Z, R)∈R n tr u λ,ν ( P , Q,Z, R) Θ(λ) e it|ν| 2 +itH(λ) ×f 0 (x • exp( P , Q, Z, R))|Pf(λ)| dν dλ d P d Q dZ d R .
Thanks to the Fubini Theorem and Young inequalities, we can write (dropping the ˜on the variables),

|f (t, x)| = (P,Q,Z,R)∈R n k t (P, Q, Z, R)f 0 (x • exp(P, Q, Z, R))dP dQ dR dZ ≤ k t L ∞ (G) (P,Q,Z,R)∈R n f 0 (x • exp(P, Q, Z, R))dP dQ dR dZ ≤ k t L ∞ (G) f 0 L 1 (G) .
Proposition 2.1 is proved.

In the next subsections, we make preliminary work by transforming the expression of k t and reducing the proof to statements equivalent to (2.3).

2.2.

Transformation of k t : expression in terms of Hermite functions. Decomposing the operator H(λ) in the basis of Hermite functions, and recalling notation (1.12) replaces (2.2) with

k t (X) = κ α∈N d Λ ν∈r * λ e it|ν| 2 +itζ(α,λ) d j=1 θ ζ j (α, λ) u λ,ν X h α,η(λ) |h α,η(λ) |Pf(λ)| dν dλ , X ∈ R n .
Using the explicit form of u λ,ν X recalled in (1.5) we find the following result. Lemma 2.2. There is a constant κ and a smooth function F such that with the above notation, we have for t = 0

k t (P, Q, tZ, R) = κ e -i |R| 2 4t t k 2 α∈N d Λ e itΦα(Z,λ) G α P, Q, η(λ) |Pf(λ)| F (λ) dλ ,
where the phase Φ α is given by

Φ α (Z, λ) := ζ(α, λ) -λ(Z) ,
with Notation (1.13), and the function G α is given by the following formula, for all (P, Q, η) ∈ R 3d :

(2.4) G α (P, Q, η) := d j=1 θ ζ j (α, λ) g αj √ η j P j , √ η j Q j while for each (ξ 1 , ξ 2 , n) in R 2 ×N, using Notation (1.10), (2.5)
g n (ξ 1 , ξ 2 ) := e -i ξ 1 ξ 2 2 R e -iξ2ξ h n (ξ 1 + ξ)h n (ξ) dξ .
Notice that (g n ) n∈N is uniformly bounded in R 2 thanks to the Cauchy-Schwarz inequality and the fact that h n L 2 (R) = 1, and hence the same holds for (G α ) α∈N d (in R 3d ).

Proof. We begin by observing that for X = (P, Q, R, Z),

I := u λ,ν X h α,η(λ) | h α,η(λ) = e -iν(R)-iλ(Z) R d e -iλ([ξ+ 1 2 P,Q]) h α,η(λ) (P + ξ)h α,η(λ) (ξ)dξ ,
with in view of (1.4)

λ ξ + 1 2 P, Q = B(λ) ξ + 1 2 P, Q = 1≤j≤d η j (λ)Q j ξ j + 1 2 P j .
As a consequence,

I = e -iν(R)-iλ(Z) 1≤j≤d R e -iηj (λ)(ξj + 1 2 Pj )Qj h αj ,ηj (λ) (P j + ξ j )h αj ,ηj (λ) (ξ j )dξ j .
The change of variables ξ j = η j (λ) ξ j gives, dropping the for simplicity,

I = e -iν(R)-iλ(Z) 1≤j≤d R e -i √ ηj (λ) Qj ξj + 1 2 √ ηj (λ)Pj h αj ξ j + η j (λ)P j h αj (ξ j )dξ j ,
which implies that

k t (P, Q, tZ, R) = κ α∈N d r(Λ) e -itλ(Z)-iν(R) e itζ(α,λ)+it|ν| 2 G α P, Q, η(λ) |Pf(λ)| dν dλ .
It is well known (see for instance Proposition 1.28 in [START_REF] Bahouri | Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren der mathematischen Wissenschaften[END_REF]) that for t = 0

(2.6)

R k e -i(ν|R)+it|ν| 2 dν = iπ t k 2 e -i |R| 2 4t ,
where (• | •) denotes the euclidean scalar product on R k . This implies that, for t = 0

|k t (P, Q, tZ, R)| 1 |t| k 2 α∈N d Λ e itΦα(Z,λ) G α P, Q, η(λ) |Pf(λ)| F (λ) dλ ,
with F the Jacobian of the change of variables f : r * λ -→ R k , which is a smooth function. Lemma 2.2 is proved.

2.3.

Transformation of the kernel k t : change of variable. We are then reduced to establishing that the kernel k t (P, Q, tZ) defined by By construction of η(λ) (which is homogeneous of degree one), we have

k t (P, Q, tZ) := α∈N d Λ e itΦα(Z,λ) G α P, Q, η(λ) |Pf(λ)| F (λ) dλ satisfies (2.7) ∀t ∈ R , k t L ∞ (G) ≤ C 1 + |t| p-
(2.8) ∀m = 0 , η(λ) = η m (γ) := 1 m η(γ) .
Let us check that if λ lies in the support of θ ζ j (α, •) , then γ lies in a fixed ring C of R p , independent of α. On the one hand we note that there is a constant C > 0 such that on the support of θ ζ j (α, λ) , the variable γ must satisfy (2.9) ∀m = 0 , (2α j + 1)η j (γ) ≤ Cm , for all α ∈ N d such that |α| = m. Since for each j we know that η j (γ) is positive and homogeneous of degree one, we infer that the function η j (γ) goes to infinity with |γ| so (2.9) implies that γ must remain bounded on the support of θ ζ j (α, λ) . Moreover thanks to (2.9) again, it is clear that the bound may be made uniform in m. Now let us prove that γ may be bounded from below uniformly. We know that there is a positive constant c such that for λ on the support of θ ζ j (α, λ) , we have

(2.10) ∀m = 0 , ζ j (α, γ) ≥ cm .
Writing γ = |γ|γ with γ on the unit sphere of R p , we find

|γ| ≥ cm ζ j (α, γ) • Defining C j := max |γ|=1 η j (γ) < ∞ ,
it is easy to deduce that if (2.10) is satisfied, then

|γ| ≥ cm (2m + d) max 1≤j≤d C j
, hence γ lies in a fixed ring of R p , independent of α = 0. This fact will turn out to be important to perform the stationary phase argument.

Then we can rewrite the expression of k t (P, Q, tZ) in terms of the variable γ, which in view of the homogeneity of the Pfaffian produces the following formula:

k t (P, Q, tZ) = Λ e itΦ0(Z,λ) G 0 (P, Q, η(λ))|Pf(λ)| F (λ) dλ + m∈N * α∈N d |α|=m m -p-d e itΦα(Z, γ m ) G α P, Q, η m (γ) |Pf(γ)| F γ m dγ .
Since the functions G α are uniformly bounded with respect to α ∈ N d and F is smooth, there is a positive constant

C such that ∀t ∈ R , k t L ∞ (G) ≤ C .
In order to establish the dispersive estimate, it suffices then to prove that

(2.11) ∀t = 0 , k t L ∞ (G) ≤ C |t| p-1 2 •

End of the proof of the dispersive estimate

In order to prove (2.11), we decompose k t into two parts, writing

k t (P, Q, tZ) = k 1 t (P, Q, tZ) + k 2 t (P, Q, tZ)
, with, for a constant c 0 to be fixed later on independently of m, (3.1)

k 1 t (P, Q, tZ) := |∇ λ Φ0(Z,λ)|≤c0 e itΦ0(Z,λ) G 0 P, Q, η(λ) |Pf(λ)| F (λ) dλ + m∈N * α∈N d |α|=m m -p-d |∇γ (Φα(Z, γ m ))|≤c0 e itΦα(Z, γ m ) G α P, Q, η m (γ) F γ m |Pf(γ)| dγ .
In the following subsections, we successively show (2.11) for k t 1 and k t 2 .

3.1. Stationary phase argument for k t 1 . To establish Estimate (2.11), let us first concentrate on k 1 t . To this end we shall, as usual in this type of problem, define for each integral of the series defining k 1 t a vector field that commutes with the phase, prove an estimate for each term and finally check the convergence of the series. More precisely, in the case when α = 0 and t > 0 (the case t < 0 is dealt with exactly in the same manner), we consider the following first order operator:

L 1 α := Id -i∇ γ (Φ α (Z, γ m )) • ∇ γ 1 + t|∇ γ (Φ α (Z, γ m ))| 2 • Clearly we have L 1 α e itΦα(Z, γ m ) = e itΦα(Z, γ m ) .
Let us admit the next lemma for the time being.

Lemma 3.1. For any integer N , there is a smooth function θ N compactly supported on a ring of R p and a positive constant C N such that defining

(3.2) ψ α (γ) := G α P, Q, η m (γ) F γ m |Pf(γ)| ,
recalling notation (2.8), we have

|( t L 1 α ) N ψ α (γ) | ≤ C N m N θ N (γ) 1 + t 1 2 ∇ γ (Φ α (Z, γ m )) 2 -N .
Returning to k 1 t , let us define (recalling that γ belongs to a fixed ring C)

C α (Z) := γ ∈ C; |∇ γ (Φ α (Z, γ m ))| ≤ c 0
and let us write for any integer N and α = 0 (which we assume to be the case for the rest of the computations)

(3.3) I α (Z) := Cα(Z) e itΦα(Z, γ m ) ψ α (γ) dγ = Cα(Z) e itΦα(Z, γ m ) ( t L 1 α ) N ψ α (γ) dγ ,
where ψ α (γ) has been defined in (3.2). Then by Lemma 3.1 we find that for each integer N there is a constant C N such that

(3.4) |I α (Z)| ≤ C N m N Cα(Z) θ N (γ) 1 + t ∇ γ (Φ α (Z, γ m )) 2 -N dγ .
Then the end of the proof relies on three steps:

(1) a careful analysis of the properties of the support of the integral, (2) a change of variables which leads to the estimate in t -(p-1)/2 , (3) a control in m in order to prove the convergence of the sum over m.

Before entering into details for each step, let us observe that by definition, we have

Φ α (Z, γ m ) = 1 m ζ(α, γ) -γ(Z) ,
with γ(Z) = (Aγ|Z) = (γ| t AZ) for some invertible matrix A. Performing a change of variables in γ if necessary, we can assume without loss of generality that A = Id. Thus we write Proof. Let us first prove that we can choose c 0 small enough so that if C α (Z) is not empty, then one has Z = 0. Equivalently, let us prove that we can choose c 0 small enough such that C α (0) = ∅. We first notice that thanks to (3.5),

(3.5) ∇ γ (Φ α (Z, γ m )) = 1 m ∇ γ ζ(α, γ) -Z .
∇ γ ζ(α, γ) = ∇ γ Φ α (0, γ m ) . Now since the function ζ(α, •) is homogeneous of degree 1, then ∇ γ ζ(α, •) is homogeneous of degree 0.
As a consequence, if ∇ζ(α, γ 0 ) = 0 with γ 0 ∈ C, then the function R t → ζ(α, tγ 0 ) is constant and identically equal to 0. This is in contradiction with the localization on a ring by the function

θ(γ) := d j=1 θ(ζ j (α, m -1 γ)) ,
which implies the existence of a positive constant c 1 such that on the support of θ,

|∇ γ ζ(α, γ)| > 2mc 1 .
This ensures the result if we assume that the constant c 0 in the definition of k 1 t is smaller than c 1 . Let us now prove that if the constant c 0 is chosen small enough, then for all γ ∈ C α (Z) we have γ•Z = 0. Indeed writing This ensures the desired result, by choosing the constant c 0 in the definition of k 1 t both smaller than c/ c and the constant c 1 defined above. Proposition 3.2 is proved.

γ • Z = γ • ∇ γ ζ(α, γ) + γ • (Z -∇ γ ζ(α, γ)) ,

3.1.2.

A change of variables: the diffeomorphism H. We can assume without loss of generality (if not the integral is zero) that C α (Z) is not empty, and in view of Proposition 3.2, we can write for any γ ∈ C α (Z) the following orthogonal decomposition (since Z = 0):

(3.6) 1 m ∇ γ ζ(α, γ) = Γ 1 Ẑ1 + Γ , with Γ 1 := 1 m ∇ γ ζ(α, γ) Ẑ and Ẑ1 := Z |Z| •
Since Γ is orthogonal to the vector Z, we infer that

(3.7) ∇ γ (Φ α (Z, γ m )) = 1 m |Z -∇ γ ζ(α, γ)| ≥ | Γ | .
Let us consider in R p an orthonormal basis ( Ẑ1 , . . . , Ẑp ). Thanks to Proposition 3.2, we have γ • Ẑ1 = 0 on the support of the integral defining I α (Z). Obviously, the vector Γ defined by (3.6) belongs to the vector space generated by ( Ẑ2 , . . . , Ẑp ). To investigate the integral I α (Z) defined in (3.3), let us consider the map H : γ → γ defined by

(3.8) C α (Z) γ -→ H(γ) := (γ • Ẑ1 ) Ẑ1 + p k=2 ( Γ • Ẑk ) Ẑk =: p k=1 γ k Ẑk . Proposition 3.3. The map H realizes a diffeomorphism from C α (Z) into a fixed compact set of R p .
Proof. It is clear that the smooth function H maps C α (Z) into a fixed compact set K of R p and that

γ 1 = γ • Ẑ1 , and for 2 ≤ k ≤ p , γ k = 1 m ∇ γ ζ(α, γ) • Ẑk .
Now let us prove that thanks to Assumption 1.4, the map H constitutes a diffeomorphism. Indeed, by straightforward computations we find that DH, the differential of H satisfies:

DH(γ) Ẑ1 , Ẑ1 = 1 DH(γ) Ẑ1 , Ẑk = 1 m D 2 γ ζ(α, γ) Ẑ1 , Ẑk for 2 ≤ k ≤ p , DH(γ) Ẑj , Ẑk = 1 m D 2 γ ζ(α, γ) Ẑj , Ẑk for 2 ≤ j, k ≤ p and DH(γ) Ẑj , Ẑ1 = 0 for 2 ≤ j ≤ p .
Proving that H is a diffeomorphism amounts to showing that for any γ ∈ C α (Z), the kernel of DH(γ) reduces to {0}. In view of the above formulas, if V = p j=1

V j Ẑj belongs to the kernel of DH(γ)

then V 1 = V • Ẑ1 = 0 and D 2 γ ζ(α, γ)V • Ẑk = 0 for 2 ≤ k ≤ p. Thus we can write D 2 γ ζ(α, γ)V = τ Ẑ1 for some τ ∈ R. Let us point out that since the function ζ(α, •) is homogeneous of degree 1, then D 2 γ ζ(α, γ)γ = 0. We deduce that 0 = D 2 γ ζ(α, γ)γ • V = γ • D 2 γ ζ(α, γ)V = τ γ • Ẑ1 .
Since for all γ ∈ C α (Z), γ • Ẑ1 = 0, we find that τ = 0 and therefore D 2 γ ζ(α, γ)V = 0. But Assumption 1.4 states that the Hessian D 2 γ ζ(α, γ) is of rank p -1, so we conclude that V is colinear to γ. But we have seen that V • Ẑ1 = 0, which contradicts the fact that γ • Ẑ1 = 0. This entails that V is null and ends the proof of the proposition.

We can therefore perform the change of variables defined by (3.8) in the right-hand side of (3.4), to obtain

|I α (Z)| ≤ C N m N K 1 (1 + t| γ | 2 ) N d γ d γ 1 .
3.1.3. End of the proof: convergence of the series. Choosing N = p -1 implies by the change of variables γ = t 1 2 γ that there is a constant C such that

|I α (Z)| ≤ C|t| -p-1 2 m p-1 ,
which gives rise to

Cα(Z) e itΦα(Z, γ m ) ψ α (γ)dγ ≤ C|t| -p-1 2 m p-1 .
We get exactly in the same way that

|∇ λ Φ0(Z,λ)|≤c0 e itΦ0(Z,λ) G 0 P, Q, η(λ) |Pf(λ)| F (λ) dλ ≤ C|t| -p-1 2 .
Finally returning to the kernel k 1 t defined in (3.1), we get

|k 1 t (P, Q, tZ)| ≤ C|t| -p-1 2 + C |t| -p-1 2 m∈N * m d-1 m -d-p m p-1 ≤ C|t| -p-1 2 ,
since the series over m is convergent. The dispersive estimate is thus proved for k 1 t .

3.2.

Stationary phase argument for k t 2 . We now prove (2.11) for k 2 t , which is easier since the gradient of the phase is bounded from below. We claim that there is a constant C such that (3.9)

|k 2 t (P, Q, tZ)| ≤ C t p-1 2

•

This can be achieved as above by means of adequate integrations by parts. Indeed, in the case when α = 0, consider the following first order operator:

L 2 α := -i ∇ γ (Φ α (Z, γ m )) • ∇ γ |∇ γ (Φ α (Z, γ m ))| 2
• Note that when α = 0, the arguments are the same without performing the change of variable λ = γ/m. The operator L 2 α obviously satisfies

L 2 α e itΦα(Z, γ m ) = t e it Φα(Z, γ m ) ,
hence by repeated integrations by parts, we get

J α (P, Q, tZ) := |∇γ (Φα(Z, γ m ))|≥c0 e itΦα(Z, γ m ) ψ α (γ) dγ = 1 t N |∇γ (Φα(Z, γ m ))|≥c0 e itΦα(Z, γ m )) ( t L 2 α ) N ψ α (γ) dλ .
Let us admit the following lemma for a while.

Lemma 3.4. For any integer N , there is a smooth function θ N compactly supported on a compact set of R p such that

( t L 2 α ) N ψ α (γ) ≤ θ N (γ) m N |∇ γ (Φ α (Z, γ m ))| N •
One then observes that if γ is in the support of the integral defining k 2 t , the lemma implies

( t L 2 α ) N ψ α (γ) ≤ θ N (γ) c N 0 m N .
This estimate ensures the result as in Section 3.1 by taking N = p -1.

3.3.

Proofs of Lemma 3.1 and Lemma 3.4. Lemma 3.1 is an obvious consequence of the following Lemma 3.5, taking (a, b) ≡ (0, 0). We omit the proof of Lemma 3.4 which consists in a straightforward modification of the arguments developed below.

Lemma 3.5. For any integer N , one can write

( t L 1 α ) N ψ α (γ) = f N,m γ, t 1 2 ∇ γ (Φ α (Z, γ m )) ,
with |α| = m, and where f N,m is a smooth function supported on C × R p with C a fixed ring of R p , such that for any couple (a, b) ∈ N p × N p , there is a constant C (independent of m) such that

|∇ a γ ∇ b Θ f N,m (γ, Θ)| ≤ C m N +|a| (1 + |Θ| 2 ) -N -|b| 2 .
Proof of Lemma 3.5. Let us prove the result by induction over N . We start with the case when N is equal to zero. Notice that in that case the function f 0,m (γ, Θ) = ψ α (γ) does not depend on the quantity Θ = t

1 2 ∇ γ (Φ α (Z, γ m )
), so we need to check that for any a ∈ N p , there is a constant C such that (3.10)

|∇ a γ ψ α (γ)| ≤ C m |a| , when |α| = m. The case when a = 0 is obvious thanks to the uniform bound on G α . To deal with the case |a| ≥ 1, we state the following technical result which will be proved at the end of this paragraph. Lemma 3.6. For any integer k, there is a constant C such that the following bound holds for the function g n defined in (2.5), n ∈ N:

∀(ξ 1 , ξ 2 ) ∈ R 2 , (ξ 1 ∂ ξ1 + ξ 2 ∂ ξ2 ) k g n (ξ 1 , ξ 2 ) ≤ Cn k .
Let us now compute ∇ a γ ψ α (γ). Recall that according to (3.2),

ψ α (γ) = G α P, Q, η m (γ) F γ m |Pf(γ)| = F γ m d j=1 ψ α,j (γ) ,
where ψ α,j (γ) := η j (γ) θ (2α j + 1) η j,m (γ) g αj η j,m (γ)P j , η j,m (γ)Q j , η j,m (γ) := 1 m η j (γ) .

We compute

∇ a γ ψ α,j (γ) = b∈N p 0≤|b|≤|a| b a ∇ b γ θ (2α j + 1) η j,m (γ) ∇ a-b γ η j (γ)g αj η j,m (γ)P j , η j,m (γ)Q j .
Let us assume first that |a -b| = 1. Then we write, for some 1 ≤ ≤ p,

∂ γ η j (γ)g αj η j,m (γ)P j , η j,m (γ)Q j = ∂ γ η j (γ)g αj η j,m (γ)P j , η j,m (γ)Q j + η j (γ) ∂ γ η j,m (γ) 2 η j,m (γ) × (ξ 1 ∂ ξ1 + ξ 2 ∂ ξ2 )g αj η j,m (γ)P j , η j,m (γ)Q j .
Next we use the fact that there is a constant C such that on the support of θ (2α j + 1) η j,m (γ) ,

η j,m (γ) ≥ 1 Cm and |∂ γ η j,m (γ)| ≤ C m ,
so applying Lemma 3.6 gives |∇ γ ψ α,j (γ)| α j . Recalling that α j ≤ m and that, for all i ∈ {1, • • • , d}, ψ α,j is uniformly bounded, this easily achieves the proof of Estimate (3.10) in the case |a| = 1 by taking the product over j. Once we have noticed that Finally let us proceed with the induction: assume that for some integer N one can write

α a1 1 • • • α a d d (α 1 + • • • + α d )
( t L 1 α ) N -1 ψ α (γ) = f N -1,m γ, t 1 2 ∇ γ (Φ α (Z, γ m ))
where f N -1,m is a smooth function supported on C × R p , such that for any couple (a, b) ∈ N p × N p , there is a constant C (independent of m) such that

(3.11) |∇ a γ ∇ b Θ f N -1,m (γ, Θ)| ≤ C m N -1+|a| (1 + |Θ| 2 ) -(N -1)-|b| 2 .
We compute for any function Ψ(γ),

t L 1 α Ψ(γ) = i ∇ γ (Φ α (Z, γ m )) • ∇ γ Ψ(γ) 1 + t|∇ γ (Φ α (Z, γ m ))| 2 + 1 + i∆(Φ α (Z, γ m )) 1 + t|∇ γ (Φ α (Z, γ m ))| 2 Ψ(γ) -2it 1≤j,k≤p ∂ γj ∂ γ k (Φ α (Z, γ m ))∂ γj (Φ α (Z, γ m ))∂ γ k (Φ α (Z, γ m )) (1 + t|∇ γ (Φ α (Z, γ m ))| 2 ) 2 Ψ(γ) .
We apply that formula to Ψ := f N -1 γ, t

1 2 ∇ γ (Φ α (Z, γ m )
) and estimating each of the three terms separately we find (using the fact that m ≥ 1),

t L 1 α f N -1 γ, t 1 2 ∇ γ (Φ α (Z, γ m )) ≤ C 1 + t|∇ γ (Φ α (Z, γ m ))| 2 -1 × m N -1+1 (1 + t|∇ γ (Φ α (Z, γ m ))| 2 ) -(N -1) + C 1 + t|∇ γ (Φ α (Z, γ m ))| 2 -1 × m N -1 (1 + t|∇ γ (Φ α (Z, γ m ))| 2 ) -(N -1) + Ct |∇ γ (Φ α (Z, γ m ))| 2 1 + t|∇ γ (Φ α (Z, γ m ))| 2 -2 × m N -1 (1 + t|∇ γ (Φ α (Z, γ m ))| 2 ) -(N -1)
thanks to the induction assumption (3.11) along with (3.10) and the fact that on C α (Z), all the derivatives of the function ∇ γ Φ α Z, γ m are uniformly bounded with respect to α and Z. A similar argument allows to control derivatives in γ and Θ, so Lemma 3.5 is proved.

Proof of Lemma 3.6. One important ingredient in the proof is the fact that for all integers k, there is a constant C such that (3.12)

| • | k h n L 2 ≤ Cn k 2 .
This comes from the well-known relations (3.13) h

n (ξ) + ξh n (ξ) = √ 2n h n-1 h n (ξ) -ξh n (ξ) = -2(n + 1) h n+1 , which imply ξh n (ξ) L 2 ξ ≤ 2(n + 1). Using h n (ξ) = -(2n + 1)h n (ξ) + ξ 2 h n (ξ)
, an integration by parts gives

ξ 2(k+1) h 2 n (ξ)dξ = ξ 2k h n (ξ) + (2n + 1)h n (ξ) h n (ξ)dξ = -h n (ξ) 2kξ 2k-1 h n (ξ) + ξ 2k h n (ξ) dξ + (2n + 1) | • | k h n 2 L 2 .
Hence

ξ 2(k+1) h 2 n (ξ)dξ ≤ -2k ξ 2k-1 h n (ξ)h n (ξ)dξ + (2n + 1) | • | k h n 2 L 2 .
Finally using (3.13) we find

| • | k+1 h n 2 L 2 ≤ -2k ξ 2k-1 h n (ξ) -(2n + 2) 1 2 h n+1 (ξ) + ξh n (ξ) dξ + (2n + 1) | • | k h n 2 L 2 ≤ 2k(2n + 2) 1 2 | • | k h n L 2 | • | k-1 h n+1 L 2 + (2n + 1) | • | k h n 2 L 2
and we obtain (3.12) by a recursive argument. Then, we consider the case when k = 1 and we write, using the formula defining g n in (2.5) and an integration by parts,

(ξ 1 ∂ ξ1 + ξ 2 ∂ ξ2 )g n (ξ 1 , ξ 2 ) = e -i ξ 1 ξ 2 2 e -iξ2ξ ξ 1 h n (ξ 1 + ξ)h n (ξ)dξ -iξ 2 e -i ξ 1 ξ 2 2 e -iξ2ξ (ξ 1 + ξ)h n (ξ 1 + ξ)h n (ξ)dξ = e -i ξ 1 ξ 2 2 e -iξ2ξ ξ 1 h n (ξ 1 + ξ)h n (ξ)dξ -e -i ξ 1 ξ 2 2 e -iξ2ξ ∂ ξ [(ξ 1 + ξ)h n (ξ 1 + ξ)h n (ξ)] dξ = -g n (ξ 1 , ξ 2 ) -e -i ξ 1 ξ 2 2 e -iξ2ξ (ξh n (ξ 1 + ξ)h n (ξ) + (ξ 1 + ξ)h n (ξ 1 + ξ)h n (ξ)) dξ . Therefore |(ξ 1 ∂ ξ1 + ξ 2 ∂ ξ2 )g n (ξ 1 , ξ 2 )| ≤ |g n (ξ 1 , ξ 2 )| + 2 ξh n (ξ) L 2 ξ h n (ξ) L 2 ξ ≤ C n.
This concludes the proof in the case when k = 1 thanks to the Cauchy-Schwarz inequality.

Let us now consider k ≥ 1 and (ξ

1 ∂ ξ1 + ξ 2 ∂ ξ2 ) k g n (ξ 1 , ξ 2 ). By induction on k, one can prove that (ξ 1 ∂ ξ1 + ξ 2 ∂ ξ2 ) k g n (ξ 1 , ξ 2 ) is a linear combination of terms of the form ξ k1 1 ξ k2 2 ∂ k3 ξ1 ∂ k4 ξ2 g n (ξ 1 , ξ 2 ) with k 1 + k 2 + k 3 + k 4
≤ k and the coefficients of the combinations are controled in terms of k or, equivalently, a linear combination of terms of the form

e -i ξ 1 ξ 2 2 e -iξ2ξ (ξ 1 + ξ) k1 ξ k2 h (k3) n (ξ 1 + ξ)h (k4) n (ξ)dξ
with coefficients controled in terms of k. Similarly, by induction on j and thanks to (3.13), one gets

h (j) n (ξ) = j p=0 a j,p,n ξ j-p n p 2 h p+n (ξ) ,
where the a j,p,n are bounded by a constant independent of n. We are left with a linear combination of terms of the form

I k1,k2,k3,k4,p,p := n p+p 2 e -i ξ 1 ξ 2 2 e -iξ2ξ (ξ 1 + ξ) k1+k3-p ξ k2+k4-p h n+p (ξ 1 + ξ)h n+p (ξ)dξ
with coefficients depending on k and controled uniformly with respect to n. By (3.12), we obtain by the Cauchy-Schwartz inequality

|I k1,k2,k3,k4,p,p | ≤ Cn p+p 2 + k 1 +k 3 -p 2 + k 2 +k 4 -p 2 ≤ Cn k .
The proposition is proved.

Optimality of the dispersive estimates

In this section, we first end the proof of Theorem 1 by proving the optimality of the dispersive estimates for groups satisfying Assumption 1.4. Then, we prove Proposition 1.7.

4.1.

Optimality for groups satisfying Assumption 1.4. Let us now end the proof of Theorem 1 by establishing the optimality of the dispersive estimate (1.19). We use the fact that there always exists λ * ∈ Λ such that (4.1)

∇ λ ζ(0, λ * ) = 0,
where the function ζ is defined in (1.12). Indeed, if not, the map λ → ζ(0, λ) would be constant which is in contradiction with the fact that ζ is homogeneous of degree 1. We prove the following proposition, which yields the optimality of the dispersive estimate of Theorem 1.

Proposition 4.1. Let λ * ∈ Λ satisfying (4.1). There is a function g ∈ D(R p ) compactly supported in a connected open neighborhood of λ * in Λ, such that for the initial data f 0 defined by

(4.2) ∀(λ, ν) ∈ r(Λ), F(f 0 )(λ, ν)h α,η(λ) = 0 for α = 0 and F(f 0 )(λ, ν)h 0,η(λ) = g(λ)h 0,η(λ) ,
there exists c 0 > 0 and x ∈ G such that

|e -it∆ G f 0 (x)| c 0 |t| -k+p-1 2 .
Proof. Let g be any smooth compactly supported function over R p , and define f 0 by (4.2). For any point x = e X ∈ G under the form X = (P = 0, Q = 0, Z, R), the inversion formula gives e -it∆ G f 0 (x) = κ λ∈Λ ν∈r * λ e it|ν| 2 +itζ(0,λ)-iλ(Z)-iν(R) g(λ)|Pf(λ)|dνdλ .

To simplify notations, we set

ζ 0 (λ) := ζ(0, λ). Setting Z = tZ * with Z * := ∇ λ ζ(0, λ * ) = 0, we get as in (2.6) e -it∆ G f 0 (x) = c 1 |t| -k 2 λ∈R p e it(λ•Z * -ζ0(λ)) g(λ)|Pf(λ)|dλ
for some constant c 1 > 0. Without loss of generality, we can assume λ * = (1, 0, . . . , 0)

(if not, we perform a change of variables λ → Ωλ where Ω is a fixed orthogonal matrix), and we now shall perform a stationary phase in the variable λ , where we have written λ = (λ 1 , λ ). For any fixed λ 1 , the phase

Φ λ1 (λ , Z) := Z • λ -ζ 0 (λ)
has a stationary point λ if and only if Z = ∇ λ ζ 0 (λ) (with the same notation Z = (Z 1 , Z )). We observe that the homogeneity of the function ζ 0 and the definition of Z * imply that In particular, for all λ 1 = 0, Hess

Z * = ∇ λ ζ 0 (1, 0, . . . , 0) = ∇ λ ζ 0 (λ 1 , 0, . . . , 0), ∀λ 1 ∈ R , hence if λ = 0,
ζ 0 (λ 1 , 0, • • • , 0) (λ 1 , 0, • • • , 0) = 0 and the matrix Hess ζ 0 (λ 1 , 0, • • • , 0) in the canonical basis is of the form Hess ζ 0 (λ 1 , 0, • • • , 0) = 0 0 0 Hess λ ,λ ζ 0 (λ 1 , 0, • • • , 0) . Using that Hess ζ 0 (λ 1 , 0, • • • , 0) is of rank p -1, we deduce that Hess λ ,λ ζ 0 (λ 1 , 0, • • • , 0) is
(f 0 )(λ)h α,η(λ) = 0 for α = 0 and F(f 0 )(λ)h 0,η(λ) = g(λ)h 0,η(λ) .
By the inverse Fourier formula, if x = e X ∈ G is such that X = (P = 0, Q = 0, tZ), then we have e -it∆ G (x) = κ e -it λ(Z) e itζ(0,λ) g(λ) |Pf(λ)| dλ .

Since ζ(0, λ) is a linear form on each connected component of Λ, there exists Z 0 in z such that for ∀λ ∈ z * ∩ supp g, -λ(Z 0 ) + ζ(0, λ) = 0 .

As a consequence, choosing Z = Z 0 , we obtain e -it∆ G (x) = κ g(λ) |Pf(λ)| dλ = 0, which ends the proof of the result.

Appendix A. On the inversion formula in Schwartz space

This section is dedicated to the proof of the inversion formula in the Schwartz space S(G) (Proposition 1.1 page 4).

Proof. We first observe that to establish (1.8), it suffices to prove that We postpone the proof of (A.2) to the end of this section. Thanks to (A.2), the order of integration does not matter and we can transform the expression of A: we use the fact that for any α ∈ N d u λ,ν X(λ,x) h α,η(λ) |h α,η(λ) = e -iν(R)-iλ(Z)

R d e -i j ηj (λ) (ξj + e -iν(R)-iλ(Z) e -iξ•Q f 0, η -1 2 (λ) Q, R, Z dQ dR dZ dξ dν dλ .

Applying the Fourier inversion formula successively on R d , R k and on R p (and identifying r(Λ) with R p × R k ), we conclude that there exists a constant κ > 0 such that A = κ f (0) , which ends the proof of (A.1).

Let us conclude the proof by showing (A.2). We choose M a nonnegative integer. According to the obvious fact that the function (Id -∆ G ) M f also belongs to S(G) (hence to L 1 (G)), we get in view of Identity (1.11)

F(f )(λ, ν)h α,η(λ) = 1 + |ν| 2 + ζ(α, λ) -M F (Id -∆ G ) M f (λ, ν)h α,η(λ) .
In view of the definition of the Fourier transform on the group G, we thus have

F(f )(λ, ν)h α,η(λ) 2 L 2 (p λ ) = 1 + |ν| 2 + ζ(α, λ) -2M × p λ G (Id -∆ G M f (x))u λ,ν X(λ,x) h α,η(λ) (ξ) dµ(x) G (Id-∆ G M f (x ))u λ,ν X(λ,x ) h α,η(λ) (ξ) dµ(x ) dξ .
Now, by Fubini's theorem, we get

F(f )(λ, ν)h α,η(λ) 2 L 2 (p λ ) = 1 + |ν| 2 + ζ(α, λ) -2M × G G
(Id -∆ G ) M f (x)(Id -∆ G ) M f (x )(u λ,ν X(λ,x) h α,η(λ) | u λ,ν X(λ,x ) h α,η(λ) ) L 2 (p λ ) dµ(x) dµ(x ) .

Since the operators u λ,ν X(λ,x) and u λ,ν X(λ,x ) are unitary on p λ and the family (h α,η(λ) ) α∈N d is a Hilbert basis of p λ , we deduce that

F(f )(λ, ν)h α,η(λ) L 2 (p λ ) ≤ 1 + |ν| 2 + ζ(α, λ) -M (Id -∆ G ) M f L 1 (G) .

Because

Assumption 1 . 4 .

 14 For each multi-index α in N d , the Hessian matrix of the map λ → ζ(α, λ) satisfies rank D 2 λ ζ(α, λ) = p -1 where p is the dimension of the center of G. Remark 1.5. As was observed in Paragraph 1.3.3, ζ(α, λ) is a smooth function, homogeneous of degree one on Λ. By homogeneity arguments, one therefore has D 2 λ ζ(α, λ)λ = 0. It follows that there always holds rank D 2 λ ζ(α, λ) ≤ p -1 , hence Assumption 1.4 may be understood as a maximal rank property.

Corollary 1 . 6 .

 16 Let G = ⊗ 1≤m≤r G m be a decomposable, 2 step stratified Lie group where the groups G m are non trivial 2-step stratified Lie groups satisfying Assumption 1.4, of radical index k m and with centers of dimension p m . Then the dispersive estimates holds with rate |t| -q ,

  1), f (t, x) = κ λ∈Λ ν∈r * λ y∈G tr (u λ,ν X(λ,x) ) * u λ,ν X(λ,y) Θ(λ) e it|ν| 2 +itH(λ) f 0 (y)|Pf(λ)| dν dλ dµ(y) = κ λ∈Λ ν∈r * λ y∈G tr u λ,ν X(λ,y) Θ(λ) e it|ν| 2 +itH(λ) f 0 (x • y)|Pf(λ)| dν dλ dµ(y) .

1 2 •

 2 To this end, let us define m := |α| = d j=1 α j , and in the case when m = 0, let us set γ := mλ ∈ R p .

3. 1 . 1 .Proposition 3 . 2 .

 1132 Analysis of the support of the integral defining I α (Z). Let us prove the following result. One can choose the constant c 0 in (3.1) small enough so that if γ belongs to C α (Z), then γ • Z = 0.

  and observing that thanks to homogeneity arguments γ • ∇ γ ζ(α, γ) = ζ(α, γ), we deduce that for any γ ∈ C α (Z) |γ • Z| ≥ |ζ(α, γ)| -|γ| |Z -∇ γ ζ(α, γ)| . Since as argued above, γ belongs to a fixed ring and ζ(α, λ) = 0 if and only if λ = 0 (as noticed in Section 1.3.3), there is a positive constant c such that for any γ ∈ C α (Z) |ζ(α, γ)| ≥ mc , which implies in view of the definition of C α (Z) that there is a positive constant c depending only on the ring C such that |γ • Z| ≥ mc -mc 0 c .

  then the phase Φ λ1 (0, Z * ) has a stationary point. From now on we choose g supported near those stationary points (λ 1 , 0), and vanishing in the neighborhood of any other stationary point. Let us now study the Hessian of Φ λ1 in λ = 0. Again because of the homogeneity of the function ζ 0 , we have [Hess ζ 0 (λ)] λ = 0, ∀λ ∈ R p .

(A. 1 )

 1 f (0) = κ λ∈Λ ν∈r * λ tr F(f )(λ, ν) |Pf(λ)| dν dλ .Indeed, introducing the auxiliary function g defined by g(x ) := f (x • x ) which obviously belongs to S(G) and satisfies F(g)(λ, ν) = u λ,ν X(λ,x -1 ) • F(f )(λ, ν), and assuming (A.1) holds, we getf (x) = g(0) = κ λ∈Λ ν∈r * λ tr F(g)(λ, ν) |Pf(λ)| dν dλ = κ λ∈Λ ν∈r * λ tr u λ,ν X(λ,x -1 ) F(f )(λ, ν) |Pf(λ)| dν dλ ,which is the desired result.Let us now focus on (A.1). In order to compute the right-hand side of Identity (A.1), we introduceA := λ∈Λ ν∈r * λ tr F(f )(λ, ν) |Pf(λ)| dν dλ = λ∈Λ ν∈r * λ x∈G α∈N d u λ,ν X(λ,x) h α,η(λ) |h α,η(λ) |Pf(λ)| f (x) dµ(x) dν dλ ,with the notation of Section 1.3. In order to carry on the calculations, we need to resort to a Fubini argument which comes from the following identity: (A.2) α∈N d λ∈Λ ν∈r * λ F(f )(λ, ν)h α,η(λ) L 2 (p λ ) |Pf(λ)| dν dλ < ∞ .

1 2 1 21 1 2d 2 (h

 1112 Pj ) Qj h α,η(λ) (P + ξ) h α,η(λ) (ξ)dξ , where we have identified p λ with R d , and this gives rise toA = λ∈Λ ν∈r * λ x∈G ξ∈R d α∈N d e -iν(R)-iλ(Z) e -i j ηj (λ) (ξj + 1 2 Pj ) Qj h α,η(λ) (P + ξ) × h α,η(λ) (ξ) |Pf(λ)| f (x) dµ(x) dξ dν dλ ,where we recall thath α,η(λ) (ξ) = d j=1 h αj ,ηj (λ) (ξ j ) with h αj ,ηj (λ) (ξ j ) = η j (λ) 1 4 h αj η j (λ) ξ j .Performing the change of variables     ξ j = η j (λ) ξ j P j = η j (λ) P j Q j = η j (λ) Q jfor j ∈ {1, . . . , d}, we obtain, dropping the on the variables,A = λ∈Λ ν∈r * λ (P,Q,R,Z)∈R n ξ∈R d α∈N d e -iν(R)-iλ(Z) e -i (ξ + 1 2 P )•Q d j=1 h αj (P j + ξ j ) h αj (ξ j ) × f η -1 2 (λ) P, η -1 2 (λ) Q, R, Z dP dQ dR dZ dξ dν dλ , with η -1 2 (λ) P := (η -(λ) P 1 , . . . , η -(λ) P d ) and similarly for Q.Then using the change of variables ξ j = ξ j + P j , for j ∈ {1, . . . , d}, givesA = λ∈Λ ν∈r * λ (ξ ,Q,R,Z)∈R n ξ∈R d α∈N d e -iν(R)-iλ(Z) e -i αj (ξ j ) h αj (ξ j ) × f η -1 2 (λ) (ξ -ξ), η -1 2 (λ) Q, R, Z dξ dQ dR dZ dξ dν dλ . Because (h α ) α∈N d is a Hilbert basis of L 2 (R d ), we have for all ϕ ∈ L 2 (R d ) ϕ(ξ) = α∈N d ξ ∈R d ϕ(ξ ) h α (ξ ) dξ h α (ξ) , which leads to A = λ∈Λ ν∈r * λ (Q,R,Z)∈R d+k+p ξ∈R d

1 +

 1 Card α ∈ N d / |α| = m = m + d -1 m ≤ C(m + 1) d-1 ,this ensures thatα∈N d λ∈Λ ν∈r * λ F(f )(λ, ν)h α,η(λ) L 2 (p λ ) |Pf(λ)| dν dλ (Id -∆ G ) M f L 1 (G) |ν| 2 + ζ(α, λ) -M |Pf(λ)| dν dλ .Hence takingM = M 1 + M 2 , with M 2 > k 2 implies that α∈N d λ∈Λ ν∈r * λ F(f )(λ, ν)h α,η(λ) L 2 (p λ ) |Pf(λ)| dν dλ (Id -∆ G ) M f L 1 (G)Noticing that ζ(α, λ) = 0 if and only if λ = 0 and using the homogeneity of degree 1 of ζ, yields that there exists c > 0 such that ζ(α, λ) ≥ c m |λ|. Therefore, we can end the proof of (A.2) by choosing M 1 large enough and performing the change of variable µ = m λ in each term of the above series.Proposition 1.1 is proved.

  The Heisenberg group. The Heisenberg group H d is defined as the space R 2d+1 whose elements can be written w = (x, y, s) with (x, y) ∈ R d × R d , endowed with the following product law:

	1.4.1. (x, y, s) • (x , y , s ) = (x + x , y + y , s + s -2(x | y ) + 2(y | x )),
	1.4. Examples. Let us give a few examples of well-known stratified Lie groups with a step 2 stratifi-
	cation. Note that nilpotent Lie groups which are connected, simply connected and whose Lie algebra
	admits a step 2 stratification are called Carnot groups.

  4 is given by step 2 stratified Lie group G for which ζ(0, λ) is a linear form on each connected component of the Zariskiopen subset Λ. Of course, the Heisenberg group and any tensor product of Heisenberg group is of that type. We then have the following result which illustrates that there exists examples of groups without any dispersion and which do not satisfy Assumption 1.4.

Proposition 1.7. Consider a step 2 stratified Lie group G whose radical index is null and for which ζ(0, λ) is a linear form on each connected component of the Zariski-open subset Λ. Then, there exists

  a1+

  ••+a d , the general case (when |a| > 1) is dealt with identically, we omit the details.

  also of rank p -1 and we conclude by the stationary phase theorem ([START_REF] Stein | Harmonic Analysis: Real-Variable Methods, Orthogonality, Oscillatory integrals[END_REF], Chap. VIII.2), choosing g so that the remaining integral in λ 1 does not vanish. 4.2. Proof of Proposition 1.7. Assume that G is a step 2 stratified Lie group whose radical index is null and for which ζ(0, λ) is a linear form on each connected component of the Zariski-open subset Λ. Let g be a smooth nonnegative function supported in one of the connected components of Λ and define f 0 by F

Acknowledgements. The authors wish to thank Corinne Blondel, Jean-Yves Charbonnel, Laurent Mazet, Fulvio Ricci and Michèle Vergne for enlightening discussions.


Remi Busseuil
email: remi.busseuil@lirmm.fr

Lyonel Barthe
email: lyonel.barthe@lirmm.fr

Gabriel Marchesan Almeida

Luciano Ost
email: ost@lirmm.fr

Florent Bruguier
email: bruguier@lirmm.fr

Gilles Sassatelli
email: sassatelli@lirmm.fr

Pascal Benoit
email: pascal.benoit@lirmm.fr

Michel Robert
email: michel.robert@lirmm.fr

Lionel Torres
email: lionel.torres@lirmm.fr

OPEN-SCALE: A SCALABLE, OPEN-SOURCE NOC-BASED MPSOC FOR DESIGN SPACE EXPLORATION

Keywords: MPSoCs, RTOS, NoC I

As complexity of embedded system increases, configurable hardware is becoming more attractive because it provides a fast and efficient basis for design development. As a consequence, one of the most promising embedded architecture consists in the replication of Processing Elements (PEs) connected through a Network-on-Chip (NoC). Such architectures provide computation parallelism, scalability, and reduced design time thanks to reusability. This paper describes the development of a scalable, distributed memory, opensource NoC-based platform called Open-Scale and its implementation into FPGA devices. The main objective of this platform is to provide a complete framework for research development on NoC-based distributed memory MPSoCs.

INTRODUCTION

The increasing complexity of application and higher performance demand make Multiprocessors System-on-Chip (MPSoCs) one valuable alternative for dealing with nowadays embedded requirements, due to their power efficiency and capability to increase system performance by using multiple processing elements (PEs).

Traditional communication infrastructure, like shared busses, will not be able to support the amount of communication required by such MPSoCs. In this direction NoC architectures are well-known solutions due to their scalability and power efficiency. NoC-based MPSoCs have gathered much attention from industry [START_REF]Tilera processors[END_REF][2] [START_REF] Rattner | Intel single-chip cloud computer[END_REF]. However, to the best of our knowledge, synthesizable NoC-based MPSoC architectures with Real-Time Operating System (RTOS) support are not available under public domain.

In this context, this paper describes the development of Open-Scale, i.e. an open-source RTL NoC-based MPSoC that executes a preemptive RTOS. MPSoC design and implementation is a complex and a time-consuming process. Further, the need for rapid prototyping and software/hardware co-design validation make the use of Open-Scale particularly attractive for detailed design space exploration of NoC-based MPSoCs, since it was totally validated in FGPA. Furthermore, Open-Scale provides a set of functions and services that can be used or even extended to allow different performance analysis (e.g. impact of using task migration [START_REF] Almeida | Evaluating the impact of task migration in multi-processor systems-on-chip[END_REF]). The main adopted strategies during the development phases and the impacts of those decisions in the overall system performance are described and evaluated, as well. Performance results using several benchmarks are also provided according to available components that can be easily chosen by the system designer.

The remaining of this paper is organized as follows: Section 2 provides an overview and trends in the field. In Section 3, hardware components of the proposed NoC-based MPSoC architecture are described. Section 4 discusses the available features of the RTOS such as mutexes, management of FIFOs, communication stack, etc. Finally, Section 5 draws conclusions and points out directions for future work.

II. RELATED WORK

NoC-based MPSoCs approaches are studied at different levels of abstraction that vary in flexibility, accuracy, and simulation speed. Most of them employ high-level models (e.g. analytical models) for proposing new mechanisms, such as dynamic mapping and task migration. However, such high-level models have to produce accurate results, allowing early design decisions. Thus, it is fundamental to adjust/calibrate these model parameters (e.g. task migration time), by using a reference platform (normally a RTL implementation).

Due to the number of modeled aspects, these reference platforms are provided to validate and to explore specific aspects that can contribute to the efficiency of the system. For instance, Joven et al. [START_REF] Joven | xENOC -An eXperimental Network-on-Chip Enviroment for Parallel Distributed Computing on NoC-based MPSoC Archtectures[END_REF] propose the xENOC, which is a framework that allows exploring the design of NoC-based MPSoC architectures. The framework comprises a tool, called NoCWizard, for RTL Verilog NoCs generation, which uses XML file as input. Such files provide the description of each system component (e.g. NoC router, PE) and application-platform mapping. In addition, embedded message passing interface (eMPI) is adopted to support parallel task communication.

In turn, in [START_REF] Meier | Lava: An open platform for rapid prototyping of mpsocs[END_REF] a rapid prototyping MPSoC based on model-drive approach called LAVA, is presented. The architecture provides a number of open-source IP cores such as PEs (Plasma [START_REF] Rhoads | Plasma -most mips i(tm)[END_REF], MB-Lite [START_REF] Kranenburg | Mb-lite: A robust, light-weight soft-core implementation of the microblaze architec-ture[END_REF], ZPU [START_REF] Harboe | Zylin -zcpu[END_REF]), a UART, a timer, and a CAN controller connected to a Wishbone bus [START_REF] Richard Herveille | Wishbone System-on-Chip (SoC) Interconnection Architecture for Portable IP Cores[END_REF]. Similar to the Joven's approach [START_REF] Joven | xENOC -An eXperimental Network-on-Chip Enviroment for Parallel Distributed Computing on NoC-based MPSoC Archtectures[END_REF], a XML file is used together with VHDL for describing the architecture. However, no synthesizable NoC is provided.

Tian [START_REF] Tian | Performance measurements of synchronization mechanisms on 16PE NOC based multi-core with dedicated synchronization and data NOC[END_REF] et al. present a NoC-based MPSoC design that comprises 16 MicroBlazes, employed as master PEs and 16 SSRAMs, which are used as slave. PEs are connected by two NoCs. One is used for data exchange, while a OCP-based NoC architecture is used to establish a synchronization between such PEs.

The HeMPS, a homogeneous NoC-based MPSoC platform, is described in [START_REF] Carara | HeMPS -A Framework for NoC-Based MPSoC Generation[END_REF]. The HeMPS architecture comprises MIPS-like processor (Plasma [START_REF] Rhoads | Plasma -most mips i(tm)[END_REF]), a local memory (RAM), a DMA controller and a NoC HERMES-based Network Interface (NI). This platform employs one Master-PE that is responsible to manage task mapping and system debug. In turn, Slave-PEs are responsible to execute application tasks. HeMPS has a preemptive microkernel that provides communication primitives such as WritePipe() and ReadPipe(), which are used to implement message passing communication.

HeMPS platform is quite similar to the NoC-based MPSoC proposed in this paper. The main differences between both hardware architectures are: (i) HeMPS employs the Plasma processor, while Open-Scale uses the Microblaze; (ii) HeMPS has a DMA controller, which is particular advantage for video streaming applications, since large data blocks can be transferred (cyclic operations) without CPU overhead. In terms of software, the RTOS supported in Open-Scale provides a considerable number of services (see Table I), which are necessary for proposing new adaptive mechanisms.

III. OPEN-SCALE -HARDWARE DESCRIPTION

A. System Overview

The Open-Scale employs a distributed memory/message passing approach and its main component is Network Processing Unit (NPU) [START_REF] Almeida | An adaptive message passing mpsoc framework[END_REF]. Figure 1 provides a general overview of the NPU internal architecture, which includes: (i) a SecretBlaze CPU, (ii) an embedded RAM, (iii) an interrupt controller, (iv) a timer, a UART, (v) a NI, (vi) a HERMES-based router [START_REF] Moraes | Hermes: an infrastructure for low area overhead packet-switching networks on chip[END_REF] and a (vii) Wishbone bus [START_REF] Richard Herveille | Wishbone System-on-Chip (SoC) Interconnection Architecture for Portable IP Cores[END_REF]. Open-Scale scalability is achieved by replicating as many NPUs as required. The NPU components are following described.

B. NPU components description

The SecretBlaze CPU [START_REF] Barthe | The secretblaze: A configurable and cost-effective open-source soft-core processor[END_REF] is a configurable open-source RISC soft-core processor developed by our research group. It implements the MicroBlaze instruction set architecture with a five-stage pipeline. Most instructions execute in a single clock cycle, achieving optimized performance for FPGA implementations. The development of the processor was mainly conducted keeping a modular approach to ensure reliability, efficiency across the whole design, while providing better design reuse opportunities in various research and educational projects.

The flexibility is one of the driving aspects of the Secret-Blaze design. On the one hand, the core provides several optional logical and integer instructions such as multiplication, division, and pattern operations, which balances computing performance and area cost to meet embedded system requirements. On the other hand, the SecretBlaze is a MMU less processor with a simplified memory sub-system that offers optional configurable data and instruction caches, implementing the pipelined Wishbone protocol for external memory interfaces [START_REF] Richard Herveille | Wishbone System-on-Chip (SoC) Interconnection Architecture for Portable IP Cores[END_REF]. However, no global cache coherency is provided by the Open-Scale platform.

The SecretBlaze uses an embedded RAM as local memory. The interrupt controller can handle up to 8 interrupts with masking, arming, and polling mechanisms. The timer is a 32-bit counter that can generate an interrupt according to a configurable time window. Besides, a UART interface, which is adjustable via software, can be used for debugging purposes. These components are interconnected by a standard open-source Wishbone bus [START_REF] Richard Herveille | Wishbone System-on-Chip (SoC) Interconnection Architecture for Portable IP Cores[END_REF]. The communication between the NPU and the NoC router is implemented in the NI, which defines HW/SW integration (e.g. bus width, bandwidth), as well as packing/unpacking the packets from/to the NoC. The adopted NoC router uses XY router based on the HERMES infrastructure [START_REF] Moraes | Hermes: an infrastructure for low area overhead packet-switching networks on chip[END_REF]. The NoC employs packet switching of wormhole type: the incoming and outgoing ports used to route a packet are locked during the entire packet transfer. The routing algorithm is an XY engine that allows deterministic routing. Each router processes one incoming FIFO per port. The size of FIFOs can be tuned for balancing area and performance.

C. Open-Scale area evaluation

Due to the numerous parameters that can be tuned on the platform, the evaluation of the implementation will not be given exhaustively. Indeed, both instruction and data cache sizes can be adjusted, as well as the size of NoC FIFOs. In addition, the processor can optionally include hardware multiplier, divider and/or barrel shifter.

To highlight the scalability of the proposed system, the size of the whole MPSoC has been measured as a function of the number of NPUs. The development platform is based on a Virtex 5 LX 110T FPGA, using optional multiplier, divider, and barrel shifter instructions [START_REF] Barthe | The secretblaze: A configurable and cost-effective open-source soft-core processor[END_REF]. These instructions are present into the Integer Unit (IU) of the SecretBlaze instance. The cache size was set to 8KB for both instruction and data. NoC FIFOs was defined to 256 32-bits 3. The area occupation is almost linear, with an increasing number of slices with order of magnitude of 2.3 and number of 6-input Look-Up Tables (LUTs) of 2.12 in average when the number of cores doubles.

IV. OPEN-SCALE -SOFTWARE DESCRIPTION

In order to keep the distributed memory structure and to preserve the scalability of the system, each NPU operates asynchronously and uses a MPI-like API for message passing communication. Global decisions are performed in a distributed fashion and no global shared-memory is used.

Due to the distributed memory characteristic of Open-Scale, applications are described using a Kahn Process Network (KPN) formalism [START_REF] Kahn | Coroutines and networks of parallel programming[END_REF], which allows parallel computation of the tasks. The KPN computation model allows deterministic behavior of the application in an asynchronous way. Furthermore, tasks placement can be optimized depending the user requirements (e.g. computation time, energy consumption).

MPI provides a comprehensive number of primitives that relate to general-purpose distributed computing; a number of works have devised lightweight implementations supporting only a subset of MPI mechanisms for embedded processors and systems. This makes sense since KPN formalism offers a sufficient support that requires only blocking read operations, which are necessary to model, for instance, data flow (e.g. video and audio) applications. Some MPI implementations are layered, and advanced communication synchronization primitives (e.g. collective) found in the upper layers make use of the simple point-to-point primitives such as MPI Send() and MPI Receive(). This enables using these collective mechanisms in an application-specific basis in case they prove necessary.

Each NPU runs a tiny preemptive RTOS that was further extended from Steve Roads Plasma RTOS [START_REF] Rhoads | Plasma -most mips i(tm)[END_REF]. Such RTOS structure is depicted in Figure 3, which comprises 4 categories: (i) basic RTOS services (e.g. function calls), (ii) communication, (iii) drivers, and (iv) libraries.

Furthermore, the RTOS provides multi-threaded preemptive execution, using a scheduler based on thread priorities that is executed periodically according to a fixed timeslot, which can be defined by the user. A round robin scheduling algorithm is executed when all tasks have the same priority. The RTOS allows the use of semaphores and mutexes, communication between local and remote tasks, and dynamic memory allocation, as well. Further, it also provides the standard C library together with a compact math library that allows floating point operations as well as software multiplications/divisions. Timer and UART drivers are also available in the platform.

A. Open-Scale RTOS development

The Open-Scale RTOS was implemented in such a way that users can easily choose which features are needed in their implementation in order to either save memory or meet performance requirements. In this scenario, new services and features were implemented in order to be compliant with the SecretBlaze architecture, while providing more efficiency in terms of management and QoS support (Table I).

Table I summarizes some services that were included in Open-Scale RTOS. As mentioned before, one of the goals of Open-Scale is to explore adaptive mechanisms (e.g. dynamic frequency scaling, task migration). For instance, to enable dynamic load balancing, the system has to be able to migrate running tasks from one to another NPU. For that reason, a run-time loading mechanism was included to allow compiled separately applications from the RTOS being dynamically uploaded at run-time.

Besides, a preemptive round-robin scheduler based on thread credits has been implemented, avoiding task execution starvation. Intra/extra-NPU communications were extended to provide more flexibility and system performance. For example, the RAW protocol was implemented in order to achieve better performance when compared to TCP/UDP (as shown in Figure 6). Further, three online system-monitoring mechanisms were included: (i) CPU utilization, (ii) FIFO filling, and (iii) CPU frequency. Once monitored information is provided, online decisions can be taken by decision-making mechanisms, like a runtime control system used for regulating NPU frequency that were added [START_REF] Almeida | PI and PID Regulation Approaches for Performance-Constrained Adaptive Multiprocessor System-on-Chip[END_REF]. Furthermore, an API, new drivers (e.g. UART, frequency scaling and timer), as well as dynamic mapping heuristics were included to provide more design space exploration alternatives

B. Memory Management

As explained in Section III.B, the Open-Scale hardware architecture does not include a memory management unit (MMU). This design choice is crucial in the development of the RTOS, once memory management in software must be carefully handled. Two major issues have been faced concerning memory management: the heap control and the dynamic task loading.

The first strategy to deal with memory management is the use of paging. In such a scenario, each task has its own virtual memory, and address translations are performed at run-time in order to access physical memory. However, a Translation Logic Buffer (TLB) provided by the MMU usually executes this process. A software run-time address translation has a heavy cost in term of computational time as translation has to be performed for each memory access. For this reason we have not adopted this solution in our design.

The second way of managing memory is using dynamic memory allocation/deallocation: whenever a new space is required, the RTOS searches for a contiguous available space in the memory large enough for storing the information. Once the information is not required any more, the memory space is deallocated. This approach creates memory fragmentation, but leads to smaller computation cost compared to software paging. Moreover, some techniques exist to defragment the memory efficiently [START_REF]File system defragmentation technique via write allocation[END_REF].

One possible solution for enabling the loading of tasks without MMU relies on a feature that is partly supported by the GCC compiler that enables to emit relocatable code (PIC: Position Independent Code). This feature, generally used for shared libraries, generates only relative jumps and accesses data locations and functions using a Global Offset Table (GOT), which is embedded into a generated ELF file. A specific post-processing tool, which operates on this format, was used for reconstructing a completely relocatable executable. Experiments show that both memory and performance overheads remain under 5% for this solution which is clearly acceptable [START_REF] Almeida | An adaptive message passing mpsoc framework[END_REF].

C. Communication Services

Open-Scale supports two types of communication: (i) Intra-NPU communication, and (ii) Inter-NPU communication.

Intra-NPU communications are handled through software FIFOs. Whenever a packet is sent to a specific task, an exception is raised to notify the receiving task that there is an incoming data. The receiving task is then scheduled for execution in order to process the received packet. In turn, inter-NPU communications are more difficult to handle due to the globally asynchronous network management that is required. A communication stack, based on classical TCP/UDP and IP standard has been adopted in [START_REF] Busseuil | A self-adaptive communication protocol allowing fine tuning between flexibility and performance in homogeneous mpsoc systems[END_REF]. Each service (e.g. task communication, task loading) is linked to a particular port. For each incoming packet of the NPU, the appropriated service associated to the destination port is executed. Concerning reliability issues, the communication stack also provides optional re-routing and CRC error checking.

Figure 4 illustrates the four layers of the RTOS communication protocol. At the physical level, the packet reception is handled by both interrupt and polling methods. The interrupt occurs when the number of elements inside the incoming FIFO reaches a given threshold, while the polling procedure is triggered at fixed timeslot from a timer interrupt.

D. Open-Scale RTOS evaluation 1) Memory Occupation

The compiled operating system is 57KB big. Users can easily choose on using or not particular features in order to reduce the RTOS size. If communication between remote tasks is not required, the RTOS size can be reduced down to 47KB.

2) Time Performance

To enhance the overall computing performance, each NPU must have the capability to process incoming data regarding the available bandwidth of the NoC. Although this assumption is completely dependent of the application's nature with its mapping onto the platform, two kinds of results can be expected when evaluating the performance. The first is the service performance provided by the RTOS while the second concerns the computation time of wellknown applications. The performance of basic RTOS operations has been measured and results are shown in Table III. The RTOS takes 237,000 clock cycles to boot (i.e. about 4.7ms at 50MHz). This time is required to initialize all services including the communication stack. The scheduler operation that consists on searching the task to be executed has been optimized. Hence, the rescheduling of a running task takes less than 500 clock cycles, while scheduling a different task takes around 1300 clock cycles since context switching is necessary. The system call latency is measured by the time that the RTOS service takes to handle an application request.

The execution performance has been measured based on three applications: (i) an advanced encryption standard application (AES), (ii) a classical integer benchmark application (Dhrystone 2.1 [START_REF] Weicker | Dhrystone benchmark: rationale for version 2 and measurement rules[END_REF]), and (iii) a video decoder (MJPEG). Processing time for these applications are shown in Figure 5. The standalone method shows results of an optimized execution without RTOS services and cache misses. RTOS execution presents a performance decrease of about 1.8 times. This can be mainly explained by the fact that there are cache misses, while RTOS services consist in less than 3% of the execution time. Furthermore, memory allocation (Section IV.B), leads to misaligned applications that produce several cache misses.

3) Communication

To measure the communication performance, a simple application consisting of two tasks (sender/receiver) that exchange packets of 200 bytes continuously. Figure 6 depicts the communication delay (in clock cycles) between two send/receive calls considering three protocols: (i) RAW, (ii) UDP, and (iii) TCP. Note that the results are classified in four categories: (i) SEND-START that represents the number of necessary clock cycles to open the socket and to send the first packet, (ii) SEND that comprises the remaining packets sending, (iii) RECEIVE-START that is used to defined the socket opening and the receiving of the first packet, and (iv) RECEIVE that comprises the time of the remaining received packets. The RAW protocol consists in bypassing the services provided by the communication stack in order to produce the highest bandwidth. For that reason, the RAW protocol provides lower communication latency results for the analyzed communication. For instance, regarding the SEND-START communication scenario is observed a reduction of 59% and 88%, respectively, when comparing the RAW to the UDP and the TCP protocols. In turn, the TCP protocol achieves the highest latency in the four send/receive communication scenarios.

In general, sending functions are more time consuming than receiving functions, except for the beginning of transactions. This phenomenon can be explained by the latency caused by the FIFOs: receiving functions must be in blocking-mode waiting for the incoming packet before receiving and processing it. In this context, the RAW protocol achieves the highest bandwidth with 1.3MB/s at 50MHz while UDP and TCP protocols achieve 476KB/s and 147KB/s respectively. However, it is important to mention that the RAW protocol does not provide any reliability once there is no checksum or CRC implementation. On the other hand, the UDP/TCP communication protocols provide the basic services such as opening/closing sockets, handshaking, and CRC error detection.

V. CONCLUSION AND FUTURE WORK

This work described a complete Open-Source framework for academic research and development of NoC-based MPSoC. Open-Scale join different features inherent to the state-of-the-art in MPSoCs design, which include: (i) customized RTOS implementation, covering important aspects like inter-task communication primitives, dynamic loader; (ii) NoC-based platform development; and (iii) hardware-software integration. These features were described and their design decisions were made targeting a scalable distributed memory MPSoC platform.

Open-Scale brings important advantages in the MPSoC design field like run-time monitoring that allows optimized decisions distribution for different NPU at software level. Therefore, making it possible to explore the use of dynamic mechanisms (e.g. dynamic task mapping not explored in this work), while assessing performance figures using different benchmark applications together with a RTOS.

Future works on this platform will include load balancing through task migration or frequency scaling, memory management and global cache-coherency, security development and reliability issues. More information about Open-Scale platform, as well as the open-source is available at: http://www2.lirmm.fr/openscale

Figure 1 -

 1 Figure 1 -NPU architecture overview.

  , each NPU possesses an internal RAM of 64KB. The area occupation results are given in Figure

Figure 2 -

 2 Figure 2 -Open-Scale hardware area occupation.

Figure 3 -

 3 Figure 3 -Structure of Open-Scale RTOS.

Figure 4 -

 4 Figure 4 -RTOS communication stack protocol.

Figure 5 -

 5 Figure 5 -Execution time for 3 applications for a standalone and Open-Scale RTOS.

Figure 6 -

 6 Figure 6 -Communication delay between send and receive calls regarding three protocols.

TABLE I

 I

	.	SERVICES INCLUDED INTO THE OPEN-SCALE RTOS
	Steve Roads Plasma RTOS	Newly Supported in Open-Scale
	1-generation of a single object file;	1-run-time dynamic applications
			loading;
	2-preemptive round-robin sheduler	2-preemptive round-robin scheduler
	based on thread priorities;	based on thread credits;
	3-intra-NPU communication based on local FIFOs;	3-intra-NPU communication based on messages exchanged by software FIFOs;
	4-Extra-NPU communication (e.g TCP protocol) through ethernet;	4-Extra-NPU communication (RAW protocol was included), as well as MPI_Send and MPI_Receive;
	5-interrupt and exception handling;	5-run-time monitoring support;
			6-decision-making mechanisms;
	6-dynamic memory allocation and	7-a run-time control system used for
	deallocation;		regulating NPU frequency;
			8-API with new primitives, etc;
	7-queues, semaphores, mutexes.	9 -development of new drives;
			10-dynamic mapping heuristics.

TRANSPORT LAYER TCP PROTOCOL: IP_ProcessTCPPacket() UDP PROTOCOL: IP_ProcessIOPacket() NETWORK LAYER IP PROTOCOL: IPReceive() / IPSendPacket() PHYSICAL LAYER HERMES NOC: FIFO Interruptions + Polling USUAL LINK OPTIONAL LINK

  Table II shows the resulting RTOS size considering different hardware/software optimizations.

	APPLICATION LAYER	OS_ReceiveTaskNPU()	OS_MPISend() OS_MPIReceive()	OS_MPISend() OS_MPIReceive()

TABLE II .

 II

		RTOS SIZE
		with HD	without HD
		optimization	optimization
	with communication layer	57.4 KB	65.4 KB
	without communitcation layer	47.6 KB	55.4 KB

TABLE III

 III

	.	RTOS OPERATION PERFORMANCE
		Start	Rescheduling	System Latency
	Time (clock cycles)	237,845	487 -1307	305