Chieh-Lei Wong

ON THE LACK OF COMPACTNESS ON STRATIFIED LIE GROUPS

Keywords: Lie groups, lack of compactness of the critical Sobolev injection, profiles, wavelets, unconditional bases

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1. Introduction 1.1. Lack of compactness. The method of profile decomposition was first introduced by Haïm Brézis and Jean-Michel Coron [START_REF] Brézis | Convergence de solutions de H-systèmes et application aux surfaces à courbure moyenne constante[END_REF], [START_REF] Brézis | Convergence of solutions of H-systems or how to blow bubbles[END_REF] (see also Michael Struwe [START_REF] Struwe | A global compactness result for boundary value problems involving limiting nonlinearities[END_REF]), with roots in the concentrated compactness method of Pierre-Louis Lions [START_REF] Lions | The concentration-compactness principle in the calculus of variations. The limit case, part 1[END_REF], [START_REF] Lions | The concentration-compactness principle in the calculus of variations. The limit case, part 2[END_REF]. In a paper by Patrick Gérard [START_REF] Gérard | Description du défaut de compacité de l'injection de Sobolev[END_REF], the defect of compactness of the Sobolev embedding Hs ֒→ L p is described in terms of a sum of rescaled and translated orthogonal profiles, up to a small term in L p . This was generalized to other Sobolev spaces by Stéphane Jaffard [START_REF] Jaffard | Analysis of the lack of compactness in the critical Sobolev embeddings[END_REF], to Besov spaces by Gabriel Koch [START_REF] Koch | Profile decompositions for critical Lebesgue and Besov space embeddings[END_REF], and finally to general critical embeddings X ֒→ Y including a wide range of functional spaces (Lebesgue L p , Sobolev Hs , Besov Bs p,q , Triebel-Lizorkin F s p,q only to name a few) by Hajer Bahouri, Albert Cohen and Gabriel Koch [START_REF] Bahouri | A general wavelet-based profile decomposition in the critical embedding of functions spaces[END_REF]. The interested reader can also refer to Kyril Tintarev and Karl-Heinz Fieseler [START_REF] Tintarev | Concentration compactness : functional-analytic grounds and applications[END_REF] for an abstract, functional analytic presentation of the concept in various settings. These profile decomposition techniques have been successfully used for studying nonlinear PDEs, namely :

• the description of bounded energy sequences of solutions of the defocusing semi-linear quintic wave equation, up to remainder terms negligible in energy norm by Hajer Bahouri and Patrick Gérard [START_REF] Bahouri | High frequency approximations of solutions to critical nonlinear wave equations[END_REF],

• the characterization of the defect of compactness for Strichartz estimates for the Schrödinger equation by Sahbi Keraani [START_REF] Keraani | On the defect of compactness for the Strichartz estimates of the Schrödinger equation[END_REF],

• the understanding of features of solutions of nonlinear wave equations with exponential growth by Hajer Bahouri, Mohamed Majdoub and Nader Masmoudi [START_REF] Bahouri | On the lack of compactness in the 2D critical Sobolev embedding[END_REF],

• the sharp estimates of the lifespan of the focusing critical semi-linear wave equation by means of the energy size of the Cauchy data by Carlos E. Kenig and Frank Merle [START_REF] Kenig | Global well-posedness, scattering and blow-up for the energy-critical, focusing, nonlinear wave equation in the radial case[END_REF],

• the study of bilinear Strichartz estimates for the wave equation by Terence Tao [START_REF] Tao | An inverse theorem for the bilinear L 2 Strichartz estimate for the wave equation[END_REF].

For more applications, refer for instance to Isabelle Gallagher and Patrick Gérard [START_REF] Gallagher | Profile decomposition for the wave equation outside a convex obstacle[END_REF], Isabelle Gallagher [START_REF] Gallagher | Profile decomposition for solutions of the Navier-Stokes equations[END_REF], Camille Laurent [START_REF] Laurent | On stabilization and control for the critical Klein-Gordon equation on a 3D compact manifold[END_REF], Hajer Bahouri and Isabelle Gallagher [START_REF] Bahouri | Weak stability of global solutions to the incompressible Navier-Stokes equations[END_REF], Hajer Bahouri, Jean-Yves Chemin and Isabelle Gallagher [START_REF] Bahouri | Stability of rescaled weak convergence for the Navier-Stokes equations[END_REF], and multiple references therein. Apart Jamel Benameur [START_REF] Benameur | Description du défaut de compacité de l'injection de Sobolev sur le groupe de Heisenberg[END_REF] who solved the case of the Heisenberg group (2008), publications were focused on the Euclidean case. This paper provides a positive answer to the natural question of extending the description of the lack of compactness in Sobolev embeddings to stratified Lie groups. Though we will further see a more precise definition, here are some examples of such Lie groups :

• Euclidean cases : Abelian fields such as R d or C d , (non Abelian) upper triangular groups, • non-flat cases : Heisenberg groups H d , Carnot groups, Lie groups of polynomial growth.

We shall also assume that our stratified Lie groups G have a Hausdorff geometrical realization. Any of the above examples are likewise.

1.2. Stratified Lie groups. Definition 1.1 (FS [START_REF] Folland | Hardy spaces on homogeneous groups[END_REF]). A Lie group (G, •) is called stratified if it is connected, simply connected and its Lie algebra g decomposes as a direct sum g = V 1 ⊕ . . . ⊕ V m with :

[V 1 , V k ] = V k+1 if 1 k < m [V 1 , V m ] = {0} .
Thus g is a m-step, nilpotent and finitely generated, as a Lie algebra, by the vector subspace V 1 . So as a manifold, G possesses a sub-Riemannian structure. The exponential map is a diffeomorphism from g -→ G.

When G is identified with g via exp, the group law on G is a polynomial map provided by the Campbell-Baker-Hausdorff formula. As a result, the Lie correspondence endows G with a richer manifold structure, and subsequently all the classic notions of differentiable functions, Haar measure, functional spaces, and so on. The left-invariant Haar measure µ G on G is induced by the Lebesgue measure on its Lie algebra g, and we then define the Lebesgue spaces on G as :

L p (G) = Borel functions f G |f | p dµ G 1 p < +∞ ,
with the standard modification when p = +∞. In particular, if left translations are defined by τ x ′ (x) = x ′ •x, the property :

∀f ∈ L 1 (G), ∀y ∈ G, G f (y • x)dµ G (x) = G f (x)dx (1.1)
results from the left-invariance of µ G . As a homogeneous group, there is a natural action of dilations (δ α ) α∈R + on elements of G, given by :

∀α ∈ R * + , δ α (x) = α ⊙ x = (αx 1,1 , . . . , αx 1,dim V 1 induced by the canonical action on V 1 , α 2 x 2,1 , . . . , α 2 x 2,dim V 2 , . . . , α m x m,1 , . . . , α m x m,dim Vm ) .
In particular, it is immediate to see that :

δ 1 (x) = 1 R ⊙ x = x , and δ α (x) -1 = (α ⊙ x) -1 = α ⊙ x -1 .
This family of nonisotropic dilations is a subgroup of Aut(G). Given a dilation δ α , a linear differential operator X is said homogeneous of degree ℓ if for any function f on G, X(f

• δ α ) = α ℓ (Xf ) • δ α .
Since we shall be dealing with a homogeneous group G endowed with a natural family of dilations, we rather define a homogeneous norm

| • | G : G -→ [0, +∞] which is a C ∞ function on G\{0} such that :    ∀x ∈ G, |x -1 | G = |x| G ∀α > 0, |α ⊙ x| G = α|x| G |x| G = 0 iff x = 0 . Let us denote by Q = m k=1 k dim R V k the homogeneous dimension of G, and introduce the L 1 -normalized dilation δ 1 t f of a function f by : ∀f, ∀x ∈ G, ∀t > 0, δ 1 t f (x) = t Q f (t ⊙ x) . (1.2)
Note that, by definition,

δ 1 t preserves the norm L 1 , that is δ 1 t f L 1 (G) = f L 1 (G) .
Remark 1.2. Depending on the context i.e. the functional spaces we are working with, we might use different normalizations for a given function. The superscript in δ • t is helpful to remind which normalization is chosen.

Elements of g can be identified with differential operators of length 1 of G which are invariant under left translations. Note that a vector field X : G -→ T G is said to be left-invariant when the following diagram commutes for all h ∈ G :

G τ h ----→ G X     X T G ----→ dτ h T G
where τ h is the left translation on G defined by τ h (x) = h • x. Then it follows that for all h in G, X • τ h = dτ h • X. This infinitesimal characterization is equivalent to say that for any smooth function f , one has X(f

• τ h ) = (Xf ) • τ h .
Let n ∈ N. Let I and k be multi-indexes in N n . For a differential operator

X I = X I 1 k 1 X I 2 k 2 . . . X In kn
where the X k i 's are taken in g, we define two distinctive notions :

• its isotropic length (or order) :

|I| = I 1 + I 2 + . . . + I n ,
• its homogeneous degree : deg

X I = n i=1 I i deg X k i , where deg X k i = j if X k i ∈ V j .
Under the identification g ≃ G, polynomials on G are polynomials on g. Let us denote by P k the vector space of homogeneous polynomials of degree k, and set P = lim -→ P k . One can also define the Schwartz space on G by S(G) = S(g). Let S ′ (G) be the space of tempered distributions on G and S ′ (G)/P the space of tempered distributions modulo polynomials on G. Duality between the two spaces is achieved by the

sesquilinear product •, • : S ′ (G) × S(G) -→ C defined by f, g = G f ḡ dµ G .
It is common to use the spectral calculus of a suitable sub-Laplacian :

∆ G = X j ∈V 1 X 2 j , (1.3)
induced by the aforementioned sub-Riemannian structure of G in the intent to define a Littlewood-Paley decomposition for functions and tempered distributions on G. Restricted to C ∞ c (G) -that is the space of smooth functions with compact support defined on G, the sub-Laplacian ∆ G is a linear differential operator, homogeneous of degree 2 and formally self-adjoint :

∀f, g ∈ C ∞ c (G), ∆ G f, g = f, ∆ G g . Its closure has a domain D = u ∈ L 2 (G) ∆ G u ∈ L 2 (G)
where ∆ G u is taken in the sense of distributions. It follows that its closure is also self-adjoint and it is actually the unique self-adjoint extension of ∆ |C ∞ c (G) . We shall continue to denote this extension by ∆ G . Moreover, the homogeneous Sobolev spaces Hs (G) are defined as :

Hs (G) = f ∈ S ′ (G)/P (-∆ G ) s 2 f ∈ L 2 (G) .
(1.4)

Characterization of the lack of compactness of the Sobolev injection

Hs (G) ֒→ L p (G). Set s Q + 1 p = 1 2 where 0 < s < Q 2 •
In the framework of the Sobolev injection (see e.g. [START_REF] Bahouri | Refined inequalities on graded Lie groups[END_REF], or [START_REF] Bahouri | Espaces de Besov et estimations de Strichartz sur le groupe de Heisenberg[END_REF] for the specific case of the Heisenberg group H d ) :

Hs (G) ֒→ L p (G) , (1.5)
both spaces are homogeneous spaces with the same scaling properties since :

h Q p f • δ h Hs (G) = f Hs (G) ,
and

h Q p f • δ h L p (G) = f L p (G) .
For any function u ∈ Hs (G), if we define the operators :

τ κ u = u • τ κ -1 ,
and

δ p h u = h Q p u • δ h ,
(G) ֒→ L p (G) with s Q + 1 p = 1 2 and 0 < s < Q 2 • Let (u n ) n>0
be a sequence of bounded functions in Hs (G). Then, up to the possible extraction of a subsequence, there exist a family of functions (φ ℓ ) ℓ∈N * in Hs (G) -the so-called profiles -as well as families of scales (h ℓ ) = (h ℓ n ) and concentration cores (κ ℓ ) = (κ ℓ n ) such that : (i) the pairs (h ℓ , κ ℓ ) are pairwise orthogonal in the sense of Definition 1.3, (ii) for any L 1, we have :

u n (x) = L ℓ=1 (h ℓ n ) s-Q p φ ℓ 1 h ℓ n ⊙ ((κ ℓ n ) -1 • x) superposition of h ℓ,n -oscillatory components +r n,L (x) , (1.8)
with lim

L→+∞ lim n→+∞ r n,L L p (G) = 0 .
The profile decomposition (1.8) is asymptotically orthogonal (or almost orthogonal) in the sense that :

u n 2 Hs (G) = L ℓ=1 φ ℓ 2 Hs (G) + r n,L 2
Hs (G) + o(1) as n → +∞ .

(1.9)

This profile decomposition recovers various versions of the concentration-compactness principle, as the ones in [START_REF] Gérard | Description du défaut de compacité de l'injection de Sobolev[END_REF], [START_REF] Lions | The concentration-compactness principle in the calculus of variations. The limit case, part 1[END_REF] or [START_REF] Lions | The concentration-compactness principle in the calculus of variations. The limit case, part 2[END_REF].

Remark 1.5. For some sequences (u n ) n>0 , it may happen that the decomposition (1.8) contains only a finite number of profiles. In particular, the sequence

(u n ) n>0 is compact in L p (G) iff ∀ℓ 1, φ ℓ = 0.
1.4. Layout of this paper.

We shall prove Theorem 1.4 by transposing to stratified Lie groups the method developed in [START_REF] Bahouri | A general wavelet-based profile decomposition in the critical embedding of functions spaces[END_REF] which is mainly based on wavelet decompositions. The authors considered critical Sobolev embeddings X ֒→ Y for generic functional spaces X and Y with same scaling properties and endowed with unconditional wavelet basis. They also emphasized on two key properties : the first one is tied to nonlinear approximation, and the other one is similar to Fatou's lemma.

To achieve our goal, we firstly exhibit a wavelet basis by using the spectral calculus of the sub-Laplacian. The mother wavelet ψ will feature some nice properties (in S(G), with infinite vanishing moments in the space variable, and compactly supported in its conjugate variable). Translated and dilated copies of ψ provide an unconditional wavelet basis for both homogeneous Besov spaces Bs p,q (G) with 1 p, q < +∞, and L p (G) with 1 < p < +∞. Consequently, the rest of this paper is divided into three parts :

• Section 2 deals with Littlewood-Paley (abbreviated as LP for short) -admissible functions ψ in order to characterize the homogeneous Besov spaces Bs p,q (G),

• in Section 3, we will clarify the unconditional convergence of the wavelet expansion in both L p (G) and Bs p,q (G) via discrete sampling techniques and Banach frames, • in Section 4, we will perform the algorithm for the profiles' extraction on G, and eventually prove Theorem 1.4. Note that in Sections 2 and 3, we shall make straightforward use of several results from FS [START_REF] Folland | Hardy spaces on homogeneous groups[END_REF] and FM [START_REF] Führ | Homogeneous Besov spaces on stratified Lie groups and their wavelet characterization[END_REF]. We intentionally omit some technical proofs which can be easily found in these references.

Acknowledgements. The author would like to thank Clotilde Fermanian-Kammerer and Hajer Bahouri for numerous fruitful discussions.

LP-admissible functions and Besov spaces

The purpose of this section is to define the homogeneous Besov spaces Bs p,q (G) via a Littlewood-Paley decomposition. In the Euclidean framework, a LP-admissible function ψ is constructed by defining a dyadic partition of unity in the Fourier side, and then apply inverse Fourier F -1 . The standard method of transposing this construction to stratified Lie groups is to replace the Fourier transform by the spectral decomposition of the sub-Laplacian ∆ G (see further Lemma 2.5), and it is essentially based on convolution. Then this construction proves to be independent of the choice of the basis of the stratum V 1 . The proper framework for such Littlewood-Paley decompositions is actually the previously introduced space of tempered distributions modulo polynomials S ′ (G)/P. The convergence of these decompositions is obtained via duality with the subspace V(G) ⊂ S(G) of functions with an infinite number of vanishing moments.

Let us recall that the noncommutative convolution product of two functions f and g on G is defined by :

f * g(x) = G f (x • y -1 )g(y)dµ G (y) = G f (y)g(y -1 • x)dµ G (y) ,
and if X is a left-invariant vector field on G, then :

X(f * g)(x) = f * Xg(x) = G f (y)(Xg)(y -1 • x)dµ G (y) .
In what follows, for a function f on G, we define f (x) = f (x -1 ) and set :

∀x ∈ G, f * (x) = f (x -1 ) ⇐⇒ f * = f . (2.1)
Then one can easily verify that for

f in L 2 (G) ∩ L 1 (G), the adjoint of the convolution operator g -→ g * f is g -→ g * f * .
2.1. The subspace V(G) of functions with an infinite number of vanishing moments.

Definition 2.1. Let n ∈ N, a function f : G -→ C is said of polynomial decay of order n if there exists a constant c > 0 such that : ∀x ∈ G, |f (x)| c (1 + |x| G ) n • Definition 2.2. Let r ∈ N, a function f has vanishing moments of order r if : ∀p ∈ P r-1 , G f (x)p(x)dx = 0 .
Under the identification of G with g, the inverse map x -→ x -1 is identified with the additive inverse map x -→ -x on g. It follows that ∀p ∈ P r-1 , p ∈ P r-1 , where p is defined as above. If f has vanishing moments of order r then :

∀p ∈ P r-1 , G f (x)p(x)dx = G f (x)p(x)dx = 0 ,
which shows that f has also vanishing moments of order r. This property of having vanishing moments is central in the wavelet analysis due to the following principle : in a convolution product such as g * δ 1 t f , vanishing moments of one factor coupled with smoothness and regularity of the other factor lead to a rapidly decay at infinity of g * δ 1 t f . Thereafter, we shall denote by V(G) the subspace in the Schwartz class S(G) of functions with an infinite number of vanishing moments. We sum up below some useful topological properties of S(G) :

Properties (FS [START_REF] Folland | Hardy spaces on homogeneous groups[END_REF] - §1 D. The Schwartz class).

• S(G) is a Fréchet space whose topology is conveniently defined by any norm of the family of norms :

∀n ∈ N, f n = sup x∈G |I| n (1 + |x| G ) (n+1)(Q+1) X I f (x) .
Convolution is continuous from S(G) × S(G) to S(G). More precisely, for every n ∈ N, there exists

c n > 0 such that : ∀ϕ, ψ ∈ S, ϕ * ψ n c n ϕ n ψ n+1 . • V(G) is a closed subspace (in particular, it is complete) of S(G), with S(G) * V(G) ⊂ V(G).
If V ′ (G) denotes the topological dual of V(G), then V ′ (G) can be canonically identified with S ′ (G)/P. • For all ψ ∈ S(G), the map S ′ (G)/P -→ S ′ (G)/P defined by u -→ u * ψ is a well-defined operator and it is continuous on S ′ (G)/P. If ψ ∈ V(G), the associated convolution operator is a well-defined and continuous operator from S ′ (G)/P into S ′ (G). which should be understood as :

∀φ, η ∈ L 2 (G), f (∆ G )φ, η = +∞ 0 f (λ)dP λ (φ, η)
where dP λ (φ, η) is the unique Borel measure associated to the pair (φ, η) is a bounded integral operator in L 2 with a convolution distribution-kernel

K f in L 2 (G) satisfying ∀η ∈ S(G), f (∆ G )η = η * K f .
In the following, we shall denote this kernel K f in some abusive way by f , that is :

∀η ∈ S(G), f (∆ G )η = η * f . (2.2)
An important property due to Andrzej Hulanicki [START_REF] Hulanicki | A functional calculus for Rockland operators on nilpotent Lie groups[END_REF] is that, for smooth and rapidly decaying functions f ∈ S(R + ), the kernel associated to f (∆ G ) is a function in the Schwartz class, namely f ∈ S(G).

At this stage, we have all the ingredients for a Littlewood-Paley type decomposition :

f = j∈Z f * ψ * j * ψ j , (2.3)
where ∀j ∈ Z,

ψ j = δ 1 2 j ψ is a dilated of ψ ∈ S(G) (with the usual convention ψ 0 = ψ). Definition 2.3. A function ψ ∈ S(G) is said LP-admissible if : ∀g ∈ V(G), g = lim n→+∞ |j| n g * ψ * j * ψ j
holds, with convergence in S(G). Duality induces the convergence of the decomposition on S ′ (G)/P :

∀u ∈ S ′ (G)/P, u = lim n→+∞ |j| n u * ψ * j * ψ j .
Before stating Lemma 2.5 which is the cornerstone for the construction of LP-admissible functions, we need the next useful preliminary result.

Lemma 2.4 (FS [START_REF] Folland | Hardy spaces on homogeneous groups[END_REF] -Proposition 1.49). Let ϑ ∈ S(G) and define ϑ t = δ 1 t ϑ, then :

∀ψ ∈ S(G), ψ * ϑ 1 t -→ c ϑ ψ in S(G) , and ∀f ∈ S ′ (G), f * ϑ 1 t -→ c ϑ f in S ′ (G) , as t → 0, where c ϑ = G ϑ(x)dx. Lemma 2.5. Let φ be a function in C ∞ (R) with support in [0, 4] such that    0 φ 1 φ ≡ 1 on 0, 1 4 . Set ψ(ξ) = φ(2 -2 ξ) -φ(ξ). Thus ψ ∈ C ∞ c (R + ) with support in the interval 1 4
, 4 and we get a dyadic partition of unity with ψ :

j∈Z ψ(2 -2j ξ) 2 = 1 a.e.
Let ∆ G be the sub-Laplacian. Let ψ be the convolution distribution-kernel as defined in (2.2) which is associated to the bounded left-invariant operator ψ(∆ G ), then ψ is LP-admissible and belongs to V(G).

Proof. The spectral theorem applied to the dyadic partition of unity with ψ gives :

j∈Z ψ(2 -2j ∆ G ) * • ψ(2 -2j ∆ G ) = 1 .
Let g ∈ V(G). First observe that due to the quadratic homogeneity of ∆ G , the convolution kernel associated to ψ(2 -2j ∆ G ) coincides with ψ j = δ 1 2 j ψ. The decomposition :

g = j∈Z ψ(2 -2j ∆ G ) * • ψ(2 -2j ∆ G ) g = j∈Z g * ψ * j * ψ j (2.4)
holds in norm L 2 . Note that, since ψ is a real-valued function, one has actually ψ * = ψ.

For any integer m ∈ N, one has : Properties.

|j| m g * ψ * j * ψ j = g * δ 1 2 m+1 φ -g * δ 1 2 -m φ ,
• ψ ∈ S(G) 1 and has an infinite number of vanishing moments 2 , hence ψ ∈ V(G).

• Any function LP-admissible built according to Lemma 2.5 satisfies the relation :

∀j, ℓ ∈ Z, |j -ℓ| > 1 =⇒ ψ * j * ψ ℓ = 0 , (2.5)
resulting from :

ψ(2 -2j ∆ G ) * • ψ(2 -2ℓ ∆ G ) = 0 .
2.3. Homogeneous Besov spaces.

Definition 2.6. Let ψ ∈ V(G) be LP-admissible. Let 1 p, q +∞ and s ∈ R. The homogeneous Besov space associated to ψ is defined by :

Bs,ψ p,q (G) = u ∈ S ′ (G)/P 2 js u * ψ * j L p (G) j∈Z ∈ ℓ q (Z) , (2.6)
with the associated norm :

u Bs,ψ p,q (G) = 2 js u * ψ * j L p (G) j∈Z ℓ q (Z)
.

(2.7) The definition of homogeneous Besov spaces requires taking L p norms of elements of S ′ (G)/P. We use the canonical embedding L p (G) ֒→ S ′ (G). For p < +∞, by using that P ∩ L p (G) = {0}, one has the embedding L p (G) ֒→ S ′ (G)/P. Given u ∈ S ′ (G)/P, we define :

• u L p (G) = u + q L p (G) when u + q ∈ L p (G)
, for a suitable q ∈ P ; note that the decomposition u + q is unique since

P ∩ L p (G) = {0}, • u L p (G) = +∞ otherwise.
By contrast, the norm • L ∞ (G) can only be defined on S ′ (G) by assigning the value +∞ to any u ∈ S ′ (G)\L ∞ (G). Note that the Hausdorff-Young's inequality u * f L p (G) u L p (G) f L 1 (G) remains valid respectively : for p < +∞ : ∀f ∈ S(G), ∀u ∈ S ′ (G)/P , and for p = +∞ : ∀f ∈ S(G), ∀u ∈ S ′ (G) .

For p < +∞, if u + q ∈ L p (G) then (u + q) * ψ = u * ψ + q * ψ ∈ L p (G).
The combination of Lemma 2.5 and Definition 2.6 shows that we shall recover the usual notion of homogeneous Besov spaces built via the spectral calculus of sub-Laplacians. The definition turns out to be independent of the choice of ψ.

Theorem 2.7 (FM [START_REF] Führ | Homogeneous Besov spaces on stratified Lie groups and their wavelet characterization[END_REF] -Theorem 3.11). Let ψ 1 , ψ 2 ∈ V(G) be LP-admissible. Let 1 p, q +∞ and s ∈ R. Then Bs,ψ 1 p,q (G) = Bs,ψ 2 p,q (G) with equivalence of norms.

We will accordingly omit the ψ superscript and simply write Bs p,q (G) for any choice of LP-admissible ψ ∈ V(G). Properties.

• Bs p,q (G) is a Banach space. • For 1 p, q +∞ and s ∈ R, one has the following continuous embeddings :

V(G) ֒→ Bs p,q (G) ֒→ S ′ (G)/P , and V(G) ֒→ Bs p,q (G) ′ = B-s p,q (
G) . For p, q < +∞, V(G) is dense in Bs p,q (G).

• The space S(G) *

|j| n ψ * j * ψ j is dense in V(G), as well as in Bs p,q (G) if p, q < +∞. And in this space, the decomposition g = j∈Z g * ψ * j * ψ j has a finite number of nonzero terms.

We now extend this Littlewood-Paley decomposition to other functional spaces.

Proposition 2.8 (FM [START_REF] Führ | Homogeneous Besov spaces on stratified Lie groups and their wavelet characterization[END_REF] -Proposition 3.14). Let 1 p, q < +∞ and ψ ∈ V(G) be LP-admissible. Then for all g in Bs p,q (G) :

g = lim n→+∞ |j| n g * ψ * j * ψ j holds in Bs p,q (G).
Proof. Consider the operators Σ n : Bs p,q (G) -→ Bs p,q (G) defined by : Σ

n g = |j| n g * ψ * j * ψ j .
This family of operators (Σ n ) n∈N is bounded in norm. The Σ n 's converge strongly to the identity operator on a dense subspace of Bs p,q (G). But by boundedness of the family, this implies the strong convergence everywhere.

Proposition 2.9 (FM [START_REF] Führ | Homogeneous Besov spaces on stratified Lie groups and their wavelet characterization[END_REF] -Proposition 3.15). Let 1 < p < +∞ and ψ ∈ V(G) be LP-admissible. Then for all g in L p (G) :

g = lim n→+∞ |j| n g * ψ * j * ψ j holds in L p (G).
Proof. Since Σ n g = g * δ 1 2 n+1 φ -g * δ 1 2 -n φ, Young's inequality implies that this sequence of operators is norm bounded. It is sufficient to show the convergence of the decomposition on the dense subspace S(G). We saw previously in Lemma 2.4 that g * δ 1 2 n+1 φ -→ n→+∞ c φ g. Moreover, for n ∈ N, one has :

g * δ 1 2 -n φ(x) = 1 2 nQ G g(y)φ 1 2 n ⊙ (y -1 • x) dy = 1 2 nQ δ 1 2 n g * φ 1 2 n ⊙ x ,
and so :

g * δ 1 2 -n φ L p (G) = 1 2 nQ G δ 1 2 n g * φ 1 2 n ⊙ x p dx 1 p = 1 2 nQ 1-1 p δ 1 2 n g * φ L p (G) .
Again, δ 1 2 n g * φ -→ n→+∞ c g φ, and in particular : 1

2 nQ 1-1 p δ 1 2 n g * φ L p (G) -→ n→+∞ 0.
Hence Σ n g -→ n→+∞ c φ g and the case p = 2 determines that c φ = 1.

Characterization of Besov spaces by the discrete wavelets

We show in this section that the characterization of Bs p,q (G) by a Littlewood-Paley theory can be discretized by sampling the convolution products f * ψ * j over a discrete set Γ ⊂ G. This is equivalent to the study of the analysis operator A ψ : S ′ (G)/P ∋ f -→ A ψ f = f, ψ j,γ j∈Z γ∈Γ associated to a discrete wavelet system ψ j,γ j∈Z γ∈Γ defined by :

∀x ∈ G, ψ j,γ (x) = τ γ δ 1 2 j ψ(x) = 2 jQ ψ(γ -1 • 2 j ⊙ x) , (3.1)
where ψ ∈ V(G) is chosen as in Lemma 2.5 and ψ * = ψ. The main goal of this section is the proof of the equivalence between Besov norms and some associated discrete norm in Theorem 3.6.

We cannot mention wavelets without some history. In 1986, Yves Meyer built a wavelet basis that suits for the simultaneous characterization of any Sobolev space Hs . Wavelet bases thus provide an elegant and unifying response to the problem of exhibiting unconditional bases for a wide range of classic functional spaces. Some useful references are [START_REF] Meyer | Ondelettes et opérateurs tome 1[END_REF] and [START_REF] Meyer | Ondelettes et opérateurs tome 2 : opérateurs de Calderón-Zygmund[END_REF].

The very first wavelet bases on stratified Lie groups were obtained in 1989 by Pierre-Gilles Lemarié [START_REF] Lemarié | Bases d'ondelettes sur les groupes de Lie stratifiés[END_REF] from a spline interpolation theory. Already the sub-Laplacian ∆ G played a major role. Later in 2006 came out continuous and discrete wavelet systems by Daryl Geller and Azita Mayeli [START_REF] Geller | Continuous wavelets and frames on stratified Lie groups[END_REF] using the spectral theory of the sub-Laplacian. So it is rather natural to expect its involvement in a characterization of homogeneous Besov spaces. It was achieved in 2012 by Hartmut Führ and Azita Mayeli [START_REF] Führ | Homogeneous Besov spaces on stratified Lie groups and their wavelet characterization[END_REF]. In this section, we mainly rely on their analysis.

In order to discretize norms over elementary tiles of G and perform a multi resolution analysis (MRA), we shall introduce the notion of regular sampling sets. For instance, lattices in R d are a special class of regular sampling sets which are also cocompact discrete subgroups. However, some Lie groups fail to admit lattices, whereas by contrast there are always sufficiently dense regular sampling sets as indicated in the following lemma.

Lemma 3.3. Given a stratified Lie group G with a Hausdorff geometrical realization, for any neighborhood U of the identity, there exists a U -dense regular sampling set.

Proof. There exists Γ ⊂ G and a relatively compact set W with non-empty interior, such that γ∈Γ γW covers G (possibly up to a set of measure zero). Then V = Wx -1 0 is a Γ-tile for some point x 0 within W. Finally, by choosing β > 0 small enough, we ensure that βV ⊂ U and then βV is a βΓ-tile.

Definition 3.4. Let Γ ⊂ G. An automorphism ̺ ∈ Aut(G) is said Γ-acceptable if it leaves Γ globally invariant, that is ̺Γ ⊆ Γ.
We shall of course choose Γ such that the dyadic dilations {δ 2 j } j∈Z and for any γ ∈ Γ, the translations τ γ are acceptable automorphisms. This definition ensures the compatibility between the group law, the nonisotropic dilations, the sampling set and the iterated wavelet system.

c jγ j∈Z γ∈Γ bs p,q = j∈Z γ∈Γ 2 j s-Q p |c jγ | p q p 1 q . (3.2)
We introduce the space of coefficients bs p,q (Γ) defined as :

bs p,q (Γ) = c jγ j∈Z γ∈Γ c jγ j∈Z γ∈Γ bs p,q < +∞ , (3.3)
which will be sometimes denoted by bs p,q when there is no possible misunderstanding on Γ. The next theorem shows that Besov norms can be expressed in terms of discrete coefficients. The constants appearing in the norm equivalence depend on the functional spaces, but it does not matter if we use the same sampling set Γ simultaneously for all spaces. Theorem 3.6 (FM [START_REF] Führ | Homogeneous Besov spaces on stratified Lie groups and their wavelet characterization[END_REF] -Theorem 5.4). There exists a neighborhood U of the identity such that, for any U -dense regular sampling set Γ, one has : ∀u ∈ S ′ (G)/P, ∀1 p, q +∞, u ∈ Bs p,q (G) ⇐⇒ u, 2 -jQ ψ j,γ j∈Z γ∈Γ ∈ bs p,q (Γ) , where ψ j,γ is defined as in (3.1). In addition, in Bs p,q (G) we have the norm equivalence :

u Bs p,q (G) ∼ j∈Z γ∈Γ 2 j s-Q p | u, 2 -jQ ψ j,γ | p q p 1 q , (3.4)
with some constants depending only on p, q, s and the regular sampling set Γ.

The accuracy of the estimate improves as the tightness of the sampling set increases.

Banach wavelet frames for Besov spaces.

In Hilbert spaces, a norm equivalence such as (3.4) is sufficient to conclude that the wavelet basis is a frame, allowing the reconstruction of u from the discrete coefficients data. For Banach spaces, we need an extended definition of frames (refer to [START_REF] Gröchenig | Describing functions : atomic decompositions versus frames[END_REF]), in order to show the invertibility of the associated frame operator.

After stating several technical lemmas, we will prove that any linear wavelet combination converges unconditionally in any L p (G) in Theorem 3.9, and also in Bs p,q (G) in Theorem 3.11 whenever its coefficients lie in bs p,q . Then for a suitable choice of sufficiently dense regular sampling sets Γ, the wavelet system 2 -jQ ψ j,γ j∈Z γ∈Γ is a Banach frame for both L p (G) and Bs p,q (G).

Definition 3.7. A basis (f n ) is an unconditional basis in a Banach space if any convergent series n a n f n converges unconditionally, that is to say the series n a σ(n) f σ(n) converges to the same limit for any permutation σ of the indexes.

Recall that the sampled convolution products can be interpreted as scalar products too, that is to assert :

f * ψ * j (2 -j ⊙ γ) = f, ψ j,γ ,
where ψ j,γ denotes the wavelet of resolution 1 2 j and at position 1 2 j ⊙ γ -see (3.1). The wavelet system is now used for synthesis.

Unconditionality in

L p (G) with 1 < p < +∞.
To check the unconditionality of wavelet decompositions in both Lebesgue and homogeneous Besov spaces, we need the next preliminary estimate. Lemma 3.8. Let η, j ∈ Z with η j and n Q + 1. Let Γ ⊂ G be a dense regular sampling set, in the sense of Definition 3.1. Then :

∀x ∈ G, γ∈Γ 2 -jQ 1 + 2 η |2 -j ⊙ γ -1 • x| G n c 2 -ηQ ,
where the constant c depends only on n and Γ.

Proof. By assumption, there exists a relatively compact open W such that µ G (γW ∩ γ ′ W) = 0 when γ = γ ′ in Γ. Then, in view of the left-invariance of the Haar measure µ G in (1.1), we have :

γ∈Γ 2 -jQ 1 + 2 η |2 -j ⊙ γ -1 • x| G n γ∈Γ 1 |W| 2 -j ⊙(γW) 1 1 + 2 η |2 -j ⊙ γ -1 • x| G n dy .
For any y ∈ 2 -j ⊙ (γW), the triangle inequality gives :

1 + 2 η |y -1 • x| G 1 + 2 η c ′ |y -1 • 2 -j ⊙ γ| G + |2 -j ⊙ γ -1 • x| G 1 + 2 η c ′ 2 -j diam(W) + |2 -j ⊙ γ -1 • x| G c ′′ 1 + 2 η |2 -j ⊙ γ -1 • x| G
by using the fact that η j. As a result :

γ∈Γ 1 |W| 2 -j ⊙(γW) 1 1 + 2 η |2 -j ⊙ γ -1 • x| G n dy 1 c ′′ n γ∈Γ 1 |W| 2 -j ⊙(γW) 1 1 + 2 η |y -1 • x| G n dy = 2 -ηQ 1 c ′′ n 1 |W| G 1 1 + |y| G n dy ,
since the γW's are pairwise disjoint. For n Q + 1, the latter integral is finite. Hence the lemma.

Theorem 3.9. Let 1 p +∞. Let η, j ∈ Z be fixed with η j. Suppose that Γ ⊂ G is a regular sampling set. For all γ ∈ Γ, we consider functions f j,γ on G satisfying the following decay condition :

∀x ∈ G, ∀η, j ∈ Z, ∀γ ∈ Γ, |f j,γ (x)| c 1 1 + 2 η |2 -j ⊙ γ -1 • x| G Q+1 ,
with some constant c 1 > 0. We define f j = γ∈Γ c jγ f j,γ with {c jγ } γ ∈ ℓ p (Γ). Then the series converges unconditionally in L p (G) with :

f j L p (G) c 2 2 (j-η)Q 2 -jQ p γ∈Γ |c jγ | p 1 p , (3.5)
for some constant c 2 independent of j, η, γ and the sequence of coefficients {c jγ } γ .

Proof. There exists a Γ-tile W such that G = α∈Γ 2 -j ⊙ (αW) . Then :

f j p L p (G) = α∈Γ 2 -j ⊙(αW) γ∈Γ c jγ f j,γ (x) p dx c p 1 α∈Γ 2 -j ⊙(αW) γ∈Γ |c jγ | 1 1 + 2 η |2 -j ⊙ γ -1 • x| G Q+1 p dx .
On each integration domain 2 -j ⊙ (αW), the triangle inequality yields the estimate :

1 + 2 η |2 -j ⊙ (γ -1 • α)| G c ′′ 1 + 2 η |2 -j ⊙ γ -1 • x| G .
Then the integrand can be majorized by the constant :

γ∈Γ |c jγ | c ′′ 1 + 2 η |2 -j ⊙ (γ -1 • α)| G Q+1 p .
Hence : |a αγ | 2 (j-η)Q . Therefore, there exists a constant c 2 such that :

f j p L p (G) (c 1 c ′′ ) p |W| α∈Γ 2 -jQ γ∈Γ |c jγ | 1 1 + 2 η |2 -j ⊙ (γ -1 • α)| G Q+1 p = (c 1 c ′′ ) p |W| α∈Γ 2 -jQ γ∈Γ |c jγ |a αγ p , with a αγ = 1 1 + 2 η |2 -j ⊙ (γ -1 • α)| G Q+1 • Lemma
f j L p (G) 2 -jQ p c 1 c ′′ |W| 1 p α∈Γ γ∈Γ |c jγ |a αγ p 1 p 2 -jQ p 2 (j-η)Q c 2 γ∈Γ |c jγ | p 1 p . 3.3.2.
Unconditionality in Bs p,q (G) with 1 p, q < +∞. Lemma 3.10. With the above notations, there exists a constant c > 0 such that ∀j, ℓ ∈ Z, ∀γ ∈ Γ, ∀x ∈ G, the following estimate holds :

|ψ j,γ * ψ * ℓ (x)|      c 2 jQ 1 + 2 j |2 -j ⊙ γ -1 • x| G Q+1 if |ℓ -j| 1 0 otherwise . ( 3.6)
Proof. Let us compute :

ψ j,γ * ψ * ℓ (x) = G 2 jQ ψ(γ -1 • 2 j ⊙ y)2 ℓQ ψ 2 ℓ ⊙ (x -1 • y) dy = G 2 jQ ψ(y)ψ ℓ-j (γ -1 • 2 j ⊙ x) -1 • y) dy = 2 jQ (ψ * ψ * ℓ-j )(γ -1 • 2 j ⊙ x)
. By (2.5), this convolution product vanishes when |j -ℓ| > 1.

In the other case i.e. ℓ -j ∈ {-1, 0, +1}, the convolution products ψ * ψ ℓ-j are functions in the Schwartz class and henceforth, |ψ j,γ * ψ

* ℓ (x)| c 2 jQ 1 + 2 j |2 -j ⊙ γ -1 • x| G Q+1 for some constant c.
The next result is crucial for the upcoming Section 4.

Theorem 3.11. Let 1 p, q < +∞. If Γ is the same regular sampling set as in Theorem 3.9, then for any sequence of coefficients c jγ j∈Z γ∈Γ ∈ bs p,q (Γ), the sum f = j,γ 2 -jQ c jγ ψ j,γ converges unconditionally in Besov norm with :

f Bs p,q (G) c j∈Z γ∈Γ 2 j s-Q p |c jγ | p q p 1 q , (3.7)
for some constant c independent of c jγ j∈Z γ∈Γ .

Proof. It is sufficient to prove the norm estimate for finite sequences of coefficients c jγ j∈Z γ∈Γ . The full statement follows from the completeness of Bs p,q (G) and the property that the Kronecker symbols δ form an unconditional basis of bs p,q (this is why p, q < +∞ is required). By definition :

f Bs p,q (G) = 2 ℓs f * ψ * ℓ L p (G) ℓ∈Z ℓ q (Z) = 2 ℓs j,γ 2 -jQ c jγ ψ j,γ * ψ * ℓ L p (G) ℓ∈Z ℓ q (Z) = 2 ℓs γ∈Γ ℓ+1 j=ℓ-1 2 -jQ c jγ ψ j,γ * ψ * ℓ L p (G) ℓ∈Z ℓ q (Z) 2 ℓs ℓ+1 j=ℓ-1 γ∈Γ 2 -jQ c jγ ψ j,γ * ψ * ℓ L p (G) ℓ∈Z ℓ q (Z)
by using (2.5). For instance, let us consider the middle term when j = ℓ. Applying successively (3.6) and Theorem 3.9 where j -η = 0, we obtain that :

2 ℓs γ∈Γ 2 -ℓQ c ℓγ ψ ℓ,γ * ψ * ℓ L p (G) 2 ℓ s-Q p c 2 γ∈Γ |c ℓγ | p 1 p
. Therefore, we get :

2 ℓs γ∈Γ 2 -ℓQ c ℓγ ψ ℓ,γ * ψ * ℓ L p (G) ℓ∈Z ℓ q (Z) c 2 ℓ∈Z γ∈Γ 2 ℓ s-Q p |c ℓγ | p q p 1 q = c 2 c jγ j∈Z γ∈Γ bs p,q .
Of course, by applying similar calculations when j = ℓ ± 1, we will find that f Bs p,q (G) c c jγ j∈Z γ∈Γ bs p,q .

Generally, for sufficiently dense regular sampling sets Γ, the invertibility of the synthesis operator S ψ,Γ : f -→ S ψ,Γ f = j,γ 2 -jQ f, ψ j,γ ψ j,γ ensures that the discrete wavelet basis obtained by dilations by powers of 2 and translations provide an universal Banach frame simultaneously for all Besov spaces Bs p,q (G) with s ∈ R and 1 p, q < +∞.

Construction of the profiles on stratified Lie groups

At this stage, we have enough tools to prove Theorem 1.4 by adapting the method in [START_REF] Bahouri | A general wavelet-based profile decomposition in the critical embedding of functions spaces[END_REF]. To sum up, we have constructed LP-admissible wavelets ψ ∈ V(G), and then confirmed the existence of unconditional bases (spanned as an iterated functions system by ψ) for both L p (G) and Bs p,q (G) in Theorems 3.9 and 3.11.

Let Γ ⊂ G be a regular sampling set such that {τ γ } γ∈Γ and {δ 2 j } j∈Z are Γ-acceptable automorphisms. We introduce a new index λ = (j, γ) for the wavelet basis, where the component j = j λ is a scale index (actually the dyadic exponent) while the other component γ = γ λ is a space index. For any element f in Hs (G) = Bs 2,2 (G), its wavelet decomposition can be written as :

f = λ∈Z×Γ d λ ψ λ ,
where ψ λ is now L p -normalized according to :

∀λ ∈ Z × Γ, ψ λ (x) = 2 j λ Q p ψ(γ -1 λ • 2 j λ ⊙ x) = τ γ λ δ p 2 j λ ψ(x) . (4.1)
By Theorem 2.7, the homogeneous Besov spaces are independent of the choice of ψ ∈ V(G). By picking ψ with support in the interval 1 2 , 1 , the ψ λ 's shall form an orthonormal basis. From (4.1), one has readily : For M > 0, let us now consider the nonlinear projector Q M defined by :

ψ λ L p (G) = ψ L p (G) ψ λ Hs (G) = ψ Hs (G) , ( 4
∀f ∈ Hs (G), Q M (f ) = λ∈E M (f ) d λ ψ λ , (4.5)
where E M (f ) is the subset of Z × Γ of cardinality M corresponding to the M largest values of |d λ |. Note that such a set always exists, but the nonlinear projection (4.5) may not be unique when some |d λ | are equal. In which case, any realization of such set suits for E M (f ). Additionally, it has been proven in [START_REF] Devore | Nonlinear approximation[END_REF] or [START_REF] Devore | Compression of wavelet decompositions[END_REF] that :

lim M →+∞ sup f Hs (G) 1 f -Q M (f ) L p (G) = 0 , (4.6) with s Q + 1 p = 1 2 • Let us emphasize that the uniform convergence (4.6) of Q M (f ) to f in L p (G) is tied
to the nonlinear nature of the operator Q M . The nonlinear projection Q M (f ), sometimes called the best M -term approximation of f , has been extensively studied -refer for instance to Ronald A. DeVore [START_REF] Devore | Nonlinear approximation[END_REF] and the numerous references therein.

The proof of Theorem 1.4 is based on a diagonal subsequence, and it is structured in three main steps. We work of course under the theorem's assumptions and consider a sequence (u n ) n>0 of bounded functions in Hs (G). Then let us define : Using the nonlinear projector defined by (4.5), one gets :

K = sup
u n = M m=1 d m,n ψ λ(m,n) + u n -Q M (u n ) ,
with, in light of (4. It is clear that E(ℓ, M ) ⊆ E(ℓ, M + 1), and the number of approximate profiles ν(M ) increases by at most one unit when going from M to M + 1.

End of the proof.

To finish the proof of Theorem 1.4, the exact profiles φ ℓ are obtained as limits in Hs (G) of the approximate profiles φ ℓ,M as M → +∞, the same way as in [START_REF] Bahouri | A general wavelet-based profile decomposition in the critical embedding of functions spaces[END_REF] by means of the invariance by scaling (4.2) and the unconditionality of the wavelet basis.

Let us now estimate the error terms in (4.8). For a given L ∈ 1, M , according to Subsections 4.1 and 4.2, we can rewrite u n as :

u n = L ℓ=1 φ ℓ λ ℓ (n) + r n,L , (4.9)
where the remainder r n,L can be split into : .

Observe that each of these summands depends on the chosen value of M , but their total sum r n,L is actually independent of M . Remark 4.1. As a final observation, note that due to multiple extractions of subsequences -and the use of the underlying axiom of choice (AC), it turns out that the profile decomposition may yet not be unique.

2. 2 .

 2 The Calderón's reproducing formula. From here, we shall use the classic notation f = F f for the Fourier transform on R. Suppose that ∆ G has a spectral resolution ∆ G = +∞ 0 λdP λ , where dP λ is the spectral projection measure i.e. a measure with values in the spectrum of ∆ G . For any bounded Borel function f on R + , the operator f (∆ G ) = +∞ 0 f (λ)dP λ

  where φ ∈ S(G) is the convolution kernel of φ(∆ G ). Since φ is in the Schwartz class, it follows by Lemma 2.4 that g * δ 1 2 m+1 φ -→ m→+∞ c φ g in S(G) for some constant c φ . Hence |j| m g * ψ * j * ψ j -→ m→+∞ c φ g in S(G), and the identity (2.4) in L 2 (G) gives c φ = 1. The Calderón's decomposition g = j∈Z g * ψ * j * ψ j converges strongly and unconditionally in norm L 2 because of the unconditional convergence of the sum j∈Z ψ(2 -2j ξ) ψ(2 -2j ξ).

3. 1 .Definition 3 . 1 .

 131 Regular sampling sets. Let G be a Lie group. A subset Γ ⊂ G is called a regular sampling set if there is a relatively compact Borel neighborhood W ⊂ G of the identity element of G satisfying γ∈Γ γW = G (up to a set of measure zero), and for all α, γ ∈ Γ, α = γ, µ G (αW ∩ γW) = 0. Such a set W is called a Γ-tile. Definition 3.2. For U ⊂ G, a regular sampling set Γ is said U -dense if there exists a Γ-tile W ⊂ U .

3. 2 .

 2 Discretization of Besov norms. Definition 3.5. Fix a discrete set Γ ⊂ G. For any family c jγ j∈Z γ∈Γ of complex numbers, we define :

. 2 )

 2 as well as f Hs (G) = {d λ } ℓ 2 (Z×Γ) . (4.3) The unconditionality of this basis implies the existence of a constant D such that for any finite subset E ⊂ Z × Γ, any coefficients (c λ ) λ∈E and (d λ ) λ∈E satisfying ∀λ, |c λ | |d λ |, one has :

n>0 u n 4 . 1 .

 n41 Hs (G) < +∞ . Reordering of the wavelet decomposition. From the wavelet decomposition u n = λ∈Z×Γ d λ,n ψ λ , the summands are reordered by decreasing moduli |d λ,n | λ∈Z×Γ such that : u n = m>0 d m,n ψ λ(m,n) .

  6) : limM →+∞ sup n>0 u n -Q M (u n ) L p (G) = 0 . (4.7)Since wavelets are normalized in Hs (G), we know that : supm 1,n |d m,n | DK,where D is the constant of (4.4).Up to a possible diagonal extraction in n of a subsequence, we can assume that, for m 1, the sequence (d m,n ) n>0 converges to a finite limit depending only on m :d m = lim n→+∞ d m,n .where the sets E(ℓ, M ) form a disjoint partition of 1, M = ν(M ) ℓ=1 E(ℓ, M ).

  λ ℓ (n) -φ ℓ λ ℓ (n) + L ℓ=1 m∈E(ℓ,M ) (d m,n -d m )ψ λ(m,n) r 1 (n,L,M ) + ν(M ) ℓ=L+1 m∈E(ℓ,M ) d m,n ψ λ(m,n) + u n -Q M (u n ) r 2 (n,L,M )

Under the norm invariance by scaling ( 4 . 2 ) 1 r 2

 4212 of the L p -normalized {ψ λ } basis, we infer that M -φ ℓ Hs (G) .Since for all ℓ 1, φ ℓ,M -→ M →+∞ φ ℓ in Hs (G), one deduces that for any fixed L 1 : combining (4.4) and the norm invariance (4.2), for all M and 1 L ν(M ) fixed, one has :L ℓ=1 m∈E(ℓ,M ) (d m,n -d m )ψ λ(m,n) ,n -d m )ψ λ(m,n) n -d m | ψ Hs (G) ,n -d m )ψ λ(m,n) (n, L, M ) Hs (G) -→ M →+∞ 0 ,which in view of (1.5) ensures that the same holds for r 1 (n, L, M ) L p (G) .Moreover, the term r 2 (n, L, M ) can be viewed as the partial sum ℓ L+1 d m,n ψ λ(m,n) . So :r 2 (n, L, M ) L p (G) D ℓ L+1 d m,n ψ λ(m,n) L p (G), and by (4.7), its convergence to 0 when L → +∞ is assured. Since M L, one obtains :lim L→+∞ sup n>0 (n, L, M ) L p (G) ) + o(1) as n → +∞ , with lim L→+∞ lim n→+∞ r n,L L p (G) = 0 ,follows as a corollary of the wavelets' mutual orthogonality and their well-defined L 2 -normalization in(1.8). That eventually concludes Theorem 1.4's proof.

  both L p and Hs norms are preserved under translations u -→ τ κ u and dilations u -→ δ p h -1 u. If u is a nonzero element of Hs (G), for any sequence of points (κ n ) n∈N going to infinity in G (i.e |κ n | G -→ of positive real numbers (h n ) n∈N converging to 0 or +∞, the two sequences τ κn u We call a scale any sequence h = (h n ) n∈N of positive real numbers, and a concentration core any sequence κ = (κ n ) n∈N of points in G. Pairs (h, κ) and ( h, κ) are said orthogonal if :

							n→+∞	+∞)
	and any sequence n∈N
	and δ p h -1 n this paper, we shall prove that these invariances under τ κn and δ p u converge weakly to 0 in Hs (G), henceforth they are not relatively compact in L p (G). In n∈N h -1 n are single responsible for the lack of
	compactness of the continuous Sobolev injection Hs (G) ֒→ L p (G).
	Before stating the main result, let us introduce the notions of scales and concentration cores.
	Definition 1.3. log	h n hn	-→ n→+∞	±∞	for the scales,	(1.6)
	or h n = hn and	1 h n	|κ -1 n • κn | G -→ n→+∞	+∞	for the concentration cores.	(1.7)
	Theorem 1.4. Consider the continuous embedding Hs

  3.8 now ensures that the Schur lemma's conditions are fulfilled for the coefficients {a αγ } with max sup

	α γ∈Γ	|a αγ |, sup γ α∈Γ

We can then write :

(4.8)

Extraction of the approximate profiles.

The exact profiles φ ℓ involved in (1.8) are inferred as limits in Hs (G) of some approximate profiles φ ℓ,i obtained by the following procedure :

(1) Initialize

(2) At step i -1, assume that we have obtained ν(i -1) functions (φ 1,i-1 , φ 2,i-1 , . . . , φ ν(i-1),i-1 ), scalespace indexes (λ 1 (n), λ 2 (n), . . . , λ ν(i-1) (n)), as well as an increasing sequence of positive integers ϕ i-1 (n) such that :

where φ ℓ,i-1

are harmless summation indexes, but the subscript indicates a translated and dilated copy of φ ℓ,i-1 . (3) Add the i-th term d i ψ λ(i,ϕ i-1 (n)) either to build a new function, either to modify slightly one of the previous functions according to the next dichotomy :

• Case 1 : Assume that we can extract ϕ i (n) from ϕ i-1 (n) such that for any ℓ ∈ 1, ν(i -1) , at least one of the two conditions below is satisfied :

In that case, with ν(i) = ν(i -1) + 1, we add a new function φ ν(i),i such that :

and we keep every previous approximate profiles, by setting : ∀ℓ ∈ 1, ν(i -1) , φ ℓ,i = φ ℓ,i-1 . • Case 2 : Suppose that for some subsequence ϕ i (n) of ϕ i-1 (n) and some ℓ ∈ 1, ν(i -1) , none of the two above conditions is satisfied. Then one can check that j(λ ℓ (ϕ i (n))) -j(λ(i, ϕ i (n))) and

G only take a finite number of values as n varies.

Therefore, up to an additional subsequence extraction, we can infer the existence of finite values  ∈ Z and γ ∈ G such that ∀n > 0 :

Set ν(i) = ν(i -1). The function φ ℓ,i-1 is now replaced by :

whereas the other profiles remain unchanged i.e. for all 1, ν(i -1)

This algorithm shows that, for any M 1, there exists ν(M ) M such that :

More explicitly, for every ℓ ∈ 1, ν(M ) , we have :


Keywords: narrowing, face, speech

From the beginning of life, face and language processing are crucial for establishing social communication. Studies on the development of face and language processing systems have yielded interesting similarities such as the observation of perceptual narrowing occurring across both domains. This article reviews several functions of human communication, and then describes how the tools used to accomplish those functions are modified by perceptual narrowing, concluding that narrowing is a characteristic common to all forms of social communication. We argue that during evolution, social communication has engaged different perceptual and cognitive systems--face, facial expression, gesture, vocalization, sound, and oral language--which have emerged at different times. These systems are interactive and linked to some extent. Narrowing can in this framework be viewed as a mechanism for infants to adapt to their native social group.

ABSTRACT

From the beginning of life, face and language processing are crucial for establishing social communication. Studies on the development of face and language processing systems have yielded interesting similarities such as the observation of perceptual narrowing occurring across both domains. This article reviews several functions of human communication, and then describes how the tools used to accomplish those functions are modified by perceptual narrowing, concluding that narrowing is a characteristic common to all forms of social communication. We argue that during evolution, social communication has engaged different perceptual and cognitive systems--face, facial expression, gesture, vocalization, sound, and oral language--which have emerged at different times. These systems are interactive and linked to some extent. Narrowing can in this framework be viewed as a mechanism for infants to adapt to their native social group. Social life requires relationships with other group members, acknowledgment of their status, and communication between individuals. Depending on the species studied, communication can be achieved through vocalization, language, faces and their expressions, or some combination.

Similarities observed across species may provide insights into the relationship between different social communication tools and networks. These observations lead to the hypothesis that the emergence of communicative tools happened during evolutionary time and our present systems reflect some aspects of this evolution.

In humans, faces and language are essential for communication, but they have been traditionally studied as separate areas with scarce interaction between the two domains even when their links are acknowledged. They have in some frameworks even been conceived as independent cognitive modules. If faces provide an early channel of communication prior to the comprehension of gestural or oral language, in newborns, postnatal exposure to the mother's voice-face combination is required for recognition of the mother's face [START_REF] Sai | The role of the mother's voice in developing mother's face preference: evidence for intermodal perception at birth[END_REF]. are on the phone--invoke analysis of facial configurations to locate relevant cues for speech decoding. The integration of audio and facial information is not only crucial to speech perception. There is also strong evidence for interference between facial and vocal information during affect recognition and identity processing [START_REF] Campanella | Integrating face and voice in person perception[END_REF]. These observations point to a close link between face and language processing which, we will argue, may reflect a fundamental characteristic of how social communication evolved and how it develops in infants and children. More specifically, we will point to functional links between gestural and oral communication in non-human primates as well as infants, which suggest that social communication is a multimodal system, involving manual and visuo-facial gestures as well as vocalization. We will argue that this multimodal system is gradually tuned during development, with narrowing manifesting itself in all the different modalities of communication.

FACE PROCESSING, LANGUAGE PROCESSING, AND DEVELOPMENT

Human adults are able to recognize familiar faces with ease and are said to be expert at processing faces. Faces form a category of stimuli that are homogenous in terms of the positioning of their internal elements and we have developed a signature way to discriminate them based on configural (i.e., relational) information, such as the particular distance between the eyes, or between lips and chin. Experience likely plays a critical role in the acquisition of this face expertise (Lee, Anzures, Quinn, Pascalis, & Slater, 2011).

Language represents a key tool for social communication since it allows for transmitting cognitively complex information that face expressions cannot. It is a complex cognitive skill requiring recursion and displacement [START_REF] Chomsky | Aspects of the theory of syntax[END_REF]), yet it is acquired by children swiftly and without instruction, whereas most adults find second language learning challenging. [START_REF] Vihman | Phonological development: The origins of language in the child[END_REF].

Development within the face and language processing systems has been closely studied and interesting similarities can be observed. [START_REF] Nelson | The development and neural bases of face recognition[END_REF] proposed that face processing develops during the first years of life from a broad non-specific system to a human-tuned face processor. Furthermore, he suggested that faces observed within the infants' visual environment shape and influence the developing face system via a process that has come to be known in the literature as perceptual narrowing: a progression whereby infants maintain ability to discriminate stimuli to which they are exposed, but lose ability to discriminate stimuli to which they are not exposed. This course of responsiveness is similar to that observed for language development.

Throughout the first year of life, initial discriminatory ability reflective of a universal sensitivity to the sounds of all human languages 'narrows' as a consequence of predominant exposure to one's native language and a general scarcity of exposure to other languages [START_REF] Werker | Influences on infant speech processing: Toward a new synthesis[END_REF]. During this time window, infants become tuned to their native language and the distribution of phonetic information in the ambient language at the expense of discriminating non-native contrasts. In other words, infants become "experts" at processing frequently experienced faces and native sounds.

Scott, Pascalis, and [START_REF] Scott | A domain-general theory of the development of perceptual discrimination[END_REF] reviewed the literature on narrowing and observed that it cuts across both visual and auditory modalities, possibly reflecting development of a common neural architecture. [START_REF] Lewkowicz | The emergence of multisensory systems through perceptual narrowing[END_REF] suggest that narrowing could be a pansensory process, that is, the same phenomenon will be observed in various senses during the same period and will be part of the development of our multisensory representation of the world. We argue that the way narrowing has been described so far in the literature fails to take into account the evolution and historical timing of when face and language processing emerged.

Our point is that what drives or motivates the development of both face and language processing is the urge to communicate. To anticipate the remainder of the article, we first describe several functions of human communication, and then explain how each of those functions are modified by perceptual narrowing, concluding that narrowing is a common characteristic of all social communication.

GESTURAL AND ORAL COMMUNICATION

Human language is described as unique even if some form of communication does exist in other species. Understanding the emergence of language during evolution is a challenge, as fossil evidence does not provide much insight into oral language. Two communication means are seen as potential precursors to human language: vocal calls and gestures. It is still a highly debated issue whether language originated in manual gestures or evolved exclusively in the vocal domain. The former hypothesis considers pointing as the initial means to communicate which later developed into a gestural "language". [START_REF] Corballis | From mouth to hand: gesture, speech, and the evolution of righthandedness[END_REF] claims that language evolved from manual gestures, gradually incorporating vocal elements, implying that language involves reciprocity in the actions of partners. The mechanism could be supported by mirror neurons, located in Broca's area in humans [START_REF] Buccino | Action observation activates premotor and parietal areas in a somatotopic manner: an fMRI study[END_REF]. This area is involved with vocalization as well as manual action and could first have been used as a neural substrate for interspecific communication, and subsequently for speech processing.

In addition, gestures, and more specifically pointing, appear to be tightly associated with language development (Kita, 2003). Ocular pointing (or deictic gaze, at 6 -9 months) and later and licking, which take on communicative significance as lip smacks, tongue smacks, and teeth chatters. It may therefore be more parsimonious to propose a direct evolutionary trajectory from primate vocalizations to human speech rather than a complex route requiring an intermediate stage of gestural communication.

Our view is that evidence of functional links between gestural and oral communication, observed in non-human primates as well as infants, suggests that communication is a multimodal system, involving manual and visuo-facial gestures as well as vocalization. Human communication may have switched to oral-dominant language for several reasons, including accessibility without seeing the other person (during the night or if far away) and accessibility while doing something else with the forelimbs such as carrying or using tools [START_REF] Corballis | From mouth to hand: gesture, speech, and the evolution of righthandedness[END_REF].

The oro-facial region would have become gradually more used than the hand in human communication.

It seems clear that different kinds of communication existed before oral language.

Vocalizations, facial expressions, and visuo-facial gestures would have been part of it. voice changes if the language was in the native tongue. This outcome suggests that voice recognition develops in pace with increasing competence in language processing. However, younger infants' ability has yet not been reported and we cannot therefore conclude that narrowing has occurred in this domain.

NARROWING ACROSS DOMAINS THAT INVOLVE SOCIAL COMMUNICATION

Faces

In addition to face recognition, infants also develop facial expression recognition skills, which further feed into their social communication abilities [START_REF] Quinn | Looking across domains to understand infant representation of emotion[END_REF]. Perceptual narrowing has been found for emotion recognition in 9-month-old infants, but only for own-race faces [START_REF] Vogel | Building biases in infancy: the influence of race on face and voice emotion matching[END_REF], suggesting that perceptual narrowing affects stimuli that are relevant to communication with our relatives. story in two languages and found that monolingual 4-and 6-month-olds were able to visually discriminate between the two languages, whereas monolingual 8-month-olds were not. The link between face and language processing is further illustrated by Lewkowicz and Hansen-Tift's (2012) study investigating infants' face scanning while infants watched and listened to a female speaking their native or a non-native language. The investigators reported developmental differences in the pattern of face scanning between 4-and 8-months of age regardless of the language spoken, but 12-month-olds behaved differently for the native and non-native language.

Audiovisual speech

The older infants looked the same amount of time at the eyes and mouth when native speech was spoken, whereas they looked longer at the mouth for the non-native language. The same pattern 2013) who showed a decrease of looking toward the mouth when the face spoke the native language and an increase when the non-native language was spoken.

Music-rhythm

Music is important for communication and may be involved in functions such as comforting, courtship, movement coordination, and social cohesion [START_REF] Brown | Biomusicology, and three biological paradoxes about music[END_REF]. It requires social skills such as vocal/gestural imitation, and involves cultural transmission. It may even be considered as an early form of oral communication that emerged before language [START_REF] Fitch | The biology and evolution of music: A comparative perspective[END_REF]. If narrowing happens for any form of communication, it should also be found for music. Indeed, it has been found that 6-month-olds are able to discriminate rhythms that are specific to their culture and those unfamiliar to them; however, 12-month-olds could only do so with a rhythm specific to their own culture [START_REF] Hannon | Tuning in to musical rhythms: Infants learn more readily than adults[END_REF]. Furthermore, early and active exposure to culturespecific music rhythms and tonalities may accelerate perceptual narrowing in music (Trainor, Marie, Gerry, Whiskin, & Unrau, 2012).

Auditory speech

Narrowing of speech perception is also well documented. Infants' speech perception becomes tuned toward their native language at around 10-12 months of age. Young infants discriminate fine phonetic differences, such as differences in voice onset time, between consonants such as /pa/ vs /ba/ [START_REF] Eimas | Speech perception in infants[END_REF]. Infants are also able to discriminate vowels (e.g., /a/ vs /i/ or /i/ vs /u/, [START_REF] Trehub | Infants' sensitivity to vowel and tonal contrasts[END_REF] However, a decline in cross-language consonant perception is observed at 10-12 months of age.

Sign language

Narrowing has additionally been observed for perception of sign language (Palmer, Fais, Golinkoff, & Werker, 2012). Hearing infants are able to discriminate ASL signs at 4-but not at 14-months of age, whereas ASL-learning infants are still able to discriminate signs at the later age. This result supports the conclusion that narrowing happens for language regardless of the support of language: gesture versus oral.

NARROWING AS A CATEGORIZATION PROCESS SERVING SOCIAL NEEDS

Narrowing is thus observed for different cognitive abilities commonly involved in communication, even though not all empirical evidence uniformly shows that narrowing occurs simultaneously across different domains (see, for instance, Hayden, Bhatt, Kangas, Zieber, & Joseph, 2012, for evidence of own-race specialization several months before language narrowing). The underlying mechanism might then not be specific to one cognitive ability, but common to all communicative tools. In terms of evolution, it will have emerged first for face and facial expression processing. It should therefore have been part of rhythm and gesture primitive language before becoming part of oral language.

An unresolved question is that the concomitance of occurrence in multiple modalities does not explain why there is narrowing. To provide informed speculation on this question, we would observe that infants are born into a social group which has developed a culture of communication that is unique, opaque (i.e., association between an oral/gestural sign and a referent may be arbitrary), and subject to evolution. first 12 months, when infants mainly interact with the mother/caregiver, they will have to rapidly learn the appropriate way of communicating when interacting within the social group. The mother/caregiver thus rapidly transmits the basic aspects of communication that are crucial to be part of the community: smiling, language characteristics, and recognition of specific faces.

The child then calibrates its various communication systems using various learning abilities including imitation. If the child is exposed to several individuals, he/she will use convergence mechanisms to calibrate the system and will end up with finely tuned representations of the faces present in the environment as well as detailed representations of the phonemes and prosodic patterns in the ambient language(s). By this account, narrowing is a categorization process that serves social needs. In the language domain, infants build a broad category inclusive of the non-native contrasts that are lost and retain tightly tuned categories for native contrasts. In the same way, in the face domain, infants build a large category for otherrace faces inclusive of multiple other-race face categories (i.e., for infants exposed mainly to Caucasian faces, this category would include Asian and African faces) and build tightly tuned categories organized around subordinate-level identity information for same-race faces (i.e., Olivier vs. Paul vs. Helene). Narrowing can therefore be conceived of as a system that allows the infant to become more efficient or specialized for the social stimuli at hand in the close environment.

CONCLUSION

In the present article, we have argued that perceptual narrowing should be observed for all forms

  thinking opens up such questions as: Is perceptual narrowing amodal? Is auditory narrowing linked to visual narrowing? One argument for the link between development of face and language processing comes from neuroanatomy. The superior temporal sulcus (STS) is associated with face processing and auditory representation of speech components (Démonet, Thierry, & Cardebat, 2005; Haxby, Hoffman, & Gobbini, 2000). Belin et al. (2011) further suggest that the posterior part of the STS may be considered as an amodal 'convergence' zone which plays a key role in integrating face and voice information. These findings provide evidence for similar, interacting, and common brain circuits for face and speech processing

  index finger pointing (deictic gesture, at 9 -11 months) have been shown to be two key stages in cognitive development that are correlated with stages in speech development. Finger pointing is associated with lexicon construction and when accompanied with word production (16-20 months), fosters morphosyntax emergence. At later stages, children start using prosodic focus, i.e., vocal pointing[START_REF] Ménard | Articulatory and acoustic correlates of contrastive focus in French: a developmental study[END_REF], or constructions involving a deictic pronoun[START_REF] Diessel | The development of relative clauses in spontaneous child speech[END_REF]. It has also been suggested that the different pointing modalities may share a common cerebral network: Results of an fMRI study of multimodal pointing show a common left parietal activation in ocular, digital, and prosodic pointing[START_REF] Loevenbruck | Pointing is 'special[END_REF]. These findings reinforce the argument of a link between gesture and language.The referential and combinatorial properties of primate vocal communication suggest, however, that language is also rooted in vocalization(Arnold & Zuberbuhler, 2008):Chimpanzees produce and understand functionally referential calls, and monkeys can combine existing calls into higher-order meaningful sequences. Furthermore, MacNeilage (1998) suggests that syllables may derive from cycles of mandibular oscillation involved in chewing, sucking,

Whereas 6 -

 6 month-olds show recognition memory for different races of human faces as well as for different monkey faces, 9-to 10-month-old infants reliably discriminate only own-species faces and own-race faces (for a review, see[START_REF] Lee | Development of face processing expertise[END_REF]. Successful social communication relies on our ability to process indexical information (i.e., all the elements that allow for person identification) about the individuals we interact with, such as their identity, age, and sex. Specialization for own-race faces improves indexical information extraction. Regarding voice recognition, Johnson, Westrek, Nazzi, and Cutler (2011) found that 7-month-olds only detected

  Lewkowicz and collaborators (Lewkowicz & Ghazanfar, 2009; Pons, Lewkowicz, Soto-Faraco, & Sebastián-Gallés, 2009) have reported a decline in responsiveness to non-native audiovisual inputs by the end of the first year of life for other species sound-face matching and non-native language. Weikum and colleagues (2007) used silent video clips of a bilingual speaker telling a

  found byKubicek et al. (

  of social communication. During evolution, our social communication has used different perceptual and cognitive systems--face, facial expression, gesture, vocalization, sound, and oral language--which have emerged at different times. These systems are interactive in adults and are linked to some extent. Their development presents similarities as infants adjust to their native social group.We suggest that the adaptation is accomplished via a specific mechanism dedicated to social cognition, which encompasses the different modalities of communication, including manual and visuo-facial gesture processing as well as vocalization processing abilities, although we are uncommitted to whether such a mechanism is present at birth or is a product of development. Future behavioral and neuroimaging studies could bring further evidence for the intertwining of the development of these social abilities. Our suggestion also has predictions in the field of neurological or developmental disorders. We predict that deficits in either the development of manual gesture processing, facial gesture processing, or vocalization processing should result in disorders of social communication. This prediction is supported by work on autism spectrum disorders suggesting that social communication strongly relies on the healthy development of these different abilities[START_REF] Adolphs | Abnormal processing of social information from faces in autism[END_REF][START_REF] Baron-Cohen | Perceptual role taking and protodeclarative pointing in autism[END_REF]).

  native dental/retroflex contrasts such as the Hindi /ʈa/ versus /ta/[START_REF] Werker | Influences on infant speech processing: Toward a new synthesis[END_REF].

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	. Crucially, infants younger than 6-8 months of age can not only
	categorically discriminate native phonetic contrasts, they can also discriminate those that fall
	outside their native language. English-learning 6-to 8-month-olds, for instance, can discriminate

  The best way to integrate within the group may be to adapt rapidly to the group's social habits and communication traditions. During the

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Lamia Belouaer

Maroua Bouzid

Malek Mouhoub
email: mouhoubm@cs.uregina.ca

Dynamic Path Consistency for Spatial Reasoning

Keywords: Dynamic qualitative spatial reasoning, dynamic temporal reasoning, constraint satisfaction problem

Dealing with spatial knowledge requires the consistency of spatial information. This consistency is usually enforced by constraint satisfaction techniques including constraint propagation through arc and path consistency. While theses techniques often assume that spatial information are static, this is in general not the case in the real world. Our goal is to propose an approach to maintain the consistency of spatial knowledge in a dynamic environment. To our best knowledge no work in spatial reasoning has addressed this issue. In this paper we use a spatial ontology called SpaceOntology to describe both objects and spatial relations namely topological and distance relations between these objects. Based on a dynamic path consistency algorithm, our proposed method maintains the consistency of spatial information after adding new instances of topological relations described by SpaceOntology of a given environment. In order to evaluate the performance of our dynamic path consistency method, we conducted several tests on instantiations of SpaceOntology in addition to randomly generated spatial constraint problems. The results of these tests demonstrate the efficiency of our method to deal with large size problems in a dynamic environment.

I. INTRODUCTION

Several studies in Qualitative Spatial Reasoning (QSR) [START_REF] Vieu | Spatial representation and reasoning in artificial intelligence, Spatial and temporal reasoning[END_REF] have been dedicated to topological relations. Most of them are based on the algebra of intervals [START_REF] Allen | Maintaining knowledge about temporal intervals[END_REF]. Qualitative spatial reasoning covers several applications, including Geographic Information Systems (GIS) [START_REF] Lesbegueries | Structuration d'information spatiale qualitative pour la recherche d'information[END_REF], language processing [START_REF] Ligozat | Un nouveau domaine en Sciences Cognitives?[END_REF], etc.

While a considerable literature has been dedicated to spatial reasoning, to our knowledge, there is no work that addressed the dynamic aspect of space in spatial reasoning and especially in qualitative spatial reasoning. However, in the real world, spatial knowledge is not static since the environment evolves over time. This evolution can be caused by external events or by the effects of executing some actions. In this paper, we focus on reasoning about qualitative spatial knowledge in a dynamic way which is relevant in many real world applications including reactive planning and path finding. Let us consider the example of an accident at a nuclear plant. Generally if something goes wrong there are always planned paths from any position in the nuclear plant to the exit according to certain security criteria. These paths are called emergency exits. In the case of a disaster that causes unexpected incidents such as earthquakes, some emergency exits may be blocked which can cause considerable damage. Moreover, in these situations the temporal dimension is very important. Indeed, the deadlines are very short. Thus, the time to compute a new path must be done very quickly. This time depends on the consistency check of spatial knowledge needed to compute a feasible path to the exit.

In this regard, several techniques based on the Constraint Satisfaction Problem (CSP) formalism have been proposed [START_REF] Euzenat | Adaptation spatiale efficace de documents SMIL, 5e congrès francophone AFRIF-AFIA Reconnaissance des Formes et Intelligence Artificielle (RFIA)[END_REF] [START_REF] Balbiani | A model for reasoning about bi-dimensional temporal relations[END_REF]. To ensure the consistency of the spatial information, these latter are first represented as a CSP. The consistency of this latter is then checked. In real world problems, the major limitation of these approaches is their ability to deal with the dynamic change of information. A CSP that is consistent at a given time can become inconsistent in the future with the change of a given information. The challenge here is to maintain the consistency of the CSP anytime there is an addition or a retraction of a constraint.

In order to address this issue, we proceed as follows.

(1) we use SpaceOntology [START_REF] Belouaer | Ontology Based Spatial Planning for Human-Robot Interaction[END_REF] to represent the spatial knowledge (2) we adapt the dynamic path consistency algorithm (initially proposed for temporal constraints) [START_REF] Mouhoub | Dynamic Path Consistency for Interval-based Temporal Reasoning[END_REF] for the case of spatial relations. In order to evaluate the performance of our dynamic path consistency method, we conducted several tests on instantiations of SpaceOntology in addition to randomly generated spatial constraint problems.The results of these tests demonstrate the efficiency of our method to deal with large size problems in a dynamic environment.

The rest of the paper is structured as follows. In the next section we present an overview of SpaceOntology with a focus on topological relations. In section III we describe in details our dynamic path consistency algorithm for topological relations. A case study illustrating our proposed technique is then covered in Section IV. Section V reports the experimental tests evaluating the performance of our dynamic method. Finally, concluding remarks and future works are listed in Section VI.

X Y Y X X Y X Y Y X Y X X Y X Y X Y Y X Y X X Y X Y left ) x I ( L) ( overlaps-left O L ( ) C x ) ( overlaps-right ) O R ( right ) R ( above overlaps-above contains inside below A ) ( (O A ) (C y ) ( I y ) (O B ) ( B )
contains inside

overlaps-below

Figure 1. ABLR relations [START_REF] Euzenat | Adaptation spatiale efficace de documents SMIL, 5e congrès francophone AFRIF-AFIA Reconnaissance des Formes et Intelligence Artificielle (RFIA)[END_REF]. X denotes the rectangle defining a region and the projections on each axis ((ox) and (oy)).

II. SPATIAL REPRESENTATION

To represent spatial knowledge, we use SpaceOntology [START_REF] Belouaer | Ontology Based Spatial Planning for Human-Robot Interaction[END_REF] described in Figure 2. The concept Space represents a global environment (a country, a city, a building, . . . ). The concept Regions is a sub-space included in a given space. A region is itself considered as a space that can be decomposed into different sub-regions. From a geometric point of view, a region is defined by the smallest rectangle corresponding to its axis-aligned bounding rectangles. A spatial relation is defined by the concept SpatialRelations which includes TopologicalRelations and DistanceRelations which could be numeric (for example, distance of 3 miles) or fuzzy (for example, close to a coffee machine). The details of the SpaceOntology definition are presented in [START_REF] Belouaer | Ontology Based Spatial Planning for Human-Robot Interaction[END_REF].

In this paper, we focus on topological relations. In SpaceOntology, a topological relation is defined using the formalism ABLR depicted in Figure 1. Formally, a topological relation is a couple r X , r Y , where:

r X ∈ {L, O L , C x , I x , O R , R} and r Y ∈ {A, O A , C y , I y , O B , B}.
SpaceOntology deduces new spatial knowledge from an initial description of space in order to complete unknown knowledge. SpaceOntology defines a set of rules translating the composition table of ABLR formalism in order to infer new topological relations. In general, the result of the inference mechanism gives a disjunction of topological relations, and even in some cases leads to the universal relation. To use this new knowledge, it must be consistent with the initial knowledge.

However, to maintain spatial information consistency is made in a static way. In other words, anytime new topological relations are added/removed checking the consistency reconsiders the whole spatial description as if it were a new environment. Thus, SpaceOntology is not exploited effectively in real applications. In the next section, we propose a technique to maintain dynamically topological relations.

III. DYNAMIC QUALITATIVE SPATIAL REASONING

Our objective is the ability to deal with topological relations in a dynamic environment (add or remove topological relation) by adapting the technique presented in [START_REF] Mouhoub | Dynamic Path Consistency for Interval-based Temporal Reasoning[END_REF] for the topological relations managed in SpaceOntology. This technique concerns the dynamic constraint satisfaction in the case of temporal networks based on Allen's primitives. According to [START_REF] Mouhoub | Dynamic Path Consistency for Interval-based Temporal Reasoning[END_REF] a Dynamic Temporal Constraint Satisfaction Problem (DTCSP) is a sequence of static TCSPs (Temporal Constraint Satisfaction Problem [START_REF] Mouhoub | Systematic versus non systematic techniques for solving temporal constraints in a dynamic environment[END_REF]): T CSP 0 , T CSP 1 , T CSP 2 , . . . , T CSP n each resulting from a change in the preceding one imposed by the outside world. More precisely T CSP i+1 is obtained by performing a change on T CSP i . The initial TCSP has an empty set of constraints. A particular case is when the initial constraint is equal to the disjunction of the 13 primitives. In this particular case, removing one or more Allen primitives from the universal relation is equivalent to adding a new constraint. The details of the constraint restriction and relaxation methods are presented in [START_REF] Mouhoub | Dynamic Path Consistency for Interval-based Temporal Reasoning[END_REF].

Similarly to [START_REF] Mouhoub | Dynamic Path Consistency for Interval-based Temporal Reasoning[END_REF], let us introduce the notions of Spatial Constraint Satisfaction Problem (SCSP) and Dynamic Spatial Constraint Satisfaction Problem (DSCSP).

A. SCSP and DSCSP

A Spatial Constraint Satisfaction Problem (SCSP) can be formalized as a directed graph called spatial network. It describes a set of variables (nodes in the spatial network) and a set of constraints (arcs in the spatial network). Each variable corresponds to a spatial entity defined in SpaceOntology (instances of the concepts Regions and Space).

Each constraint is a disjunction of the topological relations defined in SpaceOntology.

A Dynamic Spatial Constraint Satisfaction Problem (DSCSP) is a sequence of static SCSPs P 0 , P 1 , P i+1 , . . . , P n each resulting from a change of the preceding one through the addition or a retraction of a topological relation.

B. Dynamic Path Consistency Algorithm for Topological Relations

We present here an adaptation of the dynamic path consistency algorithm defined in [START_REF] Mouhoub | Dynamic Path Consistency for Interval-based Temporal Reasoning[END_REF]. The idea is to maintain the path consistency in the network at any time after adding or removing a disjunction of topological relations. The addition and the retraction of the topological relations defined in SpaceOntology is propagated to the entire spatial network using transitive closure based on the composition rules defined in SpaceOntology. In the case where the topological relations suppression results in an empty constraint, the spatial network is not consistent in this case (since it is not path consistent).

In the following, we focus on how to maintain dynamically the spatial consistency after adding inferred topological relations in the knowledge base of SpaceOntology.

The idea is to transform the initial topological knowledge described in SpaceOntology into a spatial network. The addition of inferred relations can be considered as a constraint restriction or a constraint relaxation [START_REF] Mouhoub | Dynamic Path Consistency for Interval-based Temporal Reasoning[END_REF].

1) Constraint Restriction: The constraint restriction is to add a new constraint (i.e a disjunction of topological relations) in a spatial network. The method for maintaining the consistency in this case is similar to the one proposed in [START_REF] Mouhoub | Dynamic Path Consistency for Interval-based Temporal Reasoning[END_REF].

Here, the topological relations we consider are presented on two axes. We denote by

C ij =< r H ij1 , r V ij1 > ∪ < r H ij2 , r V ij2 >, .
. . < r H ijn , r V ijn > the constraint linking two vertices i and j (H for Horizontal and V for Vertical).

To adapt and optimize the algorithm described in [START_REF] Mouhoub | Dynamic Path Consistency for Interval-based Temporal Reasoning[END_REF], we decompose the constraint into two sub-sets of constraints according to the horizontal and vertical axes decomp

(C ij ) = {C H ij , C V ij } such as C H ij = r H ij1 ∪ r H ij2 . . . ∪ r H ijn and C V ij = r V ij1 ∪ r V ij2 . . . ∪ r V ijn .
The method described by our Algorithm 1 above works as follows. Compute three intersections t H t V and t. t H is the intersection of the new sub constraint corresponding to the horizontal axis with the corresponding sub constraint, t V is the intersection of the new sub constraint corresponding to the vertical axis with the corresponding sub constraint and t is the intersection of the new constraint with the corresponding constraint in the arc and path consistent graph. If the result of one of these intersections is an empty relation then the new constraint cannot be added otherwise it will violate the path consistency of the graph. Otherwise, we replace the current constraint of the graph by the result of the intersection. We then perform the dynamic Algorithm 1 Restrict(i,j) Require:

1: {new c H , new c V } ←-decomp(new constraint) 2: {C H ij , C V ij } ←-decomp(C ij ) 3: t H ←-new c H ∩ C H ij 4: t V ←-new c V ∩ C V ij 5: t ←-new constraint ∩ C ij 6: if t H = ⊘ or t V = ⊘ or t = ⊘ then 7:
return "Constraint cannot be added" return "Constraint cannot be added" 13: end if 14: end if path consistency (AdaptedPC 2 shown below and PC [START_REF] Mouhoub | Dynamic Path Consistency for Interval-based Temporal Reasoning[END_REF]) in order to propagate the update of the constraint to the rest of the graph. If the resulting graph is not path consistent then the new constraint cannot be added. for k ←-1 to n, k = x and k = y do 6:

t H ←-C H xk ∩ C H xy .C H yk 7: t V ←-C V xk ∩ C V xy .C V yk 8: if t H = C H xk or t v = C V xk then 9: C H xk ←-t H , C V xk ←-t V 10: C H kx ←-IN V ERSE(t H ) 11: C V kx ←-IN V ERSE(t V ) 12: L ←-L ∪ {(x, k)} 13: updated list ←-updated list ∪ {(x, k)} 14: t H ←-C H ky ∪ C H kx .C H xy , t V ←-C V ky ∪ C V kx .C V xy 15: end if 16: if t H = C H ky or t V = C H ky then 17: C H yk ←-IN V ERSE(t H ) 18: C V yk ←-IN V ERSE(t V ) 19: L ←-L ∪ {(k, y)} 20: updated list ←-updated list ∪ {(y, k)} 21: end if 22:
end for 23: end while 2) Constraint Relaxation: Similarly to the case of temporal relations in [START_REF] Mouhoub | Dynamic Path Consistency for Interval-based Temporal Reasoning[END_REF], the constraint relaxation corresponds to adding some topological relations to an existing constraint.

Run the dynamic path consistency algorithm starting form the triangle containing the relaxed constraint. If at least one of the other two relations belonging to the triangle is updated, we propagate in a recursive way this change to the other constraints.

IV. CASE STUDY

Let us consider the environment depicted in Figure 3.

O O O O O O 1 2 3 4 5 6
G G 1 2

Figure 3. A Scene

There is a set of regions; offices and gates {O 1 , O 2 , . . . , O 6 , G 1 , G 2 } and the following set of topological relations between them:

• O 1 < L, I y > O 2 , O 1 < L, C y > O 4 , O 1 < I x , A > O 5 ; • O 3 < R, I y > O 2 , O 3 < L, C y > O 4 ; • O 4 < O R , A > O 6 ; • O 5 < L, C y > O 6 . O1 O2 O3 O4 O5 O6 < L, Iy > < R, Iy > < L,Cy > < L,Cy > < Ix, A > < OR, A > < L,Cy > Figure 4. Spatial Network
Figure 4 shows the spatial network formalizing all the known spatial relations. We consider this network as the initial spatial state of the world.

Let us assume that, after an earthquake at a nuclear plant, a robot is sent in it in order to do some emergency tasks. The robot is guided by a human who has partial knowledge of the environment. The human implements SpaceOntology by the following informations.

The robot mission is to isolate the region in which the office O 6 is located. For this, it must block gateways such as G 1 and G 2 . However, with partial spatial knowledge the robot cannot make a decision. Based on the initial spatial description, we can define the space described by the graph in the upper left of Figure 5. Now, by exploiting the composition rules defined in SpaceOntology [START_REF] Belouaer | Ontology Based Spatial Planning for Human-Robot Interaction[END_REF], we deduce the topological relation between O 1 and O 6 from add deduced relations check for consictency

=< L, C y > ∨ < L, A > ∨ < L, O A > ∨ < L, I y > ∨ < O L , C y > ∨ < O L , A > ∨ < O L , O A > ∨ < O L , I y > ∨ < I x , C y > ∨ < I x , A > ∨ < I x , O A > ∨ < I x , I y >.
Therefore, the result of the composition is a disjunction that cannot be exploited to make a decision. In this disjunction some information are inconsistent. When relying on our approach we can check the consistency of this knowledge without any reformulation.

To check the consistency of topological knowledge, we proceed as follows. We add DR between O 1 and O 6 and compute the intersection between < L, A > and DR. This intersection returns the following result: < L, A >. This latter relation is not in conflict with the network.

V. EXPERIMENTATION

This section presents comparative experimental tests evaluating the performance in time of a static method and our dynamic technique to maintain the consistency of the spatial knowledge that evolves over time. The static method enforces the path consistency from scratch anytime a constraint is added.

We have conducted two categories of experimental tests. The first category is performed on nine instantiations of SpaceOntolgy. The second one is conducted on randomly generated consistent DSCSPs. All the procedures are coded in Java.

A. Tests Performed on Instantiations of SpaceOntology

Each instance has a specific number of regions and topological relations as shown in Table I. For each instance new topological relations are inferred in a random order.

For each instance of SpaceOntology we formalize the initial spatial knowledge in an initial SCSP where the nodes are regions and arcs are disjunctions of topological relations. Here we compare the static and dynamic methods performances to maintain the path consistency in the network after adding inferred topological relations. The inferred relations are deduced by applying the composition rules defined in SpaceOntology. The results of the experimental tests are depicted in Table I. This table shows that maintaining consistency after adding a large number of topological relations (over 50 relationships) is faster with the dynamic method than with the static one. For example, if we add 120 topological relations, the dynamic method runs in 210.03 ms, instead of 562 ms for the static method.

B. Tests on Randomly Generated DSCSPs

The criteria used to evaluate the static and dynamic methods is the running time needed to maintain the consistency of the DSCSP. To generate a consistent SCSP we first randomly generate a numeric solution and then randomly add other information to it. For this, we consider the node representing a region defined by a rectangle. This latter is identified by a couple of intervals

(R = [(b H inf , b H sup ), (b V inf , b V sup )]
). Following the generation method described in [START_REF] Mouhoub | Systematic versus non systematic techniques for solving temporal constraints in a dynamic environment[END_REF], consistent SCSPs are randomly generated as follows.

1) Generation of numeric solution. Randomly pick N pairs (I, J) where

I = (b H I inf , b H Isup ) and J= (b V I inf , b V Isup ) are pairs of integers such that b H I inf < b H Isup and b V inf < b V sup and b H inf , b H sup , b V inf , b V sup ∈ [0, . . . , Horizon]
. This set of N pairs forms the initial solution where each couple of pairs corresponds to the projection of the rectangle on x-axis and y-axis. Horizon corresponds to a max number.

2) Generation of numeric constraints. For each couple (I, J), randomly pick an interval contained within [0, . . . , Horizon] and containing the couple (I, J). This newly generated couple defines the projection of rectangle on x-axis and y-axis. From these projections, we generate the domain of the corresponding regions. 3) Generation of constraints. Compute the basic ABLR primitives that hold between each projection pair of the initial solution. Example 1: Let us assume we want to generate a consistent SCSP with N = 3 regions and Horizon = 10.

1) A numeric solution is generated: S = {( [START_REF] Hernández | Qualitative Representation of Spatial Knowledge[END_REF][START_REF] Allen | Maintaining knowledge about temporal intervals[END_REF], [START_REF] Ligozat | Un nouveau domaine en Sciences Cognitives?[END_REF][START_REF] Belouaer | Ontology Based Spatial Planning for Human-Robot Interaction[END_REF]), ([2, 8], [START_REF] Ligozat | Un nouveau domaine en Sciences Cognitives?[END_REF][START_REF] Vieu | Spatial representation and reasoning in artificial intelligence, Spatial and temporal reasoning[END_REF])( [START_REF] Belouaer | Ontology Based Spatial Planning for Human-Robot Interaction[END_REF][START_REF] Balbiani | A model for reasoning about bi-dimensional temporal relations[END_REF], [START_REF] Hernández | Qualitative Representation of Spatial Knowledge[END_REF][START_REF] Balbiani | A model for reasoning about bi-dimensional temporal relations[END_REF])} 2) Domains for the different regions are randomly generated from the numeric solution as shown in Table II. 3) ABLR primitives are computed from the pairs of projections as described in Table III. After generating the SCSP, the solving algorithm will process the list of spatial relations in an incremental way (in order to simulate a DSCSP). We start with a DSCSP having N variables and 0 constraints. Constraints are then added one by one, in arbitrary order from the randomly generated SCSP to the DSCSP until a given number of constraints C, is reached. After adding each constraint, the solving algorithm will check the path consistency of the new DSCSP. Table IV presents the results of tests performed on DSCSP instances defined by the number of variables N, their domain size and the number of constraints C. The running time of the two methods are comparable for small and medium problems (N ≤ 80). For large problems, the dynamic method is more efficient.

For those real world applications such as real time systems where a solution needs to be returned within a given deadline an alternative is to try to solve most of the constraints (instead of all of them). For this purpose, path consistency can be used as an incomplete method that returns a solution with a quality corresponding to the percentage of satisfied constraints.

For this purpose, we conducted tests where we randomly generate DSCSPs with a number of constraints ranging from 100 to 25000. Here, we compute the percentage of non violated constraints (quality of the solution) obtained at different time instants after applying both the static and dynamic methods.

Figure 6 presents the results of these experiments. In these tests we focus on the quality of the solution. While we compute the running time needed to return the solutions, we also check the percentage of non-violated constraints. We note that, for every time instant, the solutions returned by the dynamic method are of better quality than those returned by the static method. For example, for a computation time of 300 seconds the dynamic method was able to satisfy 97% of the constraints while the static method was successful for solving only 45%.

VI. CONCLUSION

In this paper we have proposed an approach to maintain the consistency of spatial knowledge in a dynamic environment. This approach is interest for many application where qualitative spatial information need to be managed in a dynamic environment. To our best knowledge no work in spatial reasoning has addressed this issue.

Our proposed method can be extended to the other formalisms for representing the topological relations and based on Allen's algebra such as rectangles algebra [START_REF] Balbiani | A new tractable subclass of the rectangle algebra[END_REF] as well as the method of n-intersections [START_REF] Egenhofer | Point-set topological spatial relations[END_REF]. In this paper, we only use the topological aspect proposed by SpaceOntology. In the near future we plan to manage all aspects of spatial knowledge in a dynamic manner. We will also explore the case of retracting topological relations. The challenge here is to restore back all the values that have been removed due the constraint we are relaxing.

Figure 2 .

 2 Figure 2. Graphical representation of SpaceOntology's concepts.

Algorithm 2

 2 AdaptedPC Require: update list 1: P C ←f alse 2: L ←update list 3: while (L = ⊘) do 4:select and delete an (x, y) from L 5:

Figure 5 .

 5 Figure 5. A scene

Figure 6 .

 6 Figure 6. Solution quality given by the static and dynamic methods.