Diego Chamorro
email: diego.chamorro@univ-evry.fr

Anca-Nicoleta Marcoci
email: anca.marcoci@utcb.ro

Liviu-Gabriel Marcoci
email: lmarcoci@instal.utcb.ro

A Bahouri

Cohen

Functional Inequalities in Stratified Lie groups with Sobolev, Besov, Lorentz and Morrey spaces

Keywords: Improved Sobolev inequalities, Sobolev spaces, Besov spaces, Classical Lorentz spaces, Stratified Lie groups. Mathematics Subject Classification 46E35, 26D10, 46E30, 22E30

When p > 1, using as base space classical Lorentz spaces associated to a weight from the Ariño-Muckenhoupt class B p , we will study Gagliardo-Nirenberg inequalities. As a by-product we will also consider Morrey-Sobolev inequalities. These arguments can be generalized to many different frameworks, in particular the proofs are given in the setting of stratified Lie groups.

Introduction and presentation of the results

The aim of this article is to provide, in the setting of stratified Lie groups, a general proof for a particular type of improved Sobolev inequalities of the following general form

f Ẇ s 1 ,q ≤ C f θ Ẇ s,p f 1-θ Ḃ-β,∞ ∞ , (1)
where f ∈ Ẇ s,p ∩ Ḃ-β,∞ ∞ . Here we write Ẇ s,p for homogeneous Sobolev spaces and Ḃ-β,∞ ∞ for homogeneous Besov spaces (see Section 3 below for precise definitions). The parameters s, s 1 , p, q and β defining Sobolev and Besov spaces in the previous inequality are related by the conditions 1 < p < q < +∞, θ = p/q, s 1 = θs -(1 -θ)β and -β < s 1 < s, but they do not depend on the dimension and in this sense these inequalities are more general than classical Sobolev inequalities; of course the inequalities above are sharper than classical ones. Historically, the first proof in the Euclidean setting of these inequalities is due to P. Gérard, F. Oru and Y. Meyer [START_REF] Gérard | Inégalités de Sobolev Précisées. Séminaire sur les Equations aux Dérivées Partielles[END_REF] and is based on a Littlewood-Paley decomposition and interpolation results applied to dyadic blocks. Another proof of these inequalities using maximal function and Hedberg's inequality is given in [START_REF] Chamorro | Improved Sobolev Inequalities and Muckenhoupt weights on stratified Lie groups[END_REF].

q f 1-q/p Ḃs-n/q,∞ q with 1 p = 1 q - s n . (2)
Remark that in this estimate, the index q defining the Lorentz and the Besov spaces is related to the parameters p, s and the dimension n. This inequality was generalized in the Euclidean setting by D. Chamorro & P-G. Lemarié-Rieusset [START_REF] Chamorro | Real interpolation method, Lorentz spaces and refined Sobolev inequalities[END_REF] to other values of the parameter q using interpolation techniques and pointwise estimates. In a recent article, V.I. Kolyada and F.J. Pérez Lázaro [START_REF] Kolyada | On Gagliardo-Nirenberg Type Inequalities[END_REF] gave an interesting proof for inequalities of type ( 1) and ( 2) based on the use of rearrangement inequalities and the properties of the Gauss-Weierstrass kernel.

Motivated by the use of Lorentz spaces in these previous works, in our first theorem we will provide a generalization (in stratified Lie groups) of improved Sobolev inequalities of type (1) by considering weighted Lorentz-based Sobolev spaces defined as the set of measurable functions f : G -→ R such that the quantity (-∆) s 2 f Λ p (w) is bounded where for s > 0 the fractional power of the Laplacian is defined in Section 2 below, and for 1 < p < +∞ the space Λ p (w) denotes the classical Lorentz space of functions introduced in [START_REF] Lorentz | Some new functional spaces[END_REF] and [START_REF] Lorentz | On the theory of spaces Λ[END_REF] defined as

Λ p (w) = f : f Λ p (w) = +∞ 0 f * (t) p w(t)dt 1/p < +∞ ,
where w is a weight in R + and f * denotes the non-increasing rearrangement of f (see [START_REF] Bennett | Interpolation of operators[END_REF] for standard notations). Many of the properties of these spaces depend on the weight w: in particular, if w = 1 we have Λ p (w) = L p and if w(t) = t p/q-1 , with 1 ≤ q ≤ +∞, we obtain Λ p (w) = L q,p , where L q,p are the usual Lorentz spaces. In this work we will consider the weighted Lorentz space Λ p (w) such that the weight w satisfies the B p condition, the reason for this is given by the fact that M. A. Ariño and B. Muckenhoupt showed in [START_REF] Ariño | Maximal functions on classical Lorentz spaces and Hardy's inequality with weights for nonincreasing functions[END_REF] that this B p condition characterizes the boundedness of the Hardy-Littlewood maximal operator on Λ p (w) and this particular property will be intensively used in our proofs. See Section 3 for definitions and [START_REF] Ariño | Maximal functions on classical Lorentz spaces and Hardy's inequality with weights for nonincreasing functions[END_REF], [START_REF] Carro | Weighted Lorentz spaces and the Hardy operator[END_REF], [START_REF] Carro | Recent developments in the theory of Lorentz spaces and weighted inequalities[END_REF] and [START_REF] Soria | Lorentz spaces of weak-type[END_REF] for more details and properties concerning these functional spaces.

In this direction of generalization, standard Sobolev inequalities have been studied by A. Cianchi [START_REF] Cianchi | Optimal Orlicz-Sobolev embeddings[END_REF] in the context of Orlicz-Sobolev spaces, but improved inequalities of the general type (3) presented in Theorem 1 below are, to the best of our knowledge, new.

Theorem 1 Let G a stratified Lie group. Let s > 0, w ∈ B p be a weight and let f :

G -→ R be a function such that (-∆) s 2 f ∈ Λ p (w)(G) and f ∈ Ḃ-β,∞ ∞ (G).
Then we have the following version of improved Sobolev inequalities:

(-∆) s 1 2 f Λ q (w) ≤ C (-∆) s 2 f θ Λ p (w) f 1-θ Ḃ-β,∞ ∞ , (3)
where 1 < p < q < +∞, θ = p/q, s 1 = θs -(1 -θ)β and -β < s 1 < s.

The choice of the weights in the B p class is given by two important facts. First, these weights allow us to consider general functional spaces, in particular we can easily recover standard Lorentz spaces. Second, these weights ensure that maximal function is bounded in the spaces Λ p (w) for 1 < p < +∞, and this feature is crucial as the proof of Theorem 1 requires this property. Note in particular that inequality (3) is different from inequality (2) since Lorentz-Sobolev spaces are not included in the scale of Besov spaces.

Since our proof of Theorem 1 relies essentially on a pointwise inequality and on the boundedness of the Hardy-Littlewood maximal operator, it is possible to give a related result replacing classical Lorentz spaces by Morrey spaces M p,a which are a useful generalization of Lebesgue spaces. Classical Hardy-Littlewood-Sobolev inequalities were studied in this functional framework by D. Adams [START_REF] Adams | A note on Riesz potentials[END_REF] and by F. Chiarenza & M. Frasca [START_REF] Chiarenza | Morrey spaces and Hardy-Littlewood maximal function[END_REF] and our next theorem is an improvement of these inequalities.

Theorem 2 Let G be a stratified Lie group. Let s > 0, 1 < p < +∞ and 0 ≤ a < n and let f be a function such that (-∆)

s 2 f ∈ M p,a (G) and f ∈ Ḃ-β,∞ ∞ (G). Then we have (-∆) s 1 2 f M q,a ≤ C (-∆) s 2 f θ M p,a f 1-θ Ḃ-β,∞ ∞ , (4)
where

1 < p < q < +∞, θ = p/q, s 1 = θs -(1 -θ)β and -β < s 1 < s.
The plan of this article is the following. In Section 2 we present our general framework which is given by stratified Lie groups. These groups are quite natural generalization of R n but they present some particularities that should be taken into account in the computations. In Section 3 we give the precise definition of all the functional spaces used in the previous inequalities and in Section 4 we present the proof of Theorem 1. Finally, in Section 5 we give the proof of Theorem 2 and some variations of the previous results.

Stratified Lie groups: notation and basic properties

As said in the introduction, stratified Lie groups are natural generalizations of R n when considering general dilation structures. Although stratified Lie groups share common features with R n , there are some special points that must be taken into account: for example these groups are no longer abelian and this fact requires to be carefull in some computations, furthermore from the geometric point of view, the inner geometric structure of these groups can be very different from the euclidean setting. It is then necessary to recall some basic facts about stratified Lie groups, for further information see [START_REF] Folland | Subelliptic estimates and function spaces on nilpotent Lie groups[END_REF], [START_REF] Folland | Hardy Spaces on homogeneous groups[END_REF], [START_REF] Th | Analysis and geometry on groups[END_REF], [START_REF] Stein | Harmonic Analysis[END_REF] and the references given there in.

We start with the notion of homogeneous group G which is the data of R n equipped with a structure of Lie group and we will always suppose that the origin is the identity. We define a dilation structure by fixing integers (a i ) 1≤i≤n such that 1 = a 1 ≤ ... ≤ a n and by writing:

δ α : R n -→ R n (5)
x -→ δ α [x] = (α a 1 x 1 , ..., α an x n ).

We will often note αx instead of δ α [x] and α will always indicate a strictly positive real number.

Of course, the Euclidean space R n with its group structure and provided with its usual dilations (i.e. a i = 1, for i = 1, ..., n) is a homogeneous group. Here is another example: if x = (x 1 , x 2 , x 3 ) is an element of R 3 , we can fix a dilation by writing δ α [x] = (αx 1 , αx 2 , α 2 x 3 ) for α > 0. Then, the well suited group law with respect to this dilation is given by x

• y = (x 1 , x 2 , x 3 ) • (y 1 , y 2 , y 3 ) = (x 1 + y 1 , x 2 + y 2 , x 3 + y 3 + 1 2 (x 1 y 2 -y 1 x 2 )
). Remark in particular that this group law is no longer abelian. The triplet (R 3 , •, δ) corresponds to the Heisenberg group H 1 which is the first non-trivial example of a homogeneous group. The homogeneous dimension with respect to dilation structure ( 5) is given by N = 1≤i≤n a i . We observe that it is always larger than the topological dimension n since each integer a i verifies a i ≥ 1 for all i = 1, ..., n. For instance, in the Heisenberg group H 1 we have N = 4 and n = 3 while in the Euclidean case these two concepts coincide. Now we will say that a function on G\{0} is homogeneous of degree λ ∈ R if f (δ α [x]) = α λ f (x) for all α > 0. In the same way, we will say that a differential operator D is homogeneous of degree

λ if D(f (δ α [x])) = α λ (Df )(δ α [x]
), for all f in operator's domain. In particular, if f is homogeneous of degree λ and if D is a differential operator of degree µ, then Df is homogeneous of degree λ -µ. The presence of a dilation structure is one of most important features of stratified Lie groups and the homogeneity with respect to these dilations will play a useful role in our computations.

From the point of view of measure theory, homogeneous groups behave in a traditional way since Lebesgue measure dx is bi-invariant and coincides with the Haar measure, thus for any subset E of G we will note its measure as |E|. This fact also allows us to define Lebesgue spaces in a classical way (see also Section 3 below). The convolution will be a very useful tool in our computations, and for two functions f and g on G it is defined by

f * g(x) = G f (y)g(y -1 • x)dy = G f (x • y -1 )g(y)dy, x ∈ G.
However, since the group law of a stratified Lie group is not necessarly commutative, we do not have in general the identity f * g = g * f and we need to take care of this fact. Nevertheless, we have at our disposal Young's inequalities:

Lemma 2.1 If 1 ≤ p, q, r ≤ +∞ such that 1 + 1 q = 1 p + 1 r . If f ∈ L p (G) and g ∈ L r (G), then f * g ∈ L q (G) and we have the inequality f * g L q ≤ f L p g L r .
A proof is given in [START_REF] Folland | Hardy Spaces on homogeneous groups[END_REF]. A weak version of Young's inequalities will be stated in Section 3.

For a homogeneous group G = (R n , •, δ) we consider now its Lie algebra g whose elements can be conceived in two different ways: as left-invariant vector fields or as right-invariant vector fields. The left-invariant vectors fields (X j ) 1≤j≤n are determined by the formula

(X j f )(x) = ∂f (x • y) ∂y j y=0 = ∂f ∂x j + j<k q k j (x) ∂f ∂x k ,
where q k j (x) is a homogeneous polynomial of degree a k -a j and f is a smooth function on G. By this formula one deduces easily that these vectors fields are homogeneous of degree a j and we have X j (f (αx)) = α a j (X j f )(αx). We will note (Y j ) 1≤j≤n the right invariant vector fields defined in a totally similar way:

(Y j f )(x) = ∂f (y • x) ∂y j y=0 .
A homogeneous group G is stratified if its Lie algebra g breaks up into a sum of linear subspaces g = 1≤j≤k E j such that E 1 generates the algebra g and

[E 1 , E j ] = E j+1 for 1 ≤ j < k and [E 1 , E k ] = {0} and E k = {0}, but E j = {0} if j > k. Here [E 1 , E j ] indicates the subspace of g generated by the elements [U, V ] = U V -V U with U ∈ E 1 and V ∈ E j .
The integer k is called the degree of stratification of g. For example, on Heisenberg group H1 , we have k = 2 while in the Euclidean case k = 1.

We will suppose from now on that G is stratified with homogeneous dimension 1 N ≥ 4. Within this framework, we will fix once and for all the family of vectors fields

X = {X 1 , ..., X m }, such that a 1 = a 2 = . . . = a m = 1 (m < n),
then the family X is a base of E 1 and generates the Lie algebra of g, which is precisely the Hörmander's condition (see [START_REF] Folland | Hardy Spaces on homogeneous groups[END_REF] and [START_REF] Th | Analysis and geometry on groups[END_REF]) and this particular choice ensures several important properties, in particular to the family X is associated the Carnot-Carathéodory distance d which is left-invariant and compatible with the topology on G (see [START_REF] Th | Analysis and geometry on groups[END_REF] for more details) and for any x ∈ G we will denote by |x| = d(x, e) and for r > 0 we form open balls by writing B(x, r) = {y ∈ G : d(x, y) < r}. By simple homogeneity arguments we obtain that stratified Lie groups have polynomial volume growth since we have

|B(•, r)| = r N |B(•, 1)|.
The main tools of this paper depend on the properties of the gradient, the Laplacian and the associated heat kernel, but before introducing them, we make here some remarks on general vectors fields X j and Y j . Let us fix some notation: for any multi-index

I = (i 1 , ..., i n ) ∈ N n , one defines X I by X I = X i 1 1 . . . X in n and Y I by Y I = Y i 1 1 . . . Y in n
, furthermore we denote by |I| = i 1 + . . . + i n the order of the derivation of the operators X I or Y I and d(I) = a 1 i 1 + . . . + a n i n the homogeneous degree of these ones. Now, for ϕ, ψ ∈ C ∞ 0 (G) we have the equality

G ϕ(x)(X I ψ)(x)dx = (-1) |I| G (X I ϕ)(x)ψ(x)dx.
The interaction of operators X I and Y I with convolutions is clarified by the following identities:

X I (f * g) = f * (X I g), Y I (f * g) = (Y I f ) * g, (X I f ) * g = f * (Y I g). (6)
Finally, one will say that a function f ∈ C ∞ (G) belongs to the Schwartz class S(G) if the following semi-norms are bounded for all k ∈ N and any multi-index

I: N k,I (f ) = sup x∈G (1 + |x|) k |X I f (x)|.
Remark 2.1 To characterize the Schwartz class S(G) we can replace vector fields X I in the seminorms N k,I above by right-invariant vector fields Y I .

For a proof of these facts and for further details see [START_REF] Folland | Hardy Spaces on homogeneous groups[END_REF] and [START_REF] Furioli | Littlewood-Paley decomposition and Besov spaces on Lie groups of polynomial growth[END_REF].

We define now the gradient on G from vectors fields of homogeneity degree equal to one (i.e. those composing the family X) by fixing ∇ = (X 1 , ..., X m ). This operator is of course left invariant and homogeneous of degree 1. The length of the gradient is given by the formula |∇f | = (X 1 f ) 2 + ... + (X m f ) 2 1/2 . We also define the right invariant gradient ∇ = (Y 1 , ..., Y m ), and using [START_REF] Carro | Weak-type weights and normable Lorentz spaces[END_REF] we have the identity (∇f ) * g = f * ( ∇g). We define now the Laplacian we are going to work with. Let us notice that in this setting there is not a single way to build a Laplacian, see for example [START_REF] Furioli | Littlewood-Paley decomposition and Besov spaces on Lie groups of polynomial growth[END_REF]. In this article we will use the Laplacian, denoted by J , which is given from the family X in the following way

J = ∇ * ∇ = - m j=1 X 2 j . (7)
This is a positive self-adjoint, hypo-elliptic operator (since the family X satisfies the Hörmander's condition), having as domain of definition L 2 (G). Its associated heat operator on G×]0, +∞[ is given by ∂ t + J . We recall now some well-known properties of the heat operator and its associated kernel.

Theorem 3 There exists a unique family of continuous linear operators (H t ) t>0 defined on L 1 + L ∞ (G) with the semi-group property H t+s = H t H s for all t, s > 0 and H 0 = Id, such that:

1) the Laplacian J is the infinitesimal generator of the semi-group

H t = e -tJ ;
2) H t is a contraction operator on L p (G) for 1 ≤ p ≤ +∞ and for t > 0;

3) the semi-group H t admits a convolution kernel

H t f = f * h t where h t (x) = h(x, t) ∈ C ∞ (G×]0, +∞[)
is the heat kernel which satisfies the following points:

(a) (∂ t + J )h t = 0 on G×]0, +∞[, and h(x, t) = h(x -1 , t), h(x, t) ≥ 0 and G h(x, t)dx = 1, (b) h t has the semi-group property: h t * h s = h t+s for t, s > 0 and we have h(δ α [x], α 2 t) = α -N h(x, t), (c) For every t > 0, x → h(x, t) belong to the Schwartz class in G. 4) For f ∈ C ∞ (G) and for t > 0 we have J H t (f ) = H t J (f ).
For a detailed proof of these and other important facts concerning the heat semi-group see [START_REF] Folland | Hardy Spaces on homogeneous groups[END_REF] and [START_REF] Saka | Besov spaces and Sobolev spaces on a nilpotent Lie group[END_REF].

To close this section we recall the definition of the Laplacian's fractional powers. If s > 0 we write

J s f (x) = 1 Γ(k -s) +∞ 0 t k-s-1 J k H t f (x)dt, (8)
for all f ∈ C ∞ (G) with k an integer greater than s. The interaction between this operator and the heat kernel is given by the following lemma.

Lemma 2.2 If 1 ≤ p ≤ +∞, for s > 0 and for t > 0 we have the estimate

J s 2 h t L p ≤ Ct - s+N(1-1 p ) 2 .
See [START_REF] Saka | Besov spaces and Sobolev spaces on a nilpotent Lie group[END_REF] for a proof.

Functional spaces

We give in this section the precise definition of all the functional spaces involved in Theorems 1 and 2. In a general way, given a norm • X , we will define the corresponding functional space X(G) by {f ∈ S ′ (G) : f X < +∞}. The constant that appear in this paper such as C may change from one occurrence to the next. • weak-Lebesgue spaces L p,∞ (G). We define them as the set of all measurable functions f :

G -→ R such that f L p,∞ = sup α>0 {α • |{x ∈ G : |f (x)| > α}| 1/p } is finite.
We will need the following version of Young's inequality where weak L p spaces are involved:

Lemma 3.1 Let p, q, r > 1. If f ∈ L p,∞ (G) and if g ∈ L r (G), then f * g ∈ L q (G) with 1 + 1 q = 1 p + 1 r and we have the inequality f * g L q ≤ f L p,∞ g L r .
See a proof of this Lemma in [START_REF] Grafakos | Classical and Modern Fourier Analysis[END_REF], Theorem 1.4.24.

• Sobolev spaces Ẇ s,p (G). If 1 < p < +∞ and for s > 0 we have f Ẇ s,p = J s 2 f L p , while if p = s = 1 we will note f Ẇ 1,1 = ∇f L 1 . We recall classical Sobolev inequalities in this setting:

f L N N-1 = ∇f L 1 and f L q = f Ẇ s,p ,
where 1 < p < q and -N q = s -N p .

• weak Sobolev spaces Ẇ s,p ∞ (G). These spaces are defined just as classical Sobolev spaces, but we replace the L p norm by the weak L p one as follows:

f Ẇ s,p ∞ = J s 2 f L p,∞ with 1 < p < +∞ and s > 0.
• Besov spaces Ḃs,q p (G). There are many different (and equivalent) ways to define these spaces in the setting of stratified Lie groups. In this article we will mainly use the thermic definition given by

f Ḃs,q p = +∞ 0 t (m-s/2)q ∂ m H t f ∂t m (•) q L p dt t 1/q
, for 1 ≤ p, q ≤ +∞, s > 0 and m an integer such that m > s/2. For Besov spaces of indices (-β, ∞, ∞) which appear in all the improved Sobolev inequalities we have:

f Ḃ-β,∞ ∞ = sup t>0 t β/2 H t f L ∞ .
Recall that for 0 < s < 1 we have the inequality

J s 2 f Ḃ-β-s,∞ ∞ ≤ M f Ḃ-β,∞ ∞ , ( 10
)
where M is a universal constant. It is not difficult to see that if 0 < p < q < +∞, then we have the inclusion of classes B p ⊂ B q . We define the Lorentz spaces Λ p (w) with 1 ≤ p < +∞ by the formula

• Lorentz spaces Λ p (w)(G). Let f : G -→ R
f Λ p (w) = +∞ 0 (f * (t)) p w(t)dt 1 p .
As said in the introduction, the choice of the B p class is due to the fact that this class of weights characterizes the boundedness of the Hardy-Littlewood maximal operator M B , given for a measurable function f by

M B f (x) = sup B∋x 1 |B| B |f (y)|dy, where B is an open ball, (11)
on the spaces Λ p (w) for 1 < p < +∞:

M B f Λ p (w) ≤ C f Λ p (w)
, where C is depending on the quantity

[w] Bp = sup r>0 r p +∞ r w(t) t p dt r 0 w(t)dt .
For more properties of these weights and the associated classical Lorentz spaces see [START_REF] Ariño | Maximal functions on classical Lorentz spaces and Hardy's inequality with weights for nonincreasing functions[END_REF], [START_REF] Carro | Weighted Lorentz spaces and the Hardy operator[END_REF], [START_REF] Soria | Lorentz spaces of weak-type[END_REF] and [START_REF] Carro | Recent developments in the theory of Lorentz spaces and weighted inequalities[END_REF].

• Lorentz-Sobolev spaces Λs,p (w)(G). Once we have fixed the base space Λ p (w), the homogeneous Lorentz-Sobolev spaces are easy to define and are given for 1 < p < +∞ and for s > 0 in the following way

f Λs,p (w) = +∞ 0 (J s 2 f ) * (t) p w(t)dt 1 p
.

• weak Lorentz spaces Λ p,∞ (w)(G). Let w a weight in R + . For 0 < p < +∞, the weak Lorentz space Λ p,∞ (w) is the class of all measurable functions f : G -→ R such that

f Λ p,∞ (w) = sup t>0 f * (t)W 1/p (t) < +∞,
where W (t) = t 0 w(s)ds. The weak Lorentz spaces were introduced in [START_REF] Carro | Weighted Lorentz spaces and the Hardy operator[END_REF] and further investigated in [START_REF] Carro | Boundedness of some integral operators[END_REF], [START_REF] Carro | Weak-type weights and normable Lorentz spaces[END_REF] and [START_REF] Carro | The Hardy-Littlewood maximal function and weighted Lorentz spaces[END_REF]. The problem of characterizing when the weak type Lorentz spaces Λ p,∞ (w), 0 < p < +∞ are Banach spaces was studied in [START_REF] Soria | Lorentz spaces of weak-type[END_REF].

• weak Lorentz-Sobolev spaces Λs,p,∞ (w)(G). For 1 < p < +∞ the homogeneous weak Lorentz-Sobolev spaces are given by

f Λs,p,∞ (w) = sup t>0 (J s 2 f ) * (t) t 0 w(s)ds 1/p .
• Morrey spaces M p,a (G). For 1 < p < +∞ and 0 ≤ a < N , we define Morrey spaces as the space of locally integrable functions such that f M p,a = sup

x 0 ∈R n sup 0<r<+∞ 1 r a B(x 0 ,r) |f (x)| p dx 1 p < +∞.
Morrey spaces are indeed a generalization of Lebesgue spaces since when a = 0 we have M p,0 ≃ L p . The use of Morrey and Morrey-Sobolev spaces in this article is due to the fact that the Hardy-Littlewood maximal operator ( 11) is also bounded in such spaces. See more details in [START_REF] Chiarenza | Morrey spaces and Hardy-Littlewood maximal function[END_REF] in the framework of R n and [START_REF] Akbulut | Boundedness of fractional maximal operator and their higher order commutator in the generalized Morrey spaces on Carnot groups[END_REF] in the setting of stratified Lie groups. See also [START_REF] Sawano | Generalized fractional integral operators and fractional maximal operators in the framework of Morrey spaces[END_REF] and the references given there in for other interesting generalizations.

• Morrey-Sobolev spaces Ṁs,p,a (G). For 0 < s and 1 < p < +∞ with 0 ≤ a < N we consider the homogeneous Morrey-Sobolev spaces Ṁs,p,a by the quantity

f Ṁs,p,a = J s 2 f M p,a .
4 Proof of Theorem 1

We will prove here, in the framework of stratified Lie groups, the inequality

f Λs 1 ,q (w) ≤ C f θ Λs,p (w) f 1-θ Ḃ-β,∞ ∞ , where f : G -→ R is a function such that f ∈ Λs,p (w)(G) ∩ Ḃ-β,∞
∞ (G) with 1 < p < q < +∞, θ = p/q, s 1 = θs -(1 -θ)β and -β < s 1 < s. We will always assume here that w is a weight in the Ariño-Muckenhoupt class B p . The reason for this particular choice of weights relies on the fact that we will need the boundedness of the Hardy-Littlewood maximal operator on Lorentz Λ p (w) spaces and this is ensured by the condition w ∈ B p . See [START_REF] Ariño | Maximal functions on classical Lorentz spaces and Hardy's inequality with weights for nonincreasing functions[END_REF] and [START_REF] Carro | Recent developments in the theory of Lorentz spaces and weighted inequalities[END_REF] for details.

By the definition of Lorentz-Sobolev spaces given in Section 3, this inequality can be rewritten in the following way

J s 1 2 f Λ q (w) ≤ C J s 2 f θ Λ p (w) f 1-θ Ḃ-β,∞ ∞ .
For the proof of this inequality, we will use a variant of Hedberg's inequality. Indeed, since 0 < s 1 < s, we use the characterization of the positive powers of the Laplacian given in (8) and we have for

k > s/2 > s 1 /2 J s 1 2 f (x) = 1 Γ(k -s 1 /2) +∞ 0 t k-s 1 2 -1 J k H t f (x)dt = 1 Γ(k -s 1 /2) T 0 t k-s 1 2 -1 J k H t f (x)dt + +∞ T t k-s 1 2 -1 J k H t f (x)dt ,
where T will be defined below. In particular we have

|J s 1 2 f (x)| ≤ 1 Γ(k -s 1 /2) T 0 t k-s 1 2 -1 |J k H t f (x)|dt + +∞ T t k-s 1 2 -1 |J k H t f (x)|dt . ( 12
)
For the first integral of the right-hand side of the previous formula we will use the following fact.

Lemma 4.1 Let f ∈ S ′ (G) and ϕ ∈ S(G). We denote by M ϕ (f ) the maximal function of f (with respect to ϕ) which is given by the expression

M ϕ f (x) = sup 0<t<+∞ {|f * ϕ t (x)|}, with ϕ t (x) = t -N/2 ϕ(t -1/2 x).
If the function ϕ is such that |ϕ(x)| ≤ C(1 + |x|) -N -ε for some ε > 0, then we have the following pointwise inequality

M ϕ f (x) ≤ CM B f (x),
where M B f (x) is the Hardy-Littlewood maximal function defined by [START_REF] Chamorro | Improved Sobolev Inequalities and Muckenhoupt weights on stratified Lie groups[END_REF].

For a proof of this lemma see [START_REF] Grafakos | Classical and Modern Fourier Analysis[END_REF] or [START_REF] Folland | Hardy Spaces on homogeneous groups[END_REF]. With this lemma in mind, and since k > s/2, we remark that we have the identity

J k H t f (x) = J k-s 2 h t * J s 2 f (x)
. Now, by homogeneity we obtain J k-s 2 (h t )(x) = t -k+ s 2 J k-s 2 h t (x) and if we denote ϕ t by ϕ t (x) = J k-s 2 h t (x) we have that ϕ t (x) = t -N/2 ϕ(t -1/2 x), moreover, since the heat kernel h t is a smooth function, with the previous notation we obtain |ϕ(x)| ≤ C(1 + |x|) -N -ε . Then we can write

J k H t f (x) = t -k+ s 2 ϕ t * J s 2 f (x),
and applying the Lemma 4.1 we have the following pointwise inequality for the first term of ( 12):

|J k H t f (x)| = t -k+ s 2 M B J s 2 f (x).
Now, for the second integral of the right-hand side of ( 12) we simply use the fact that

J k f Ḃ-β-2k,∞ ∞ ≃ f Ḃ-β,∞ ∞
and the thermic definition of Besov spaces to obtain

|J k H t f (x)| = |H t J k f (x)| ≤ Ct -β-2k 2 J k f Ḃ-β-2k,∞ ∞ .
With these two inequalities at hand, we apply them in [START_REF] Chamorro | Some functional inequalities on polynomial volume growth Lie groups[END_REF] and one has

|J s 1 2 f (x)| ≤ C Γ(k -s 1 /2) T 0 t k-s 1 2 -1 t -k+ s 2 M B J s 2 f (x)dt + +∞ T t k-s 1 2 -1 t -β-2k 2 J k f Ḃ-β-2k,∞ ∞ dt ≤ C Γ(k -s 1 /2) T s-s 1 2 M B J s 2 f (x) + T -β-s 1 2 J k f Ḃ-β-2k,∞ ∞ .
We fix now the parameter T by the condition

T =   J k f Ḃ-β-2k,∞ ∞ M B J s 2 f (x)   2 β+s
, and we obtain the following inequality

|J s 1 2 f (x)| ≤ C Γ(k -s 1 /2) M B J s 2 f 1- s-s 1 β+s (x) J k f s-s 1 β+s Ḃ-β-2k,∞ ∞ .
Since s-s 1 β+s = 1 -θ and using again the fact

J k f Ḃ-β-2k,∞ ∞ ≃ f Ḃ-β,∞ ∞ we have |J s 1 2 f (x)| ≤ C Γ(k -s 1 /2) M B J s 2 f θ (x) f 1-θ Ḃ-β,∞ ∞ . ( 13
)
Once we have obtained this pointwise inequality, we will use the following properties of the nonincreasing rearrangement function.

Lemma 4.2 If f, g : G -→ R are two measurable functions, we have

1) if |g| ≤ |f | a.e. then g * ≤ f * , 2) if 0 < θ, then (|f | θ ) * = (f * ) θ .
For a proof see Proposition 1.4.5 of [START_REF] Grafakos | Classical and Modern Fourier Analysis[END_REF]. Recalling that θ = p/q and applying these facts to the inequality (13) we obtain

(J s 1 2 f ) * (t) q ≤ C (M B J s 2 f ) * (t) p f q-p Ḃ-β,∞ ∞ . (14)
Multiplying the previous inequality by a weight w from the Ariño-Muckenhoupt class B p and integrating with respect to the variable t we obtain +∞ 0

(J s 1 2 f ) * (t) q w(t)dt ≤ C +∞ 0 (M B J s 2 f ) * (t) p w(t)dt f q-p Ḃ-β,∞ ∞ ,
and then, by the definition of classical Lorentz spaces given in Section 3 we have

J s 1 2 f Λ q (w) ≤ C M B (J s 2 f ) θ Λ p (w) f 1-θ Ḃ-β,∞ ∞ .
Now, since the weight w belongs to the class B p with 1 < p < +∞, we have that the Hardy-Littlewood maximal operator is bounded on the space Λ p (w) and we obtain

M B (J s 2 f ) Λ p (w) ≤ J s 2 f Λ p (w) ,
and finally we have the desired inequality for classical Lorentz spaces:

J s 1 2 f Λ q (w) ≤ C J s 2 f θ Λ p (w) f 1-θ Ḃ-β,∞ ∞ .
Now we will state in the following corollaries some interesting consequences of this previous theorem.

Corollary 4.1 Let w ∈ B p be a weight and let f :

G -→ R be a function such that f ∈ Λs,p,∞ (w)(G)∩ Ḃ-β,∞ ∞ (G).
Then we have the following version of improved Sobolev inequalities of weak type:

f Λs 1 ,q,∞ (w) ≤ C f θ Λs,p,∞ (w) f 1-θ Ḃ-β,∞ ∞ , where 1 < p < q < +∞, θ = p/q, s 1 = θs -(1 -θ)β and -β < s 1 < s.
Proof . We start again with the pointwise inequality ( 14):

(J s 1 2 f ) * (t) q ≤ C (M B J s 2 f ) * (t) p f q-p Ḃ-β,∞ ∞ .
Now, we multiply both parts of this inequality by W (t) and we take the supremum in the variable t:

J s 1 2 f q Λ q,∞ (w) = sup t>0 W (t) (J s 1 2 f ) * (t) q ≤ C sup t>0 M B (J s 2 f ) * (t) p W (t) f q-p Ḃ-β,∞ ∞ ≤ C M B (J s 2 f ) p Λ p,∞ (w) f q-p Ḃ-β,∞ ∞ ,
since it is known (see e.g. [START_REF] Soria | Lorentz spaces of weak-type[END_REF]) that for w ∈ B p the Hardy-Littlewood maximal operator M B is bounded on Λ p,∞ (w), therefore we obtain that

J s 1 2 f Λ q,∞ (w) ≤ C J s 2 f θ Λ p,∞ (w) f 1-θ Ḃ-β,∞ ∞ .
Now we will study other variations of the previous results by considering a different type of weights. To be more precise, we will study two-weighted inequalities and in what follows, for v and w two weigths and for t > 0, we will denote by V (t) and W (t) the quantities

V (t) = t 0 v(s)ds and W (t) = t 0 w(s)ds.
Our first two-weighted improved Lorentz-Sobolev inequality is given in the following corollary.

Corollary 4.2 Let 1 < p < q < +∞ and let (v, w) be a pair of positive weights satisfying the following properties

sup t>0 W (t) 1/p V (t) 1/p < +∞ and sup t>0 +∞ t w(s) s p ds 1/p t 0 v(s)s p ′ V (s) p ′ ds 1/p ′ < +∞. If f : G -→ R is a function such that f ∈ Λs,p (v) ∩ Ḃ-β,∞
∞ with s > 0, then we have a two-weighted version of improved Sobolev inequalities

f Λs 1 ,q (w) ≤ C f θ Λs,p (v) f 1-θ Ḃ-β,∞ ∞ ,
where 1 < p < q < +∞, θ = p/q, s 1 = θs -(1 -θ)β and -β < s 1 < s.

This inequality is interesting since it is possible, under some hypotheses, to consider different weights in the left-hand side and in the right-hand side of the inequality.

Proof . Using the pointwise inequality [START_REF] Chiarenza | Morrey spaces and Hardy-Littlewood maximal function[END_REF] and the fact that the Hardy-Littlewood maximal operator

M B : Λ p (v) -→ Λ p (w)
is bounded for such weights (see [START_REF] Sawyer | Boundedness of classical operators on classical Lorentz spaces[END_REF] for details) we obtain the desired inequality.

If we are allowed to change the weights that define the Lorentz spaces in the previous inequalities, it is then also possible to change, with specific conditions on the weights, the parameters of these spaces. In the following corollary we gather some results where we consider different Lorentz spaces in the right-hand side of the inequality. Indeed, the first point is a generalization of the previous corollary and we will consider in the right-hand side Lorentz-Sobolev spaces of type Λs,q 0 (v) instead of Λs,p (v) where 1 < q 0 ≤ p < +∞. The second point allows us to study the case when 1 < p < q 0 < +∞ and finally, the third point treats the case when 0 < q 0 < 1.

Corollary 4.3 Let 0 < q 0 < +∞, s > 0, let f : G -→ R be a measurable function and let (v, w) be a pair of weights. 1) If 1 < q 0 ≤ p < +∞ and if (v, w) are satisfying the following conditions sup t>0 W (t) 1/p V (t) 1/q 0 < +∞ [START_REF] Cianchi | Optimal Orlicz-Sobolev embeddings[END_REF] and

sup t>0 t 0 w(s) s p ds 1/p t 0 v(s)s q ′ 0 V (s) q ′ 0 < +∞, (16)
then, if f ∈ Λs,q 0 (v)(G) ∩ Ḃ-β,∞ ∞ (G), we have the following inequality

f Λs 1 ,q (w) ≤ C f θ Λs,q 0 (v) f 1-θ Ḃ-β,∞ ∞ ,
where 1 < p < q < +∞, θ = p/q, s 1 = θs -(1 -θ)β and -β < s 1 < s.

2) If 1 < p < q 0 < +∞ and (v, w) are satisfying

+∞ 0 W (s) V (s)
r/q 0 w(s)ds

1/r < +∞ and   +∞ 0   +∞ s w(t) t p dt 1/p t 0 v(t)t q 0 ′ V (t) q 0 ′ dt 1/p ′   r v(s)s q 0 ′ V (s) q 0 ′ ds   1/r < +∞,
where r is given by 1 r = 1 p -1 q and 1 q 0 + 1

q 0 ′ = 1. Then, if f ∈ Λs,q 0 (v)(G) ∩ Ḃ-β,∞ ∞ (G), we have f Λs 1 ,q (w) ≤ C f θ Λs,q 0 (v) f 1-θ Ḃ-β,∞ ∞ ,
where 1 < p < q < +∞, θ = p/q, s 1 = θs -(1 -θ)β and -β < s 1 < s. 3) If 0 < q 0 < 1 and 1 < p < +∞ and if (v, w) are satisfying [START_REF] Cianchi | Optimal Orlicz-Sobolev embeddings[END_REF] and

sup t>0 t V (t) 1/q 0 +∞ t w(s) s p ds 1/p < +∞, then, assuming that f ∈ Λs,q 0 (v)(G) ∩ Ḃ-β,∞ ∞ (G), we obtain f Λs 1 ,q (w) ≤ C f θ Λs,q 0 (v) f 1-θ Ḃ-β,∞ ∞ ,
where 1 < p < q < +∞, θ = p/q, s 1 = θs -(1 -θ)β and -β < s 1 < s.

Proof. From the pointwise inequality ( 14) we obtain that

J s 1 2 f Λ q (w) ≤ C M B (J s 2 f ) θ Λ p (w) f 1-θ Ḃ-β,∞ ∞ .
Now, under all these hypotheses on the weights v and w, we have that the Hardy-Littlewood maximal operator M B : Λ q 0 (v) -→ Λ p (w) is bounded (see [START_REF] Sawyer | Boundedness of classical operators on classical Lorentz spaces[END_REF] and [START_REF] Carro | Boundedness of some integral operators[END_REF]) and then we obtain

f Λs 1 ,q (w) ≤ C f θ Λs,q 0 (v) f 1-θ Ḃ-β,∞ ∞ .
We have also the following two-weighted version of improved Sobolev inequalities of weak type:

Corollary 4.4 Let 1 < p < +∞, 0 < q 0 < +∞. Let (v, w) be a pair of weights such that

sup t>0 W (t) 1/p t t 0 V -1/q 0 (s)ds < +∞, (17)
and let f :

G -→ R be a function such that f ∈ Λs,q 0 ,∞ (v)(G)∩ Ḃ-β,∞ ∞ (G). Then we have the following inequality f Λs 1 ,q,∞ (w) ≤ C f θ Λs,q 0 ,∞ (v) f 1-θ Ḃ-β,∞ ∞ ,
where 0 < q 0 < +∞, 1 < p < q < +∞, θ = p/q, s 1 = θs -(1 -θ)β and -β < s 1 < s.

Proof . It is enough to follow the same lines of the Corollary 4.1 to obtain

J s 1 2 f q Λ q,∞ (w) ≤ C M B (J s 2 f ) p Λ p,∞ (w) f q-p Ḃ-β,∞ ∞ ,
since the pair of weights (v, w) satisfies the condition [START_REF] Folland | Subelliptic estimates and function spaces on nilpotent Lie groups[END_REF] it implies that the operator M B : Λ q 0 ,∞ (v) -→ Λ p,∞ (w) is bounded (see [START_REF] Soria | Lorentz spaces of weak-type[END_REF]) and we obtain

J s 1 2 f q Λ q,∞ (w) ≤ C J s 2 f p Λ q,∞ (v) f q-p Ḃ-β,∞ ∞ ,
which is the desired inequality.

Generalizations

In this section we give some generalizations of Theorems 1 and we prove Theorem 2. These generalizations are made possible since the techniques developed in our proofs are based on general harmonic analysis arguments and since many of the tools used in this article are available in other frameworks. Indeed, the spectral theory associated to the Laplace operator, the boundedness of the Hardy-Littlewood maximal operator and the use of appropiate weights in order to define well suited functional spaces are intensively studied and many interesting properties were generalized to different settings.

Morrey spaces

We prove now Theorem 2 in the setting of stratified Lie groups. Morrey spaces were studied in this framework by many authors, see for example the articles [START_REF] Akbulut | Boundedness of fractional maximal operator and their higher order commutator in the generalized Morrey spaces on Carnot groups[END_REF], [START_REF]The Campanato, Morrey and Hölder spaces on spaces of homogeneous type[END_REF] and the references there in.

As said in the introduction, once we have at our disposal the fact that the Hardy-Littlewood maximal operator is bounded in the convenient functional framework, it is possible to improve Sobolev inequalities in the following way. The starting point of our proof is the pointwise inequality (13):

|J s 1 2 f (x)| ≤ C Γ(k -s 1 /2) M B J s 2 f θ (x) f 1-θ Ḃ-β,∞ ∞ .
Since θ = p/q we have for r > 0 and for 0 ≤ a < N the inequalities

1 r a B(x 0 ,r) |J s 1 2 f (x)| q dx ≤ C 1 r a B(x 0 ,r) M B J s 2 f p (x)dx f q(1-θ) Ḃ-β,∞ ∞ 1 r a B(x 0 ,r) |J s 1 2 f (x)| q dx 1/q ≤ C 1 r a B(x 0 ,r) M B J s 2 f p (x)dx 1/q f 1-θ Ḃ-β,∞ ∞ ,
from which we derive the estimate

J s 1 2 f M q,a ≤ C M B J s 2 f θ M p,a f 1-θ Ḃ-β,∞ ∞ .
In order to conclude, we use the fact that the Hardy-Littlewood maximal operator is bounded in Morrey spaces and we obtain

J s 1 2 f M q,a ≤ C J s 2 f θ M p,a f 1-θ Ḃ-β,∞ ∞ ,
which is the desired inequality stated in Theorem 2.

Remark 5.1 The boundedness of the Hardy-Littlewood maximal operator was studied for generalized Morrey spaces in [START_REF] Akbulut | Boundedness of fractional maximal operator and their higher order commutator in the generalized Morrey spaces on Carnot groups[END_REF], [START_REF]Hardy-Littlewood maximal operator, singular integral operators and Riesz potentials on generalized Morrey spaces[END_REF] and [START_REF] Sawano | Generalized fractional integral operators and fractional maximal operators in the framework of Morrey spaces[END_REF]. As long as this boundedness property is satisfied it should be possible to generalize Theorem 2. Indeed, from the pointwise inequality (13) it should be easy (taking into account the necessary precautions) to reconstruct the corresponding norms in order to obtain an improved Sobolev-like inequality.

Nilpotent Lie groups

We consider now a more general framework than the one given by stratified Lie groups. Indeed, going one step further in the process of generalization, it is possible to consider nilpotent Lie groups since all the tools used in the proof of Theorem 1 are available in these settings.

We recall for the sake of completness this framework. Let G be a connected unimodular Lie group endowed with its Haar measure dx. Denote by g the Lie algebra of G and consider a family (that will be fixed from now on) of left-invariant vector fields on G X = {X 1 , ..., X k }, satisfying the Hörmander condition2 . We endow the group G with a metric structure by considering the Carnot-Carathéodory metric associated with X. See [START_REF] Th | Analysis and geometry on groups[END_REF] for details. We will denote x the distance between the origin e and x and y -1 • x the distance between x and y. For r > 0 and x ∈ G, denote by B(x, r) the open ball with respect to the Carnot-Carathéodory metric centered in x and of radius r, and by V (r) = B(x,r) dx the Haar measure of any ball of radius r. When 0 < r < 1, there exists d ∈ N * , c l and C l > 0 such that, for all 0 < r < 1 we have

c l r d ≤ V (r) ≤ C l r d .
The integer d is the local dimension of (G, X). When r ≥ 1, two situations may occur, independently of the choice of the family X: either G has polynomial volume growth and there exist D ∈ N * , c ∞ and C ∞ > 0 such that, for all r ≥ 1 we have

c ∞ r D ≤ V (r) ≤ C ∞ r D ,
or G has exponential volume growth, which means that there exist c e , C e , α, β > 0 such that, for all r ≥ 1 we have c e e αr ≤ V (r) ≤ C e e βr .

When G has polynomial volume growth, the integer D is called the dimension at infinity of G. Recall that nilpotent groups have polynomial volume growth and that a strict subclass of the nilpotent groups consists of stratified Lie groups where d = D.

Once we have fixed the family X, we define the gradient on G by ∇ = (X 1 , ..., X k ) and we consider a Laplacian J on G defined in the same way as in ( 7)

J = - k j=1 X 2 j ,
which is a positive self-adjoint, hypo-elliptic operator since X satisfies the Hörmander's condition, see [START_REF] Th | Analysis and geometry on groups[END_REF]. Its associated heat operator on ]0, +∞[×G is given by ∂ t + J and we will denote by (H t ) t>0 the semi-group obtained from the Laplacian J . It is worth noting that many of the properties given in Theorem 3 remain true for the heat semi-group H t in this general setting. For more details concerning nilpotent Lie groups see the books [START_REF] Th | Analysis and geometry on groups[END_REF], [START_REF] Folland | Hardy Spaces on homogeneous groups[END_REF], [START_REF] Stein | Harmonic Analysis[END_REF] and the articles [START_REF] Furioli | Littlewood-Paley decomposition and Besov spaces on Lie groups of polynomial growth[END_REF], [START_REF] Saka | Besov spaces and Sobolev spaces on a nilpotent Lie group[END_REF], [START_REF] Chamorro | Some functional inequalities on polynomial volume growth Lie groups[END_REF] and the references there in. Fractional powers of the Laplacian can be defined in a completely similar way using the expression [START_REF] Carro | Weighted Lorentz spaces and the Hardy operator[END_REF]. It is then possible to define all the functional spaces given in Section 3 in the framework of nilpotent Lie groups.

With all these preliminaries, we see that we have at our disposal all the ingredients needed in order to perform the computations done in Sections 4, and thus Theorem 1 can be generalized to the setting of nilpotent Lie groups.

•

  Lebesgue spaces L p (G). For a measurable function f : G -→ R and for 1 ≤ p < +∞ we define Lebesgue space by the norm f L p = G |f (x)| p dx 1/p , while for p = +∞ we have f L ∞ = ess sup x∈G |f (x)|. Let us notice that we also have the following characterization using the distribution function f p L p = p +∞ 0 α p-1 |{x ∈ G : |f (x)| > α}|dα.

  be a measurable function. We define f * , the non-increasing rearrangement of the function f , by the expression f * (t) = inf{α ≥ 0 : |{x ∈ G : |f (x)| > α}| ≤ t}. We will say that a nonnegative locally integrable function w : R + -→ R + belongs to the Ariño-Muckenhoupt class B p for 1 ≤ p < +∞, if there exists C > 0 such that +∞ r r t p w(t)dt ≤ C r 0 w(t)dt, for all 0 < r < +∞.

The lower bound N ≥ 4 corresponds to the homogeneous dimension of the Heisenberg group H 1 , which is the simplest non-trivial stratified Lie group.

which means that the Lie algebra generated by the family X is g.

Acknowledgments. A part of this work was performed while the second and third authors visited the University of Evry Val d'Essonne. We express our gratitude to the Laboratoire de Mathématiques et Modélisation d'Evry (LaMME) of the University of Evry Val d'Essonne for the hospitality and excellent conditions. The second named author was partially supported by POSDRU/159/1.5/S/137750.


Hanane Hemi
email: hanane.hemi@uqtr.ca

Nacer K M'sirdi
email: nacer.msirdi@lis-lab.fr

Aziz Naamane
email: aziz.naamane@lis-lab.fr

Open Circuit Voltage of a Lithium ion Battery Model adjusted by data fitting

Keywords:

.

I. INTRODUCTION

Lithium ion (Li-ion) batteries are nowadays, the favorite energy storage system due to several advantages compared to other technologies. Li-ion battery is an electro-chemical cell that transforms chemical energy into electrical energy. In fact, Li-ion batteries have a high power/energy density, with high cell voltage. They have a high life span, and a low self discharge rate [START_REF] Tian | State of charge estimation of lithium-ion batteries using an optimal adaptive gain nonlinear observer[END_REF], [START_REF] Waldmann | Li plating as unwanted side reaction in commercial li-ion cells-a review[END_REF]. This type of battery is more and more used in hybrid and electrical vehicle (HEV) systems due to several advantages. It has been compared favourably to other batteries.

A. The Context

In Hybrid and Electrical Vehicles (HEV), a Li-ion battery is considered as a secondary source of power used to optimize energy consumption. During a driving cycle, Liion battery can assist a primary source during acceleration and can be charged either per it or by recovering braking energy during deceleration. Then, it plays a significant role in the overall system efficiency. It allows also to reduce the consumption of the primary power source (as for example, the consumption of a hydrogen for a fuel cell vehicle).

However, ambient and storage system temperatures have an negative effect on battery behavior (on current, voltages and State of Charge) and also on its life span.

Studying influence of environmental conditions on the battery performance is essential to minimize its possible degradation and avoid thermal runaway. Then, a good knowledge of the system's heating and heat transfer by the battery, becomes of a crucial importance.

Modeling the behaviour of a Li-ion battery becomes then paramount to study a battery performance, depending on several external or internal conditions. In another hand, a behaviour model is useful to estimate and predict the current and future battery state variables. Its use is also interesting to manage the control for a safe battery operation modes. In another hand we will be able to manage the batteries use such to keep them always in ideal conditions to avoid their aging an faults.

In fact, the ambient temperature has a significant impact on the battery State Of Charge (SOC). Then estimation and / or prediction of the battery SOC will be harder and then it will be harder too to know the optimal charge/discharge current range to control the use of the batteries. However, knowing that OCV and SOC are dependent on ambient temperature, it is important to model this dependence by studying data.

B. Literature models

The Electrical Equivalent Circuit Models (ECM) are intuitive and easy to handle and are composed by electrical components such as voltage sources, resistors, and capacitors [START_REF] Li | A practical lithiumion battery model for state of energy and voltage responses prediction incorporating temperature and ageing effects[END_REF], [START_REF] Belhani | Battery models for estimation of state of charge by sliding mode observer[END_REF]. These models have been used most in electrical engineering simulation, and especially for HEV applications. Indeed, due to their electrical nature which makes them possible to be connected directly to the electrical network [START_REF] Shafiei | Battery modeling approaches and management techniques for plug-in hybrid electric vehicles[END_REF], [START_REF] Hu | A comparative study of equivalent circuit models for li-ion batteries[END_REF].

The electrical Equivalent Circuit Models (ECM) can be classified to three categories [START_REF] Benabdelaziz | Battery dynamic energy model for use in electric vehicle simulation[END_REF]. The first one is the Thevenin based Electrical Model (TEM) [START_REF] Chen | Accurate electrical battery model capable of predicting runtime and iv performance[END_REF]. This model is basic and uses series resistor R, an RC parallel network and in addition, a voltage source OCV [START_REF] He | Online model-based estimation of state-of-charge and open-circuit voltage of lithium-ion batteries in electric vehicles[END_REF]. The second one is the Impedance based Electrical Model (IEM). This model kind is an AC equivalent impedance model, describing the behaviour in the frequency domain. It has been obtained by an electrochemical impedance spectroscopy [START_REF] Chen | Accurate electrical battery model capable of predicting runtime and iv performance[END_REF], [START_REF] Dai | Impedance characterization and modeling of lithium-ion batteries considering the internal temperature gradient[END_REF]. The last one is the Runtime Based Electrical Model (REM). This model simulates battery runtime and DC voltage response for a constant discharge current [START_REF] Chen | Accurate electrical battery model capable of predicting runtime and iv performance[END_REF].

C. Contribution

In this part of our work, we focus our study on the influence of ambient temperature on battery performance as a representation model parameters. In particular, we analyze the impact of the temperature on the battery Open Circuit Voltage (OCV ).

This paper is structured in three sections and a conclusion.The Section II describes the experimental data and Test Analysis. Section III presents the Lithium Ion battery Proposed Model. Section IV presents the conclusions on our results.

II. EXPERIMENTAL DATA AND TEST ANALYSIS

We consider, in our study, the experimental data provided by the CALCE battery group of a A123 battery cell [START_REF]CALCE: center for advanced life cycle engineering[END_REF].

In this paper, we selected to study the A123 battery cell which is a LiFePO4 battery. Tab.I shows the battery characteristics given by battery manufacturer [START_REF]A123systems high power lithium ion apr18650m1a datasheet[END_REF]. The database is proposed by the CALCE battery group [START_REF]CALCE: center for advanced life cycle engineering[END_REF], [START_REF] Xing | State of charge estimation of lithium-ion batteries using the open-circuit voltage at various ambient temperatures[END_REF]. They have used these data to define a battery description using look u tables.

The main motivation of this choice is that this type of battery is typically used in HEV applications. Fig. 1: Test bench for battery tests experiments [START_REF] He | State of charge estimation for li-ion batteries using neural network modeling and unscented kalman filter-based error cancellation[END_REF] The objective of our work in this step is to study the relationship between the OCV , SOC and the ambient temperature. In order to ensure this, a low-current OCV method is used for this test.

The Low-current OCV test consists in using a small current to charge and discharge the battery and then measure the output voltage which is approximately equal to OCV . This test have been performed, by the CALCE battery group, for an ambient temperature varying from -10 • C to 50 • C with interval of 10 • C.

Fig. 1 presents the test bench for battery tests experiments [START_REF] He | State of charge estimation for li-ion batteries using neural network modeling and unscented kalman filter-based error cancellation[END_REF]. Test bench detail has been given by the CALCE battery group in the References [START_REF]CALCE: center for advanced life cycle engineering[END_REF], [START_REF] He | State of charge estimation for li-ion batteries using neural network modeling and unscented kalman filter-based error cancellation[END_REF]. We can remark that, in Fig. 2 and3, there are three main regions: two region with a quasi exponential decrease (from 0 to 0.1Ah and from 0.9 to 1.1 Ah) and a middle region which has a quasi linear decrease. The same temperature effects are observed when the battery is in the charge mode.

III. LITHIUM ION BATTERY PROPOSED MODEL

Li-ion battery is considered as a part of the HEV system model. Therefore, to predict a HEV behavior in real time. The choice of the battery model is important for a real time HEV operation, and especially, the battery SOC model must be relevant to estimate and predict the BMS (Battery Management System) calculation. For these reasons, a good model is a key factor for control in real time operating. Thus, fast battery models are preferred compared with complex models. In fact, a battery used in HEV applications is a pack composed by several battery cells organized in series and parallel depending on the battery pack design. The electrical equivalent circuit model has a simple design and also has a fast computation time in both cells or pack battery models. However, the electrochemical model describes a single cell and is not suitable for real time applications, specially for HEV applications, due its complex equations which are hard to implement and then required a long time of computation.

A. Lithium Ion battery standard model

Fig. 4 shows the ideal battery curve during a constant current discharge [START_REF] Raszmann | Modeling stationary lithium-ion batteries for optimization and predictive control[END_REF]. The Shepherd model describes by a mathematical equation, the relationship between the voltage (V Batt (t)), current (i(t)) for a constant current discharge. Eq.1 represents this relationship [START_REF] Raszmann | Modeling stationary lithium-ion batteries for optimization and predictive control[END_REF], [START_REF] Li | Study of battery modeling using mathematical and circuit oriented approaches[END_REF]. In fact, the battery cell voltage is composed of an internal resistance R and the open-circuit voltage (OCV ) given in Eq.2.

V Batt (t) = V 0 - K.Q Q -i.t .i(t) -R.i(t) (1)
V oc (t) = V 0 - K.Q Q -i.t .i(t) (2)
where V 0 is the constant voltage (V ), Q is the maximum capacity (Ah), i.t is the discharged capacity (Ah), K is the polarization constant (Ah) -1 and R is internal resistance. To improve a battery mathematical model to fit to battery charge and discharge curves, the Shepherd model is modified by adding some terms. Eq.3 and Eq.4 show expression of the OCV at discharge and charge modes respectively [START_REF] Raszmann | Modeling stationary lithium-ion batteries for optimization and predictive control[END_REF], [START_REF] Gallo | Optimization of experimental model parameter identification for energy storage systems[END_REF], [START_REF] Li | Study of battery modeling using mathematical and circuit oriented approaches[END_REF]. If the current is positive, then the battery is in discharge mode, OCV = V oc.Discharge , as shown in Eq.3. If the current is negative, then the battery is in the charge mode, OCV = V oc.Charge , as presented in Eq.4.

V oc.Discharge = V 0 - K.Q Q -i.t .i * - K.Q Q -i.t .i.t+A * exp(-B.i.t) (3)
V oc.Charge = V 0 - K.Q 0.1 * Q -i.t .i * - K.Q Q -i.t .i.t+A.exp(-B.i.t) (4
) where A is the exponential voltage coefficient (in V ), B is the exponential capacity (in (Ah) -1 and i * is the filtered current (in A).

This model is quite adequate but it can be enhanced to be more precise and take into account the non linearity.

B. Lithium Ion battery proposed Model

In Li-ion battery, the relationship between OCV and SOC is a non linear function. Therefore, the battery chosen must have a NonLinear Equivalent RC Circuits to Model (NL-ECM) to describe this non linear relationship function. The NL-ECM model proposed here for the battery is composed by an OCV, a capacitance to model polarization effect (C p ), a propagation resistor to model propagation resistance (R p ), a diffusion resistor as a function of current (R d ), an ohmic resistance (R o ), and battery voltage (V Batt ). The Fig. 5 presents the proposed battery Li-ion NL Equivalent Circuit Model (NL-ECM). Based on the measurement presented in section II, we have shown the impact of temperature on battery behavior. In fact, the V oc measured during discharge mode, is different for each temperature. This difference is emphasized in OCV and the capacity maximum and the exponential and the nominal zones (in the Fig. 2 and Fig. 3). These curves fitting has been deployed to fit data to a function. In this paper, we have chosen a rational polynomials function to find a relationship between OCV and SOC. This function has a numerator polynomial with a second degree and a denominator polynomial of a third degree. The degrees of the numerator and the denominator polynomial have been chosen to get the best curve fitting and minimum number of coefficients.

The Eq.5 represents the OCV function which produces the best fit. In addition, this function has been tested for several ambient temperatures.

V oc (SOC) = p1 * SOC 2 + p2 * SOC + p3 SOC 3 + q1 * SOC 2 + q2 * SOC + q3 (5)
where p i are the coefficients of the numerator polynomial, q j are the coefficients of the denominator polynomial, and SOC is given by Eq.6. The battery model shown in Fig. 6 is used in simulation using MATLAB/Simulink and SimPower Systems software packages, for validation tests. As mentioned in section II, the test used in this simulation is the Low-current OCV test. In fact, it consist by using a small current to charge and discharge the battery. In this case, the current used to discharge a battery is equal to 0.05 A as shown in Fig. 6a. Therefore, the output voltage measured is approximately equal to V Batt = V oc as presented in Fig. 6b. The OCV equation (Eq.5) is implemented as shown in the Fig. 6c. Fig. 7 shows the OCV measured and simulated comparison for ambient temperature -10 • C 10 • C 20 • C and 50 • C. The simulation results shows a small error between simulations and measurements for each temperature.

SOC = i.t Q (6)
We remark in all the validations tests that the simulated curves using our model are in excellent concordance with the real data ones. The modeling errors are very small.

IV. CONCLUSIONS

The paper presented a data curve fitting used to find a relationship between OCV and SOC for different temperatures. This equation had been implemented and simulated using MATLAB/Simulink and SimPower Systems software packages. The comparison between simulation and measurement shows a small error due to a complex dynamic of battery. In The next step of this study, the function of OCV will be improved to minimize the error and then the battery model behavior at different temperatures will be developed and completed.

Fig. 2 presentsFig. 2 :Fig. 3 :

 223 Fig. 2: OCV vs time at discharge mode

Fig. 4 :

 4 Fig.4: Ideal battery curve during discharge[START_REF] Gallo | Optimization of experimental model parameter identification for energy storage systems[END_REF]

Fig. 5 :

 5 Fig. 5: Proposed NL-ECM battery Li-ion model

Fig. 6 :

 6 Fig. 6: Battery model

Fig. 7 :

 7 Fig. 7: V oc measured and simulated comparison

TABLE I :

 I Battery characteristics given by battery manufacturer[START_REF]A123systems high power lithium ion apr18650m1a datasheet[END_REF]

	Type	LiFePO4
	Nominal capacity	1.1 Ah
	Nominal voltage	3.3	V
	Upper cut-off voltage	3.6	V
	Lower cut-off voltage	2	V
	Maximum continuous discharge current	30	A

ACKNOWLEDGMENT

The authors would like to thank Professor Michael Pecht, from the CALCE battery group, for sharing their experimental data and exchanges, to let us being able to use them for the validation of our model. The "CALCE" is the Center for Advanced Life Cycle Engineering. See https://web.calce.umd.edu/batteries/index.h

*This work is supported by SASV group of the LIS UMR CNRS 7020


Nacer K M'sirdi

Antoine Monneau

Aziz Naamane

Adaptive Notch Filters for Prediction of Narrow Band Signals

Keywords: Adaptive Notch Filters, Narrow band Signals, Frequency Tracking, Spectrum Analysis, Non-stationary Signals, ANC

This paper proposes efficient structures of Adaptive Notch Filters (ANF) implemented as constrained ARMA models. These ANF suit very well for fast frequencies tracking and evolutionary spectrum analysis for narrow band and sinusoidal signals in additive noise. Lastly, non-stationary ANF are considered and an estimation procedure implemented.

I. INTRODUCTION

A. Objective

The main interest in this contribution, is the retrieval of the parameters of sinusoidal signal components (amplitudes, phases and frequencies), in noise. We assume (slowly) varying frequencies and varying amplitudes and phases. This is the case when the process is perturbed by the wind (cars, planes, flying robots, wind turbines ...) or the sea waves, swell (boats, submarine robots, wave turbines...). Other examples are the vibrations of processes in mechanics, hydraulics or pneumatics. It is often crucial to detect and predict the vibration signals in order to compensate or tackle their effects.

Recently introduced adaptive control techniques use compensation of the narrow band perturbation signals based on ANF [START_REF] Landau | Adaptive regulation rejection of unknown multiple narrow band disturbances (a review on algorithms and applications)[END_REF], [START_REF] Landau | Benchmark on adaptive regulation rejection of unknown/time-varying multiple narrow band disturbances[END_REF], [START_REF] Tomei | Multi-sinusoidal disturbance rejection for discrete-time uncertain stable systems[END_REF], [START_REF] Airimitoaie | Robust Design and Tuning of Active Vibration Control Systems[END_REF]. Autoregressive (AR) and ARMA (Autoregressive with Mean Average) modeling techniques have shown powerful potentialities. They are well suited for recursive signal spectral analysis, interesting for time varying signal spectrum [START_REF] Widrow | Adaptive noise cancelling: Principles and applications[END_REF], [START_REF] Landau | Identification of a Heat Exchanger Using an Apple II Computer[END_REF], [START_REF] Landau | Techniques de modélisation récursives pour l'analyse spectrale paramétrique adaptative[END_REF].

However, when the noise level is too high and / or when the signal model has poles near or on the unit circle, serious (numerical or stability) problems arise [START_REF] Landau | Identification of a Heat Exchanger Using an Apple II Computer[END_REF].

B. Previous work

In the narrow-band signals case, successful applications have been realized with AR models [START_REF] Widrow | Adaptive noise cancelling: Principles and applications[END_REF], [START_REF] Glover | Adaptive noise canceling applied to sinusoidal interferences[END_REF] but, they need high order autoregressive FIR filter. Very high order AR models has been shown to be asymptotically equivalent to an Adaptive Notch Filter (ANF). Additionally the noise effect is reduced by increasing the order of the AR filter. This adds the expense of computational cost [START_REF] Sirdi | RML ALGORITHM FOR RETRIEVAL OF SINUSOIDS WITH CASCADED NOTCH FIL-TERS[END_REF]. This has been studied in extensive simulations [START_REF] Glover | Adaptive noise canceling applied to sinusoidal interferences[END_REF], [START_REF] Ra0 | Adaptive notch filtering for retrieval of sinusoids in noise[END_REF], [START_REF] Landau | Techniques de modélisation récursives pour l'analyse spectrale paramétrique adaptative[END_REF].

Thus, the use of the a priori knowledge of a narrowband feature of the signal, lead us to the design of Notch Filter Structure (NFS) with constrained poles and zeros. The Adaptive Notch Filters (ANF) have been proposed to provide the frequencies estimation and tracking [START_REF] Nehorai | A minimal parameter adaptive notch filter with constrained poles and zeros[END_REF] [START_REF] Sirdi | RML ALGORITHM FOR RETRIEVAL OF SINUSOIDS WITH CASCADED NOTCH FIL-TERS[END_REF], [START_REF] Sirdi | Behaviour analysis of adaptive ARMA predictors with non-stationary inputs[END_REF]. These estimators can be used by implementation in a cascaded blocks of second order cells.

Some strategies has been proposed to track time varying parameters (like frequencies amplitudes and phases), which may be used in line [START_REF] Sirdi | An rml algorithm for retrieval of sinusoids with cascaded notch filters[END_REF]. ANF(s) are used to estimated the frequency components in a first step. Then, in a second step, after frequencies tracking the estimated frequencies are used to provide the estimation of amplitudes and phases. This can also be implemented in a recursive manner.

The proposed recursive algorithm consists in two steps. The first step involves a Recursive Maximum Likelihood algorithm to adapt the cascaded filters parameters [START_REF] Landau | Techniques de modélisation récursives pour l'analyse spectrale paramétrique adaptative[END_REF], which will provide the frequencies estimates. The second step uses the last estimates, and then the estimations of amplitudes and phases are given by a Recursive Extended Least Squares (RELS) algorithm .

A theoretical analysis of these algorithms, have been presented in [START_REF] Landau | Identification of a Heat Exchanger Using an Apple II Computer[END_REF], [START_REF] Ra0 | Adaptive notch filtering for retrieval of sinusoids in noise[END_REF] and the applicable estimation methods are reviewed in [START_REF] Ra0 | Adaptive notch filtering for retrieval of sinusoids in noise[END_REF], [START_REF] Nehorai | A minimal parameter adaptive notch filter with constrained poles and zeros[END_REF] [14], [START_REF] Sirdi | RML ALGORITHM FOR RETRIEVAL OF SINUSOIDS WITH CASCADED NOTCH FIL-TERS[END_REF], [START_REF] Sirdi | Adaptive comb filters for enhancement of quasi periodic signals[END_REF]. These last structures introduce in Parameter Adaptation Algorithms (PAA) some nonlinearity which may give some problems in the transient period of the estimation. The cascade implementation of second order sub models, have been claimed possible.

C. Contribution of this work

In this paper clarifies the rationale behind the Adaptive Notch Filters that we have proposed in the past and revisited in this paper. We give a complete theoretical analysis and prove the convergence of the two estimation algorithms:n one to estimate the frequency components by an ARML algorithm and the second to estimate the amplitudes and phases by an RLS algorithm with exponential weighting and a forgetting factor. These algorithms are applied for adaptive prediction of the swell perturbations on a boat.

In section II of this paper we present the Recursive estimation algorithms and the filter structure. We then develop an implementation of the cascaded ANF. The Prediction Error Method (PEM) is applied. This application leads to an Approximate Recursive Maximum Likelihood (ARML) algorithm for frequencies estimation. In a second step, using these estimates, a Recursive Least Squares (RLS) algorithm is employed for the estimation of amplitudes and phases.

The theoretical analysis of these algorithms and their performance evaluation is presented in section III.

Section for will be dedicated to effective Prediction of narrow band signals and parameters estimation. The narrow band signals, considered are the sea waves. After the ANF for the signal Prediction, we estimate the frequencies and the deduce the estimations of the amplitudes and phase. The algorithms Validation is done by using the swell signals perturbing a boat motion.

II. RECURSIVE ESTIMATION ALGORITHMS AND FILTER

STRUCTURE

The proposed recursive estimation procedure consists of two stages, as the parameters to estimated are in non linear combination. The first step processed involves an ANF structure implemented in cascaded form for the estimation of the signal frequencies. An Approximate Recursive Maximum Likelihood (ARML) method is developed and used. The second step uses Recursive Instrumental Variables (RIV) to avoid non linearities and get an appropriate model for linear prediction techniques. This leads us a model linear in its parameters (to be estimated). An adaptive estimation algorithm, based on Weighted Recursive Least Squares (WRLS), is used to provide the amplitudes and phases estimations.

The involved signals may be modeled as follows where v k is the noise disturbance, C (k) and β i (k) are, respectively, the time varying amplitude and phase of the frequency component f i (k). All the parameters are susceptible to vary in time. T e is the sampling period.

y k = p i=1 C i (k).Sin(2.πf i .T e .k + β i (k)) + v k (1)
A. Amplitudes and Phases Estimation with RIV-WRLS

In this step, we assume that the frequencies of the signal are known (well estimated by the proposed ARLM algorithm) and we develop an estimator for the amplitudes and phases of each frequency component. We use a Weighted Recursive Least Squares algorithm, as in [START_REF] Nehorai | A minimal parameter adaptive notch filter with constrained poles and zeros[END_REF], to estimate the amplitude and phase of each sinusoid. The frequencies estimations are provided by the ANF, as will be developed in the next subsection in a previous step.

The signal Model of equation ( 1), can be decomposed in Fourier Series. It may be written as linear combination of the Fourier basis elements (which are the Instrumental Variables (IV) to be used in estimation):

y k = p l=1 (g l Cos(2πf l T e .k + h l Sin(2πf l T e .k)) + v k (2)
where the amplitudes C l and phases β l for (l = 1...p) are given from the Fourier series parameters by

C l = g 2 l + h 2 l and tan(β l ) = g l /h l (3)
It is worthwhile to note that the Fourier series parameters are identifiable. We can then define the parameter θ(k) and the regression (by Recursive Instrumental Variable RIV) ψ (k)

vectors as follows          θ(k) = [g 1 , g 2 , ..., g p , h 1 , h 2 , ..., h p ] T Φ k = [C, S] T with C = [Cos(2πf l .T e .k) .., Cos(2πf p .T e .k)] S = [Sin(2πf l .T e .k) , ..., Sin(2πf p .T e .k)] (4)
The parameters are then estimated using an WRLS algorithm as follows:

               y 0 k = θ(k -1) T .Φ k ε 0 (k) = y k -θ(k -1) T .Φ k G k = 1 λ0 G k-1 - G k-1 Φ k Φ T k G k-1 λ0+Φ T k G k-1 Φ k θ(k) = θ(k -1) + G k .Φ k .ε 0 (k) C l = g 2 l + h 2 l tan(β l ) = g l /h l (5)
where y 0 k is the a priori prediction and 0 (k) is the a priori prediction error. In equation ( 5) filtered versions of the signal y k can be used; For example, we can use the prediction (of the ANF) produced by the frequency estimation stage. Doing this we will get the RIV coupled with the signal Output Prediction Error Method (RIV-OPEM).

We can note, also, that the RIV regression vector will verify the Persistent Excitation Condition (PEC). The use of a constant λ 0 as an exponential forgetting factor will help for the alertness of the amplitudes and phases estimation by WRLS. The adaptation gain G k will not decrease to zero and remain positive. Then estimation will remain correct in case of slowly time varying parameters.

B. Frequency estimation whit cascaded ANF

Adaptive Notch Filters are very well suited for estimation of the sinusoidal component frequencies. The i th sinusoidal component will be filtered by the i th second order ANF:

H i (z -1 ) = 1 + a i z -1 + z -2 1 + r.a i z -1 + r 2 z -2 (6)
The last equation presents a second order notch filter. Let us note that the transfer function does not depend on amplitudes and phases. Those informations are in the initial conditions. The notch bandwidth is defined by the parameter 0<r<1. The parameter a i depends on the notch frequency

a i = -2cos(2πf i T e ) (7)
(for i = 1...p). As they are p sinusoidal components in equation ( 1), let us consider the notch filter transfer function as p cascaded second order Notch cells:

H(z -1 ) = p i=1 H i (z -1 ) (8)
With no loss of generality we assume identical bandwidth parameter r for the notches. Thus independence between frequencies yields independence between the parameters of each second order cell. Then each cell can be adapted independently of the others after pre-filtering the signal by the others. For comb filter, relations between the frequencies can be used to reduce the computational burden. If all the frequency components are independent we can write

y j k = p i = 1 i = j H i (q -1 )y k (9)
If we suppose that the cascaded filters H i (z -1 ) with central frequency f i (for i = l...p and i = j) have converged then the filter H j (z -1 ) will remove one of the j th sinusoidal components, remaining after the filters H i (z -1 ), for i = 1...p and i = j.

y j k = p i = 1 i = j 1 + a i q -1 + q -2 1 + r.a i q -1 + r 2 q -2 .y k (10)
This means that in the signal y j k the p-1 sinusoidal components have been removed by the corresponding notches and then the prediction error ε k is obtained by removing the j th one:

ε k = H j (q -1 ) y j k =
1 + a j q -1 + q -2 1 + r.a j q -1 + r 2 q -2 y j k [START_REF] Nehorai | A minimal parameter adaptive notch filter with constrained poles and zeros[END_REF] Thus, there will exit a unique global minimum for the Mean Squared Error (MSE) Criterion, as each notch remove one frequency component. Following the approach of the Output Prediction Error Method (OPEM), for the estimation algorithm, we obtain as the estimation gradient:

ψ j k-1 = - dε k da j = (1 -r)(1 -rq -2 ) 1 + r.a j q -1 + r 2 q -2 . y j k-1 (12)
The RML algorithm may be summarized by the following equations, where F j k is the adaptation gain and 0 < λ < 1 the forgetting

     f or j = 1, ..., p, do âj k = âj k-1 + F j k-1 .ψ j k-1 .ε k F j k = F j k-1 /(λ + ψ j k-1 .F j k-1 .ψ j k-1 ) (13)
The computational complexity of the RML algorithm comes essentially from the computation of the gradient and the successive filtering (equations [START_REF] Ra0 | Adaptive notch filtering for retrieval of sinusoids in noise[END_REF][START_REF] Nehorai | A minimal parameter adaptive notch filter with constrained poles and zeros[END_REF][START_REF] Sirdi | Behaviour analysis of adaptive ARMA predictors with non-stationary inputs[END_REF]. A first approximation can be done to reduce computational complexity. It yields an Approximate RML (ARML) algorithm. We can then use the formula (using equation II-B instead of 12):

ψ j k-1 s j k-1 -rε k-1 1 + r.a j q -1 + r 2 q -2 . y j k (14) with the signals s j k computed as follows      f orj = (p -1), ..., 1do y j k = p l=j+1 H l (q -1 ).y k s j k = y j k -y j-1 k ( 15
)
Another suggestion is to adapt the Notch filter second order cells successively for i = 1, .., p in a cascaded way or recur-sively. Each cell get as input the output error computed from the previous cells only, in a recursive manner. The bandwidth or what we call the de-biasing parameter r k must be a positive.

In our experience we choose this parameter exponentially time varying, from 0 or r 0 to one (or to r f ), according to the following expression:

r k = r d .r k-1 + (1 -r d ).r f .

III. ANALYSIS AND PERFORMANCE EVALUATION

A. Convergence of the ANF (OPEM RML)

1) Second order ANF with an RML algorithm: Let us first consider the case of a single cell H j (z -1 ) (see ( 6)) driven by the signal defined in (1) with p = 1.

Assume, either if p > 1, that the preceding filters H i (z -1 ) (for i = 1.. j -1, j + 1, .. n and i = j) have, all of them, converged to their optimal value. The bandwidth, of the filter H i (z -1 ) are assumed large enough to remove (or attenuate) the other frequencies (f i respectively i = j).

Thus ỹj k is composed by a single frequency f j plus an additive independent noise η k .

ỹj k = C j (k).Sin(2.πf j .T e .k + β j (k)) + η k (16)
Tacking for the signal an AR model with two complex conjugate poles on the unit circle and applying the RML Parameter Estimation algorithm, the a posteriori error can be written:

ε k = Âj (rq -1 ) A j (rq -1 ) .(â j -a j ).ψ j k-1 + η k A j (rq -1 ) (17)
with ỹj k = -a j ỹj k-1 -ỹj k-2 + η k A j (rq -1 ) = 1 + r.a j q -1 + r 2 q -2 Âj (rq -1 ) = 1 + r.â j q -1 + r 2 q -2 (18) It is well known, in the general case, that the RML algorithm needs a stability monitoring. For the ANF this procedure can be removed in virtue of the following Lemma. We use a time varying de-biasing parameter. Lemma 1. Let r k be a positive (bandwidth) parameter exponentially time varying, from 0 to one, according to the following expression:

r k = r d .r k-1 + (1 -r d ).r f (19)
.

Then there exit positive constants, less than 1, (r 0 , r d , r f ) such that Â(r k , q -1 ) is infinitely often stable.

Proof: The bandwidth (or de-biasing) parameter r k acts like a contraction of the poles, attracting them toward the center of the unit circle. In addition r k starts at r 0 , for k=0, and goes to r f , when k goes to infinity. We can then take r f less than the inverse of the maximum modulus of the unstable poles of Â(r k , q -1 ). Now applying the theoretical background, on convergence of adaptation algorithms, presented in [START_REF] Landau | A feedback system approach to adaptive filtering[END_REF] and particularly the Theorem 4 of [START_REF] Landau | A feedback system approach to adaptive filtering[END_REF] leads to the following Theorem to establish the convergence of the ANF OPRM RML algorithm.

Theorem 2. : Under the following assumptions:

• Â(r k , q -1 ) is infinitely often stable (r k .â j (k) < 2)

• for some âj fixed ψ j k , ε k and η k Aj (rq -1 ) are quasi stationary signal.

• η k is independent of ψ j k-1 and • the transfer function Â(rz -1 )

A(rz -1 ) -1 2 is SPR (Strictly Positive Real). Then we get, for the estimation algorithm, the following:

• D c ={a such as E{(â j -a j ).ψ j k-1 } = 0} is the convergence domain • lim( k -η k ) = 0 WP1 (With probability one) • lim( âj (k)) D c WP1 (With probability one)
To ensure the SPR condition we consider the following lemma.

Lemma 3. : There always exist an r such that Â(r.q -1 ) A(r.q -1 ) is a Strictly Positive Real transfer function (SPR).

The last condition on r is less restrictive than the one for the stability. We can then conclude that the Lemmas 1 and 2, with the last theorem, allow us to assert the global stability of the ANF if r is appropriately chosen such that Â(r.q -l ) is always stable during estimation progress.

2) Second order ANF with an ELS algorithm: If the ELS estimation method is applied, the estimation error becomes [START_REF] Sirdi | An rml algorithm for retrieval of sinusoids with cascaded notch filters[END_REF]:

ε k = 1 A j (rq -1 ) .(â j -a j ).ψ j k-1 + η k (20)
There exits an upper limit l for the de-biasing parameter in order to get the SPR condition. This limit l depends on the signal frequency and limits the accuracy of the estimation. The SPR condition on r is more restrictive than the stability one. Then Theorem 2 cannot be applied in all cases.

Thus the local stability cannot be established although the global stability can be ensured.

3) Uniqueness of the RML estimates: In theorem 2 we have got asymptotically WPl:

ε k = η k Aj (rq -1 )
Equations 4 and 14 yield

ε k = Âj (q -1 ) Âj (rq -1 ) .ỹ j k = Âj (q -1 ) Âj (rq -1 ) . η k A j (q -1 ) (21)
It then follows from 17 and 18 that:

Âj (q -1 ) Âj (rq -1 ) = A j (q -1 ) A j (rq -1 ) (22)
and then, â1 = a 1 with probability one (WP1) if r = 1.

Theorem 4. : : If the order is correctly chosen (the number of the frequency components is matched), then the RML estimates for ANF are unique

The main results concerning ANF are then:

1) The global convergence can be ensured with an appropriate choice of r the de-biasing parameter without any stability monitoring.

2) The RML estimates are unique.

For p cascaded cells adapted along the same procedure, each cell will converge near to a local minimum. The filtering by ANFs will make these minima distinct, removing, for each cell, one of the other frequencies. This will not be the case if the implementation of the Notch cells is made in parallel instead of in cascaded form.

B. Convergence of Amplitudes and phases estimation, RIV-WRLS

The Exponential Convergence of the estimation of the amplitudes and the phases is guaranteed by the WRLS algorithm properties, as shown in the following theorem which uses the concept of persistent excitation see eg [START_REF] Widrow | Adaptive noise cancelling: Principles and applications[END_REF], [START_REF] Landau | Techniques de modélisation récursives pour l'analyse spectrale paramétrique adaptative[END_REF].

Theorem 5. Convergence of the WRLS algorithm

The WRLS estimation scheme of equations (II-A) with an exponential forgetting factor λ is exponentially stable, under persistent excitation condition.

Proof. The proof of this theorem can be conducted in the same lines as in [START_REF] Johnson | Exponential convergence of rls with exponential forgetting factor[END_REF].

The regression vectors (RIV) are composed by p sinusoids having different frequencies and phases. Then the Persistent Excitation Condition (PEC) is satisfied.

Equation (23) yields by lemma 1 of [START_REF] Johnson | Exponential convergence of rls with exponential forgetting factor[END_REF]. The regressors are composed by p sinusoids having different frequencies. Then the persistent excitation condition is satisfied:

0 < γ 1 .I < j+s k=j Φ k Φ T k < γ 2 .I < ∞ (23)
Equation ( 23) yields by lemma 1 of [START_REF] Johnson | Exponential convergence of rls with exponential forgetting factor[END_REF]:

0 < γ 1 .(λ -1 -1) (λ -(s+1) -1) .I < F -1 k-1 < γ 2 1 -λ s+1 .I + o(λ k ) (24)
The adaptation gain is bounded and will never be zero (the alertness property of WRLS and its tracking capabilities). Finally reasoning as in [START_REF] Johnson | Exponential convergence of rls with exponential forgetting factor[END_REF] we obtain:

θk 2 < λ s+1 -1 γ 1 (λ -1 -1) .λ k .λ max (F -1 0 ). θ0 2 (25)
This means that the parameter estimation error will be always strictly decreasing. Then the convergence of the estimations.

Comparative Simulation results will be presented in the final version of this paper.

IV. SIMULATIONS AND EXPERIMENTAL VALIDATION

In order to test the proposed ARML and RLS for predicting narrow band signals, we consider simulation of non-stationary signal with 3 frequency components and an additive centered and normal white noise v k (with variance σ 2 v = 1) and signal to noise ratio SNR = 25dB.

y k = 3 i=1 C i (k).Sin(2.πf i .T e .k + β i (k)) + v k (26)
Frequencies, amplitudes and phases of the three components are slowly time varying to prove the tracking capability of the estimation. The sampling rate is F e = 2Hz and the 3 frequencies are set initially to f 1 = 0.2Hz, f 2 = 0.3Hz, f 3 = 0.4Hz and vary as in Figure [START_REF] Sirdi | An rml algorithm for retrieval of sinusoids with cascaded notch filters[END_REF]. The frequencies deduced, on line, from the estimated parameters using equations 7. The following figure (3) illustrates the good convergence of the on line frequency estimations, after a short transient period (less than 20s). We observe that the frequency estimation converges after 36 sampling points (18s) and tracks simultaneous linear variations of the instantaneous frequencies.

B. Amplitude and phase estimation by WRLS

Using previous estimated frequencies, this second stage estimates amplitudes C i and phases β i of each one of the three components, using the proposed Weighted Recursive Least Squares algorithm. The regression vector and the parameters vector to be estimated are defined in the equation ( 4) and the setimations are given by equations (5 and II-A). The initial value of the adaptation gain is set to G 0 = 100 and the forgetting factor lambda λ 0 is set to 0.99. The parameters vector θ is initially set to 0. Components of θ : g l and h l are used to compute C i and β i using the last equation of 5.

The following figures illustrate the convergence of WRLS for amplitude/phase estimation. The figure 4 depicts the evolution of components amplitudes estimates versus time. Fig. 5: Phases estimates versus time The figure 4 shows the evolution of components phases estimates versus time. We observe that the amplitude/phase estimation converges after 50 points (25s) and tracks simultaneous linear variations of amplitudes and phases. The last figure shows that the prediction is quite good for a signal with 6 frequency components.

V. CONCLUSION

The main interest of the ANF in Cascade form is to simplify computation of the estimated frequencies and in case of independently time varying frequencies. A second order filter model appears to be faster to track variation than a higher order one.

The main results in this paper demonstrate the good performances observed in simulation for the ANF. This confirm the theoretical analysis and convergence proves (for RML and ELS PAAs). The exponential convergence of the estimation of the amplitudes and phases has been also proved. The proposed (OPEM-RML) algorithm is asymptotically consistent and robust faced with the neglected dynamics and noise. In addition, for case of time varying signals, its tracking capabilities insure very good estimations and tracking of the parameters time variation.

Several estimation algorithms may be applied with these structures, like RELS, ARML, RIV (Recursive Extended Least Squares, Approximate Recursive Maximum Likelihood, Recursive Instrumental Variables with WRLS). Some special procedures may be employed to increase the convergence rate (by adjusting the bandwidth parameter r) and to enhance the estimation consistency and reduce the algorithm complexity.

The obtained estimation accuracy is better with the RML Adaptation Algorithm than with the Recursive Extended Least Squares method for which an upper limit of the de-biasing parameter is crucial in order to have a convergence without local instabilities.

The exponential convergence of the RIV-WRLS estimates of amplitudes and phases is proved by use of the theoretical background on persistent excitation. These results are very important for the time varying systems or in case of neglected dynamics due to the resulting robustness of the proposed algorithms. The implementation form studied here is computationally attractive and robust also when frequency, amplitude or phase are time varying.

Fig. 1 :

 1 Fig. 1: Signal (blue) and its ANF prediction (green) A. Frequencies estimation by cascaded ANF and ARML The first stage estimates the three cascaded Notch Filter parameters. The parameters estimates âi (0) are set to zero at the beginning. All ANF bandwidths are initially set at r 0 = 0.1 and exponentially increases to r f = 0.8 with a factor of r d = 0.99 (see equation 19). The figure 2 depicts the evolution of the ANF parameters estimation âi (k) versus time (t = kT e ). The dotted lines represent the true values of these parameters. The convergence is fast and estimations are precise and accurate. This results in a good (one step ahead) prediction of the signal as shown as in Figure (1).

Fig. 2 :

 2 Fig. 2: Notch Filters parameters estimates versus time

Fig. 3 :

 3 Fig. 3: Frequencies estimates versus time

Fig. 4 :

 4 Fig. 4: Amplitudes estimates versus time

Fig. 6 :

 6 Fig. 6: Phases estimates for 6 components

ACKNOWLEDGMENT

The first author would like to thank I.D. Landau, for discussions and the suggestion to publish these results. He is also grateful for all the past collaborations they had. He addresses also many thanks to Odile Macchi for the collaborations. This work started in the LaG of ENSIEG in Grenoble, and is, now, founded by the SASV group of the LIS (UMR CNRS 7020).