Yaozhong Hu
email: hu@math.ku.edu

Samy Tindel
email: tindel@iecn.u-nancy.fr

SMOOTH DENSITY FOR SOME NILPOTENT ROUGH DIFFERENTIAL EQUATIONS

Keywords: . 2000 Mathematics Subject Classification. 60H07, 60H10, 65C30 fractional Brownian motion, rough paths, Malliavin calculus, hypoellipticity

In this note, we provide a non trivial example of differential equation driven by a fractional Brownian motion with Hurst parameter 1/3 < H < 1/2, whose solution admits a smooth density with respect to Lebesgue's measure. The result is obtained through the use of an explicit representation of the solution when the vector fields of the equation are nilpotent, plus a Norris type lemma in the rough paths context.

Introduction

Let B = (B 1 , . . . , B d ) be a d-dimensional fractional Brownian motion with Hurst parameter 1/3 < H < 1/2, defined on a complete probability space (Ω, F , P). Remind that this means that all the component B i of B are independent centered Gaussian processes with covariance

R H (t, s) := E B i t B i s = 1 2 (s 2H + t 2H -|t -s| 2H ). (1)
In particular, the paths of B are γ-Hölder continuous for all γ ∈ (0, H). This paper is concerned with a class of R m -valued stochastic differential equations driven by B, of the form

dy t = d i=1 V i (y t ) dB i t , t ∈ [0, T ], y 0 = a, (2)
where T > 0 is a fixed time horizon, a ∈ R m stands for a given initial condition and (V 1 , . . . , V d ) is a family of smooth vector fields of R m . Stochastic differential systems driven by fractional Brownian motion have been the object of intensive studies during the past decade, both for their theoretical interest and for the wide range of application they open, covering for instance finance [START_REF] Guasoni | No arbitrage under transaction costs, with fractional Brownian motion and beyond[END_REF][START_REF] Willinger | Stock market prices and long-range dependence[END_REF] or biophysics [START_REF] Kou | Generalized Langevin equation with fractional Gaussian noise: subdiffusion within a single protein molecule[END_REF][START_REF] Odde | Stochastic dynamics of the nerve growth cone and its microtubules during neurite outgrowth[END_REF] situations. The first aim in the theory has thus been to settle some reasonable tools allowing to solve equations of type [START_REF] Baudoin | Operators associated with a stochastic differential equation driven by fractional Brownian motions[END_REF]. This has been achieved, when the Hurst parameter H of the underlying fBm is > 1/2, thanks to methods of fractional integration [START_REF] Nualart | Differential equations driven by fractional Brownian motion[END_REF][START_REF] Zähle | Integration with respect to fractal functions and stochastic calculus I[END_REF], or simply by means of Young type integration (see e.g [START_REF] Gubinelli | Controlling rough paths[END_REF]). When one moves to more irregular cases, namely H < 1/2, the standard method by now in order to solve equations like (2) relies on rough paths considerations, as explained for instance in [START_REF] Friz | Multidimensional stochastic processes as rough paths: theory and applications[END_REF][START_REF] Gubinelli | Controlling rough paths[END_REF][START_REF] Lyons | System control and rough paths[END_REF].

A second natural step in the study of fractional differential systems consists in establishing some properties about their probability law. Some substitute for the semigroup property governing L(y t ) in the Markovian case (namely when H = 1/2) have been given in [START_REF] Baudoin | Operators associated with a stochastic differential equation driven by fractional Brownian motions[END_REF][START_REF] Neuenkirch | Trees and asymptotic developments for fractional diffusion processes[END_REF], in terms of asymptotic expansions in a neighborhood of t = 0. nSome considerable efforts have also been made in order to analyze the density of L(y t ) with respect to Lebesgue measure. To that respect, in the regular case H > 1/2 the situation is rather clear: the existence of a density is shown in [START_REF] Nualart | Malliavin calculus for stochastic differential equations driven by a fractional Brownian motion[END_REF] under some standard nondegeneracy conditions, the smoothness of the density is established in [START_REF] Hu | Differential equations driven by Hölder continuous functions of order greater than 1/2[END_REF] under elliptic conditions on the coefficients, and this result is extended to the hypoelliptic case in [START_REF] Baudoin | A version of Hörmander's theorem for the fractional Brownian motion[END_REF]. In all, this set of results replicates what has been obtained for the usual Brownian motion, at the price of huge technical complications.

In the irregular case H < 1/2, the picture is far from being so complete. Indeed, the existence part of the density results have been thoroughly studied under elliptic and Hörmander conditions (see [START_REF] Cass | Densities for rough differential equations under Hörmander's condition[END_REF][START_REF] Friz | Multidimensional stochastic processes as rough paths: theory and applications[END_REF] for a complete review). However, when one wishes to establish the smoothness of the density, some strong moment assumptions on the inverse of the Malliavin derivative of y t are usually required. These moment estimates are still an important open question in the field, as well as the smoothness of density for random variables like y t .

The current paper proposes to make a step in this direction, and we wish to prove that L(y t ) can be decomposed as p t (z) dz for a smooth function p t in some special non trivial examples of equation [START_REF] Baudoin | Operators associated with a stochastic differential equation driven by fractional Brownian motions[END_REF]. Namely, we will handle in the sequel the case of nilpotent vector fields V 1 , . . . , V d (see Hypothesis 4.1 for a precise description), and in this context we shall derive the following density result: Theorem 1.1. Suppose that the vector fields V i , 1 = 1, 2, . . . , d are smooth with all derivatives bounded, and that they are n-nilpotent in the sense that their Lie brackets of order n vanish for some positive integer n. We also assume that V 1 , . . . , V d satisfy Hörmander's hypoelliptic condition (their Lie brackets generate R m at any point x ∈ R m ), and that all the Lie brackets of order greater or equal to 2 are constant. Then for all t > 0 the probability law of the random variable y t , defined by (2) admits a smooth density with respect to Lebesgue measure.

Notice that the hypoelliptic assumption is quite natural in our context. Indeed, it would certainly be too restrictive to consider a family of vector fields V 1 , . . . , V d being nilpotent and elliptic at the same time. Moreover, some interesting examples of equations satisfying our standing assumptions will be given below. It should be stressed however that the basic aim of this article is to prove that smoothness of density results can be obtained for rough differential equations driven by a fractional Brownian motion in some specific situations, even if the general hypoelliptic case is still an important open problem. We refer to [START_REF] Baudoin | Gaussian bounds for the density of solutions of stochastic differential equations driven by fractional Brownian motions[END_REF] for another case, based on skew-symmetric properties, where a similar theorem holds true.

In order to prove Theorem 1.1, two main ingredients have to be highlighted: (i) Working under the nilpotent assumptions described above enables to use a Strichartz type representation for the solution to our equation, given in terms of a finite chaos expansion. This allows to derive some bounds for the moments of both y t and its Malliavin derivative, which is the main missing tool on the way to smoothness of density results for rough differential equations in the general case. (ii) With the integrability of Malliavin derivative in hand, we shall follow the standard probabilistic way to prove smoothness of density under Hörmander's conditions, for which we refer to [START_REF] Hairer | On Malliavin's proof of Hörmander's theorem[END_REF][START_REF] Malliavin | Stochastic calculus of variation and hypoelliptic operators[END_REF][START_REF] Norris | Simplified Malliavin calculus[END_REF]. To this purpose, the second main ingredient is a Norris type lemma, which has to be extended (in the rough path context) to controlled processes. It should be mentioned at this point that a similar result has been proven recently (an independently) in [START_REF] Hairer | Regularity of laws and ergodicity of hypoelliptic SDEs driven by rough paths[END_REF]. These two ingredients will be developed in the remainder of the article.

Here is how our article is structured: Some preliminaries on rough differential equations and fractional Brownian motion are given in Section 2. Section 3 is devoted to the proof of our Norris type lemma for controlled processes in the sense of [START_REF] Gubinelli | Controlling rough paths[END_REF]. Finally, Malliavin calculus tools and their application to density results for the random variable y t are presented at Section 4. Notation: In the remainder of the article, c, c 1 , c 2 will stand for generic positive constants which may change from line to line. We also write a b (resp. a ≍ b) when a ≤ c b (resp. a = c b) for a universal constant c.

Rough differential equations and fractional Brownian motion

Generalized integrals will be needed in the sequel in order to define and solve equations of the form (2), and also to get an equivalent of Norris lemma in our context. Though all those elements might be obtained within the landmark of usual rough paths setting [START_REF] Friz | Multidimensional stochastic processes as rough paths: theory and applications[END_REF][START_REF] Lyons | System control and rough paths[END_REF] we have chosen here to work with the algebraic integration framework, which (from our point of view) is more amenable to handy calculations.

In this section, we recall thus the main concepts of algebraic integration. Namely, we state the definition of the spaces of increments, of the operator δ, and its inverse called Λ (or sewing map). We also recall some elementary but useful algebraic relations on the spaces of increments. The interested reader is sent to [START_REF] Gubinelli | Controlling rough paths[END_REF] for a complete account on the topic, or to [START_REF] Deya | Rough Volterra equations 2: convolutional generalized integrals[END_REF][START_REF] Gubinelli | Rough evolution equations[END_REF] for a more detailed summary.

2.1.

Increments. The extended integral we deal with is based on the notion of increments, together with an elementary operator δ acting on them.

The notion of increment can be introduced in the following way: for two arbitrary real numbers ℓ 2 > ℓ 1 ≥ 0, a vector space V , and an integer k ≥ 1, we denote by

C k ([ℓ 1 , ℓ 2 ]; V ) the set of continuous functions g : [ℓ 1 , ℓ 2 ] k → V such that g t 1 •••t k = 0 whenever t i = t i+1
for some i ∈ {0, . . . , k -1}. Such a function will be called a (k -1)-increment, and we will set

C * ([ℓ 1 , ℓ 2 ]; V ) = ∪ k≥1 C k ([ℓ 1 , ℓ 2 ]; V ). To simplify the notation, we will write C k (V ), if there is no ambiguity about [ℓ 1 , ℓ 2 ].
The operator δ is an operator acting on k-increments, and is defined as follows on C k (V ):

δ : C k (V ) → C k+1 (V ), (δg) t 1 •••t k+1 = k+1 i=1 (-1) i g t 1 ••• ti •••t k+1 , (3)
where ti means that this particular argument is omitted. Then a fundamental property of δ, which is easily verified, is that δδ = 0, where δδ is considered as an operator from

C k (V ) to C k+2 (V ). We will denote ZC k (V ) = C k (V ) ∩ Kerδ and BC k (V ) = C k (V ) ∩ Imδ.
Some simple examples of actions of δ, which will be the ones we will really use throughout the article, are obtained by letting g ∈ C 1 (V ) and h ∈ C 2 (V ). Then, for any t, u, s ∈ [ℓ 1 , ℓ 2 ], we have (δg) st = g tg s and (δh

) sut = h st -h su -h ut . (4)
Our future discussions will mainly rely on k-increments with k = 2 or k = 3, for which we will use some analytical assumptions. Namely, we measure the size of these increments by Hölder norms defined in the following way: for f ∈ C 2 (V ) let

f µ = sup s,t∈[ℓ 1 ,ℓ 2 ] |f st | |t -s| µ and C µ 2 (V ) = {f ∈ C 2 (V ); f µ < ∞} . (5)
Using this notation, we define in a natural way C µ 1 (V ) = {f ∈ C 1 (V ); δf µ < ∞}. In the sequel, we also handle norms including supremums, of the form

f µ,∞ = f µ + f ∞ , and C µ,0 1 (V ) = {f ∈ C 1 (V ); f µ,∞ < ∞}. (6)
In the same way, for h ∈ C 3 (V ) we set

h γ,ρ = sup s,u,t∈[ℓ 1 ,ℓ 2 ] |h sut | |u -s| γ |t -u| ρ , (7)
h µ = inf i h i ρ i ,µ-ρ i ; h = i h i , 0 < ρ i < µ ,
where the last infimum is taken over all sequences {h i , i ∈ N} ⊂ C 3 (V ) such that h = i h i and over all choices of the numbers ρ i ∈ (0, µ). Then • µ is easily seen to be a norm on C 3 (V ), and we define

C µ 3 (V ) := {h ∈ C 3 (V ); h µ < ∞} . Eventually, let C 1+ 3 (V ) = ∪ µ>1 C µ 3 (V )
, and note that the same kind of norms can be considered on the spaces ZC 3 (V ), leading to the definition of the spaces ZC µ 3 (V ) and ZC 1+ 3 (V ). In order to avoid ambiguities, we sometimes denote in the following by N [ • ; C κ j ] the κ-Hölder norm on the space C j , for j = 1, 2, 3. For ζ ∈ C j (V ), we also set

N [ζ; C 0 j (V )] = sup s∈[ℓ 1 ;ℓ 2 ] j ζ s V .
The invertibility of δ under Hölder regularity conditions is an essential tool for the construction of our generalized integrals, and can be summarized as follows:

Theorem 2.1 (The sewing map). Let µ > 1. For any h ∈ ZC µ 3 (V ), there exists a unique Λh ∈ C µ 2 (V ) such that δ(Λh) = h. Furthermore,

Λh µ ≤ 1 2 -2 µ N [h; C µ 3 (V )]. (8)
This gives rise to a continuous linear map Λ :

ZC µ 3 (V ) → C µ 2 (V ) such that δΛ = id ZC µ 3 (V ) . Proof.
The original proof of this result can be found in [START_REF] Gubinelli | Controlling rough paths[END_REF]. We refer to [START_REF] Deya | Rough Volterra equations 2: convolutional generalized integrals[END_REF][START_REF] Gubinelli | Rough evolution equations[END_REF] for two simplified versions.

The sewing map creates a first link between the structures we just introduced and the problem of integration of irregular functions: Corollary 2.2 (Integration of small increments). For any 1-increment g ∈ C 2 (V ) such that δg ∈ C 1+ 3 , set h = (id -Λδ)g. Then, there exists f ∈ C 1 (V ) such that h = δf and

δf st = lim |Πst|→0 n i=0 g t i t i+1 ,
where the limit is over any partition Π st = {t 0 = s, . . . , t n = t} of [s, t] whose mesh tends to zero. The 1-increment δf is the indefinite integral of the 1-increment g.

We also need some product rules for the operator δ. For this recall the following convention:

for g ∈ C n ([ℓ 1 , ℓ 2 ]R l,d ) and h ∈ C m ([ℓ 1 , ℓ 2 ]; R d,p ) let gh be the element of C n+m-1 ([ℓ 1 , ℓ 2 ]; R l,p
) defined by (gh) t 1 ,...,t m+n-1 = g t 1 ,...,tn h tn,...,t m+n-1 [START_REF] Driscoll | Smoothness of Density for the Area Process of Fractional Brownian Motion[END_REF] for t 1 , . . . , t m+n-1 ∈ [ℓ 1 , ℓ 2 ]. With this notation, the following elementary rule holds true:

Proposition 2.3. Let g ∈ C 2 ([ℓ 1 , ℓ 2 ]; R l,d ) and h ∈ C 1 ([ℓ 1 , ℓ 2 ]; R d ). Then gh is an element of C 2 ([ℓ 1 , ℓ 2 ]; R l ) and δ(gh) = δg h -g δh.
2.2. Random differential equations. One of the main appeals of the algebraic integration theory is that differential equations driven by a γ-Hölder signal x can be defined and solved rather quickly in this setting. In the case of an Hölder exponent γ > 1/3, the required structures are just the notion of controlled processes and the Lévy area based on x.

Indeed, recall that we wish to consider an equation of the form

dy t = d i=1 V i (y t ) dx i t , t ∈ [0, T ] , y 0 = a, ( 10
)
where a is a given initial condition in R m , x is an element of C γ 1 ([0, T ]; R d ), and (V 1 , . . . , V d ) is a family of smooth vector fields of R m . Then it is natural that the increments of a candidate for a solution to [START_REF] Fernique | Fonctions aléatoires gaussiennes, vecteurs aléatoires gaussiens[END_REF] should be controlled by the increments of x in the following way: Definition 2.4. Let z be a path in C κ 1 (R m ) with 1/3 < κ ≤ γ, and set δx := x 1 . We say that z is a weakly controlled path based on x if z 0 = a with a ∈ R m , and δz ∈ C κ 2 (R m ) has a decomposition δz = ζx 1 + r, that is, for any s, t ∈ [0, T ],

δz st = ζ j s x 1,j st + r st , (11)
where we have used the summation over repeated indices convention, and with ζ 1 , . . . , ζ d ∈ C κ 1 (R m ), as well as r ∈ C 2κ 2 (R m ). The space of weakly controlled paths will be denoted by Q x κ,a (R m ), and a process z ∈ Q x κ,a (R m ) can be considered in fact as a couple (z, ζ). The space Q x κ,a (R m ) is endowed with a natural semi-norm given by

N [z; Q x κ,a (R m )] = N [z; C κ 1 (R m )] + d j=1 N [ζ j ; C κ,0 1 (R m )] + N [r; C 2κ 2 (R m )], (12)
where the quantities N [g; C κ j ] have been defined in Section 2.1. For the Lévy area associated to x we assume the following structure: Hypothesis 2.5. The path x : [0, T ] → R d is γ-Hölder continuous with 1 3 < γ ≤ 1 and admits a so-called Lévy area, that is, a process

x 2 ∈ C 2γ 2 (R d,d ), which satisfies δx 2 = x 1 ⊗ x 1 , namely δx 2,ij sut = [x 1,i ] su [x 1,j
] ut , for any s, u, t ∈ [0, T ] and i, j ∈ {1, . . . , d}.

To illustrate the idea behind the construction of the generalized integral assume that the paths x and z are smooth and also for simplicity that d = m = 1. Then the Riemann-Stieltjes integral of z with respect to x is well defined and we have

t s z u dx u = z s (x t -x s ) + t s (z u -z s )dx u = z s x 1 st + t s (δz) su dx u for ℓ 1 ≤ s ≤ t ≤ ℓ 2 .
If z admits the decomposition ( 11) we obtain

t s (δz) su dx u = t s ζ s x 1 su + ρ su dx u = ζ s t s x 1 su dx u + t s ρ su dx u . (13)
Moreover, if we set

x 2 st := t s x 1 su dx u , ℓ 1 ≤ s ≤ t ≤ ℓ 2 ,
then it is quickly verified that x 2 is the Lévy area associated to x. Hence we can write

t s z u dx u = z s x 1 sz + ζ s x 2 st + t s ρ su dx u .

Now recast this equation as

t s ρ su dx u = t s z u dx u -z s x 1 st -ζ s x 2 st , (14)
and apply the increment operator δ to both sides of this equation. For smooth paths z and x we have

δ z dx = 0, δ(z x 1 ) = -δz x 1 ,
by Proposition 2.3 (recall also our convention (9) on products of increments). Hence applying these relations to the right hand side of ( 14), using the decomposition [START_REF] Ferreiro | Lévy area for Gaussian processes: A double Wiener-Itô integral approach[END_REF], the properties of the Lévy area and again Proposition 2.3, we obtain

δ ρ dx sut = δz su x 1 ut + δζ su x 2 ut -ζ s δx 2 sut = ζ s x 1 su x 1 ut + ρ su x 1 ut + δζ su x 2 ut -ζ s x 1 su x 1 ut = ρ su x 1 ut + δζ su x 2 ut . Summarizing, we have derived the representation δ ρ dx sut = ρ su x 1 ut + δζ su x 2 ut .
As we are dealing with smooth paths we have δ ρ dx lies into the space ZC 1+ 3 and thus belongs to the domain of Λ due to Proposition 2.1. (Recall that δδ = 0.) Hence, it follows

t s ρ su dx u = Λ st ρ x 1 + δζ x 2 ,
and inserting this identity into (13) we end up with

t s z u dx u = z s x 1 st + ζ s x 2 st + Λ st ρ x 1 + δζ x 2 .

Since in addition

ρ x 1 + δζ x 2 = -δ(z x 1 + ζ x 2 ),
we can also write this as

z u dx u = (id -Λδ)(z x 1 + ζ x 2 ).
Thus we have expressed the Riemann-Stieltjes integral of z with respect to x in terms of the sewing map Λ, the couple (x 1 , x 2 ) and of increments of z. This can now be generalized to the non-smooth case. Note that Corollary 2.2 justifies the use of the notion of integral.

Proposition 2.6. For fixed 1 3 < κ ≤ γ, let x be a path satisfying Hypothesis 2.5 on an arbitrary interval

[0, T ]. Furthermore, let z ∈ Q x κ,α ([ℓ 1 , ℓ 2 ]; R d
) such that the increments of z are given by [START_REF] Ferreiro | Lévy area for Gaussian processes: A double Wiener-Itô integral approach[END_REF]. Define ẑ by ẑℓ 1 = α with α ∈ R and

δẑ st = (id -Λδ)(z i x 1,i + ζ ji x 2,ij ) st ( 15
)
for ℓ 1 ≤ s ≤ t ≤ ℓ 2 . Then J (z * dx) := ẑ is a well-defined element of Q x κ, α([ℓ 1 , ℓ 2
]; R) and coincides with the usual Riemann integral, whenever z and x are smooth functions.

Moreover, the Hölder norm of J (z * dx) can be estimated in terms of the Hölder norm of the integrator z. (For this and also for a proof of the above Proposition, see e.g. [START_REF] Gubinelli | Controlling rough paths[END_REF].) This allows to use a fixed point argument to obtain the existence of a unique solution for rough differential equations.

Theorem 2.7. For fixed 1 3 < κ < γ, let x be a path satisfying Hypothesis 2.5 on an arbitrary interval

[0, T ]. Consider a given initial condition a in R m and (V 1 , . . . , V d ) a family of C 3 vector fields of R m , bounded with bounded derivatives. Let f µ,∞ = f ∞ + δf µ be the usual Hölder norm of a path f ∈ C 1 ([0, T ]; R l ).
Then we have:

(1) Equation ( 10) admits a unique solution y in Q x κ,a ([0, T ]; R m ) for any T > 0, and there exists a polynomial P T : R 2 → R + such that

N [y; Q x κ,a ([0, T ]; R m )] ≤ P T ( x 1 γ , x 2 2γ ) (16) holds. (2) Let F : R m × C γ 1 ([0, T ]; R d ) × C 2γ 2 ([0, T ]; R m,m ) → C γ 1 ([0, T ]; R m ) be the mapping defined by F a, x 1 , x 2 = y,
where y is the unique solution of equation ( 10). This mapping is locally Lipschitz continuous in the following sense: Let x be another driving rough path with corresponding Lévy area x2 and ã be another initial condition. Moreover denote by ỹ the unique solution of the corresponding differential equation. Then, there exists an increasing function

K T : R 4 → R + such that y -ỹ γ,∞,T ≤ K T ( x 1 γ , x1 γ , x 2 2γ , x2 2γ ) (17) × |a -ã| + x 1 -x1 γ + x 2 -x2 2γ holds.
The theorem above is borrowed from [START_REF] Friz | Multidimensional stochastic processes as rough paths: theory and applications[END_REF][START_REF] Gubinelli | Controlling rough paths[END_REF][START_REF] Lyons | System control and rough paths[END_REF], and we send the reader to these references for more details on the topic.

2.3.

Fractional Brownian motion. We shall recall here how the abstract Theorem 2.7 applies to fractional Brownian motion. We will also give some basic notions on stochastic analysis with respect to fBm, mainly borrowed from [START_REF] Nualart | Malliavin Calculus and Related Topics[END_REF], which will turn out to be useful in the sequel.

As already mentioned in the introduction, on a finite interval [0, T ] and for some fixed H ∈ (1/3, 1/2), we consider (Ω, F , P) the canonical probability space associated with fractional Brownian motion with Hurst parameter H. That is, Ω = C 0 ([0, T ]; R d ) is the Banach space of continuous functions vanishing at 0 equipped with the supremum norm, F is the Borel sigma-algebra and P is the unique probability measure on Ω such that the canonical process B = {B t , t ∈ [0, T ]} is a d-dimensional fractional Brownian motion with Hurst parameter H. Specifically, B has d independent coordinates, each one being a centered Gaussian process with covariance given by (1).

2.3.1. Functional spaces. Let E be the set of the space of d-dimensional elementary functions on [0, T ]:

E = f = (f 1 , . . . , f d ); f j = n j -1 i=0 a j i 1 [t j i ,t j i+1 ) , 0 = t 0 < t j 1 < • • • < t j n j -1 < t j n j = T, for j = 1, . . . , d . (18)
We call H the completion of E with respect to the semi-inner product

f, g H = d i=1 f i , g i H i , where 1 [0,t] , 1 [0,s] H i := R(s, t), s, t ∈ [0, T ].
Then, one constructs an isometry

K * H : H → L 2 ([0, 1]; R d ) such that K * H 1 [0,t 1 ] , . . . , 1 [0,t d ] = 1 [0,t 1 ] K H (t 1 , •), . . . , 1 [0,t d ] K H (t d , •) ,
where the kernel K H is given by

K H (t, u) = c H u t 1 2 -H (t -u) H-1 2 + 1 2 -H u 1 2 -H t u v H-3 2 (v -u) H-1 2 dv 1 {0<u<t} , (19)
with a strictly positive constant c H , whose exact value is irrelevant for our purpose.

Notice that this kernel verifies R H (t, s)

= s∧t 0 K H (t, r)K H (s, r) dr. Moreover, observe that K *
H can be represented in the following form: for ϕ = (ϕ 1 , . . . , ϕ d ) ∈ H, we have

K * H ϕ = (K * H ϕ 1 , . . . , K * H ϕ d ), with K * H ϕ = K * H ϕ 1 , . . . , K * H ϕ d , where [K * H ϕ i ] t = d H t 1/2-H D 1/2-H T - u -(1/2-H) ϕ i t ,
for a strictly positive constant d H . In particular, each H i is a fractional integral space of the form I

1/2-H T - (L 2 ([0, T ])) and C 1/2-H 1 ([0, T ]) ⊂ H i .

Malliavin derivatives.

Let us start by defining the Wiener integral with respect to B: for any element f in E whose expression is given as in [START_REF] Hu | Calculs formels sur les EDS de Stratonovitch[END_REF], we define the Wiener integral of f with respect to B as

B(f ) := d j=1 n j -1 i=0 a j i (B j t j i+1 -B j t j i ) .
We also denote this integral as T 0 f (t)dB t , since it coincides with a pathwise integral with respect to B.

For θ : R → R, and j ∈ {1, . . . , d}, denote by θ [j] the function with values in R d having all the coordinates equal to zero except the j-th coordinate that equals to θ. It is readily seen that E B 1

[j] [0,s) B 1 [k] [0,t) = δ j,k R s,t .
This definition can be extended by linearity and closure to elements of H, and we obtain the relation

E [B(f ) B(g)] = f, g H , valid for any couple (f, g) ∈ H 2 .
In particular, B(•) defines an isometric map from H into a subspace of L 2 (Ω). It should be pointed out that (Ω, H, P) defines an abstract Wiener space, on which chaos decompositions can be settled. We do not develop this aspect of the theory for sake of conciseness, but we will use later the fact that all L p norms are equivalent on finite chaos.

We can now proceed to the definition of Malliavin derivatives, for which we need an additional notation: Notation 2.8. For n, p ≥ 1, a function f ∈ C p (R n ; R) and any tuple (i 1 , . . . i p ) ∈ {1, . . . , d} p , we set ∂ i 1 ...ip f for ∂ p f ∂x i ...∂xp . With this notation in hand, let S be the family of smooth functionals F of the form

F = f (B(h 1 ), . . . , B(h n )), (20)
where h 1 , . . . , h n ∈ H, n ≥ 1, and f is a smooth function with polynomial growth, together with all its derivatives. Then, the Malliavin derivative of such a functional F is the H-valued random variable defined by

DF = n i=1 ∂ i f (B(h 1 ), . . . , B(h n )) h i .
For all p > 1, it is known that the operator D is closable from L p (Ω) into L p (Ω; H) (see e.g. [START_REF] Nualart | Malliavin Calculus and Related Topics[END_REF]Chapter 1]). We will still denote by D the closure of this operator, whose domain is usually denoted by D 1,p and is defined as the completion of S with respect to the norm

F 1,p := (E(|F | p ) + E( DF p H )) 1
p . It should also be noticed that partial Malliavin derivatives with respect to each component B j of B will be invoked: they are defined, for a functional F of the form (20) and j = 1, . . . , d, as

D j F = n i=1 ∂ i f (B(h 1 ), . . . , B(h n ))h [j] i ,
and then extended by closure arguments again. We refer to [26, Chapter 1] again for the definition of higher derivatives and Sobolev spaces D k,p for k > 1.

Levy area of fBm.

There are many ways to define the Levy area B 2 associated to fBm, and the reader is referred to [START_REF] Friz | Multidimensional stochastic processes as rough paths: theory and applications[END_REF]Chapter 15] for a complete review of these. The recent paper [START_REF] Ferreiro | Lévy area for Gaussian processes: A double Wiener-Itô integral approach[END_REF] is however of special interest for us, since it enables a direct definition of B 2 by Wiener chaos techniques. It can be summarized in the following way: Proposition 2.9. Let 1/3 < H < 1/2 be a fixed Hurst parameter. Then the fBm B belongs almost surely to any space C γ 1 for γ < H, and it gives raise to an increment B 2 ∈ C 2γ 2 which satisfies Hypothesis 2.5. Furthermore, for any 0 ≤ s < t ≤ T , B 2 st is an element of the second chaos associated to B, and

E B 2 st p ≤ c p (t -s) 2Hp , p ≥ 1.
Moreover, the iterated integrals of B can be obtained as limits of Riemann type integrals. Indeed, for k ≥ 1 and 0 ≤ s < t ≤ T , consider the simplex

S k ([s, t]) = {(u 1 , . . . , u k ); s ≤ u 1 < • • • < u k ≤ t} . ( 21
)
For a given partition Π of [0, T ], we also denote by B Π the linearization of B based on Π.

Combining the results of [START_REF] Ferreiro | Lévy area for Gaussian processes: A double Wiener-Itô integral approach[END_REF][START_REF] Friz | Multidimensional stochastic processes as rough paths: theory and applications[END_REF], the following proposition holds true:

Proposition 2.10. Let k ≥ 1, and for a sequence of partitions (Π n ) n≥1 , set B n := B Πn . For 0 ≤ s < t ≤ T and (i 1 , . . . , i k ) ∈ {1, . . . , d} k , we consider then

B k,n,i 1 ,...,i k st = S k ([s,t]) dB n,i 1 u 1 • • • dB n,i k u k ,
understood in the Riemann sense. Then there exists a sequence of partitions (Π n ) n≥1 such that B k,n,i 1 ,...,i k converges almost surely and in L 2 , as an element of C kγ 2 for any γ < H, to an element called B k,i 1 ,...,i k . When k = 1, we obtain the increment δB of our fBm. When k = 2, the limit corresponds to the increment B 2 of Proposition 2.9.

As a corollary of the previous considerations, we have the Proposition 2.11. Assume 1/3 < H < 1/2. Then Theorem 2.7 applies almost surely to the fBm paths, enhanced with the Levy area B 2 . We are thus able to solve equation

dy t = d i=1 V i (y t ) dB i t , t ∈ [0, T ] , y 0 = a, (22)
under the conditions of Theorem 2.7.

A Norris type lemma

Norris' lemma [START_REF] Norris | Simplified Malliavin calculus[END_REF] is one of the basic ingredients in order to obtain smoothness of densities for solutions to stochastic differential equations under hypoelliptic conditions, and was already extended to fBm with Hurst parameter H > 1/2 in [START_REF] Baudoin | A version of Hörmander's theorem for the fractional Brownian motion[END_REF]. We shall extend in the current section this lemma to the rough paths context. A preliminary step along this direction consists of proving the following elementary lemma:

Lemma 3.1. Let 0 < α < ρ < 1, and consider b ∈ C([0, T ]). Then for any 0 < η < 1, we have b α,∞ ≤ C α,ρ η b ρ,∞ + η -1/(ρ-α) b L 1 , (23)
where we recall that b α,∞ has been defined by [START_REF] Cass | Densities for rough differential equations under Hörmander's condition[END_REF].

Proof. Recall that b α has been defined by [START_REF] Borell | Tail probabilities in Gauss space. Vector space measures and applications[END_REF]. Thus, for 0 < α < ρ < 1, we have

b α = sup s,t |δb st | |t -s| ρ α ρ |δb st | 1-α ρ ≤ 2 1-α ρ b 1-α ρ ∞ b α ρ ρ .
Thus, for an arbitrary constant η > 0, we have

b α ≤ C α,ρ η b ρ + η -α ρ-α b ∞ ≤ C α,ρ η b ρ + η -α ρ-α b ∞ ,
thanks to Young's inequality. Therefore for an arbitrary constant 0

< η ≤ 1 b α ≤ C α,ρ η b ρ + η -α ρ-α b ∞ .
Invoke now the interpolation inequality [3, formula (3.17)] in order to get

b α ≤ C α,ρ η b ρ + η -α ρ-α γ b ρ + γ -1 1+α b L 1 = C α,ρ η b ρ + η -1/(ρ-α) b L 1 ,
where we have chosen γ = η α ρ-α . Invoking again [3, formula (3.17)] in order to go from • α to • α,∞ norms and this finishes our proof.

We can now turn to the announced Norris type lemma, whose proof is an adaptation of [START_REF] Baudoin | A version of Hörmander's theorem for the fractional Brownian motion[END_REF] to the case of controlled processes. Proposition 3.2. Assume B is a fractional Brownian motion with H > 1/3. Let z be a controlled path in Q B γ (R m ), with decomposition

δz i st = ζ i,j s B 1,j st + r i st . ( 24
)
We assume that 1/3 < α < γ < H and that the quantity

E[N p [z; Q B γ (R m )]
] is finite for all p ≥ 1. Set δy st = J (z * dB) according to Proposition 2.6. Then there exists q > 0 such that, for every p > 0 we can find a strictly positive constant c p such that P ( y γ,∞ < ε, and z α,∞ > ε q ) < c p ε p .

Proof. In order to avoid cumbersome indices, we shall prove our result in the case of 1-dimensional processes. Generalization to the multidimensional setting is a matter of trivial considerations. We also work on the interval [0, 1] instead of [0, T ] for the sake of notational simplicity. As a last preliminary observation, note that if z admits the decomposition [START_REF] Neuenkirch | Trees and asymptotic developments for fractional diffusion processes[END_REF], then according to [START_REF] Guasoni | No arbitrage under transaction costs, with fractional Brownian motion and beyond[END_REF] we have

δy = z B 1 + ζ B 2 + y ♯ , where y ♯ = Λ r B 1 + δζ B 2 .
Similarly to what is done in [START_REF] Baudoin | A version of Hörmander's theorem for the fractional Brownian motion[END_REF], we consider two time scales δ ≪ ∆ ≪ 1. We assume moreover that ∆/δ = r with r ∈ N. We use a partition {t n ; n ≤ 1/δ} of [0, 1] with t n = nδ, so that t N r = N∆. Some increments below will then be frozen on the time scale ∆, in order to take advantage of some averaging properties of the process B.

Step1: Coarse graining on increments. Consider then n such that (N -1)r ≤ n ≤ Nr -1, so that (N -1)∆ ≤ t n ≤ N∆ -1. According to [START_REF] Guasoni | No arbitrage under transaction costs, with fractional Brownian motion and beyond[END_REF], we have

δy tnt n+1 = z tn B 1 tnt n+1 + ζ tn B 2 tnt n+1 + y ♯ tnt n+1 = z N ∆ B 1 tnt n+1 -δz tn,N ∆ B 1 tnt n+1 + ζ tn B 2 tnt n+1 + y ♯ tnt n+1 ,
where y ♯ is an increment in C 3γ 2 . Thus

z N ∆ B 1 tnt n+1 = δy tnt n+1 + δz tn,N ∆ B 1 tnt n+1 -ζ tn B 2 tnt n+1 -y ♯ tnt n+1 . (25)
Set now

X N = N r-1 n=(N -1)r B 1 tnt n+1 4 and Y N = X 1/4 N . Then |z N ∆ | 4 |X N | = N r-1 n=(N -1)r z N ∆ B 1 tnt n+1 4 .
Furthermore, invoking relation ( 25), we get

|z N ∆ B 1 tnt n+1 | ≤ y γ δ γ + z γ B γ δ γ ∆ γ + ζ ∞ B 2 2γ δ 2γ + y ♯ 3γ δ 3γ .
Raising this inequality to power 4 and summing over (N -1)r ≤ n ≤ Nr -1, we obtain

|z 4 N ∆ X N | ≤ δ 4γ-1 ∆ y γ + z γ B γ ∆ γ + ζ ∞ B 2 2γ δ γ + y ♯ 3γ δ 2γ 4
, and therefore

|z N ∆ |Y N ≤ δ γ-1/4 ∆ 1/4 y γ + z γ B γ ∆ γ + ζ ∞ B 2 2γ δ γ + y ♯ 3γ δ 2γ . Sum now over N (recall that 1 ≤ N ≤ 1/∆) in order to get 1/∆ N =1 |z N ∆ |Y N ≤ δ γ-1/4 ∆ -3/4 y γ + z γ B γ ∆ γ + ζ ∞ B 2 2γ δ γ + y ♯ 3γ δ 2γ . ( 26
)
Step 2: Behavior of a 4 th order variation. Throughout the proof, we shall use the notations , ≍ given in the introduction. For K ≥ 1, set XK = K n=1 |B 1 tnt n+1 | 4 . We shall prove that E[ XK ] ≍ Kδ 4H , and Var( XK ) Kδ 8H .

Indeed, a simple scaling argument shows that XK

(L) = δ 4H XK , with XK = K n=1 |B 1 n,n+1 | 4 .
Introduce now the k-th Hermite polynomial H k (see [START_REF] Nualart | Malliavin Calculus and Related Topics[END_REF] for a definition and properties of these objects), and notice in particular that H 2 (x) = x 2 -1 and H 4 (x) = x 4 -6x 2 + 3. This enables to decompose XK as

XK = K n=1 H 4 (B 1 n,n+1 ) + 6H 2 (B 1 n,n+1 ) + 3K. ( 28
)
Recall now that for a centered Gaussian vector (U,

V ) in R 2 such that E[U 2 ] = E[V 2 ] = 1 we have E[H k (U)] = 0, and E [H k (U) H l (V )] = k! (E[U V ]) k 1 (k=l) . (29)
Plugging this identity in [START_REF] Nualart | Malliavin calculus for stochastic differential equations driven by a fractional Brownian motion[END_REF], this immediately yields E[ XK ] = 3K, which is our first assertion in [START_REF] Nualart | Differential equations driven by fractional Brownian motion[END_REF]. In addition, the second part of (29) entails

Var( XK ) = 2 K n 1 ,n 2 =1 12 α 4 n 1 ,n 2 + α 2 n 1 ,n 2 := 2S K ,
where we have set

α n 1 ,n 2 = E[B 1 n 1 ,n 1 +1 B 1 n 2 ,n 2 +1 ] = 1 2 |n 2 -n 1 + 1| 2H + |n 2 -n 1 -1| 2H -2|n 2 -n 1 | 2H .
Summarizing, we have obtained that

Var( XK ) = δ 8H Var( XK ) = 2 δ 8H S K . (30)
We will now prove that S K K. Indeed, write first

S K = 1≤n 1 ≤K 12 α 4 n 1 ,n 1 + α 2 n 1 ,n 1 + 2 1≤n 1 <n 2 ≤K 12 α 4 n 1 ,n 2 + α 2 n 1 ,n 2 := S 1 K + 2S 2 K .
Then, since α n 1 ,n 1 = 1, it is readily checked that S 1 K K. Moreover, the term S 2 K can be decomposed into -4 is finite whenever H < 3/4. Gathering our bounds on S 1 K , S 21 K and S 22 K , we obtain S K K, and plugging this bound into [START_REF] Strichartz | The Campbell-Baker-Hausdorff-Dynkin formula and solutions of differential equations Jour[END_REF], we end up with Var( XK ) K δ 8H , which is our claim. This finishes the proof of [START_REF] Nualart | Differential equations driven by fractional Brownian motion[END_REF].

S 2 K = 1≤n 1 ≤K-1 12 α 4 n 1 ,n 1 +1 + α 2 n 1 ,n 1 +1 + 1≤n 1 ,n 2 ≤K, n 2 -n 1 ≥2 12 α 4 n 1 ,n 2 + α 2 n 1 ,n 2 := S 21 K +S 22 K . Notice now that α n 1 ,n 1 +1 = -[1 -2 -2(H-1/2) ], which easily yields S 21 K ≤ c H K. As far as S 22 K is concerned, write for n 2 -n 1 ≥ 2 α n 1 ,n 2 = H(2H -1) 1 0 dr r -r |(n 2 -n 1 ) + u| 2H-2 du, which immediately yields α n 1 ,n 2 |(n 2 -n 1 ) -1| 2H-2 . Thus S 22 K K-2 n 1 =1 K n 2 =n 1 +2 |(n 2 -n 1 ) -1| 8H-8 + 3|(n 2 -n 1 ) -1| 4H-4 K-2 n 1 =1 K-n 1 +1 m=1 m 4H-4 ≤ K ∞ m=1 m 4H-4 K, since ∞ m=1 m 4H
Step 3: Concentration inequalities for Y N . Let us recall that X N is in the 4 th chaos of the fBm B. Hence, a result by Borell [START_REF] Borell | Tail probabilities in Gauss space. Vector space measures and applications[END_REF] entails

P |X N -E[X N ]| [Var(X N )] 1/2 ≥ u ≤ c 1 e -c 2 u 1/2 , u ≥ 0,
for two universal constant c 1 , c 2 > 0. With [START_REF] Nualart | Differential equations driven by fractional Brownian motion[END_REF] in hand, this yields

P |X N -3∆δ 4H-1 | ≥ ∆ 1/2 δ 4H-1/2 u ≤ c 1 e -c 2 u 1/2 , u ≥ 0. ( 31
)
We now wish to produce a concentration inequality for

Y N = X 1/4
N . Since X N is a small random quantity of order ∆δ 4H-1 , let us use the inequality

|b 1/4 -a 1/4 | ξ -3/4 |b -a|, where ξ ∈ (a ∧ b, a ∨ b).
Apply this with ξ = 3 2 ∆δ 4H-1 in order to get

P |Y N -3 1/4 ∆ 1/4 δ H-1/4 | ≥ c ∆ -3/4 δ -3(4H-1)/4 ∆ 1/2 δ 4H-1/2 u ≤ A 1 + A 2 , (32)
with

A 1 = P |X N -3∆δ 4H-1 | ≥ c ∆ 1/2 δ 4H-1/2 u, X N ≥ 3 2 ∆δ 4H-1 A 2 = P X N ≤ 3 2 ∆δ 4H-1 .
Furthermore, a straightforward application of [START_REF] Yamato | Stochastic differential equations and nilpotent Lie algebras[END_REF] gives

A 1 ≤ c 1 e -c 2 u 1/2 , and A 2 ≤ c 1 e -c 2 (∆/δ) 1/4 .
Plugging these inequalities into (32), we end up with the following concentration inequality for Y N .

P |Y N -3 1/4 ∆ 1/4 δ H-1/4 | ≥ c ∆ -1/4 δ H+1/4 u ≤ c 1 e -c 2 u 1/2 + c 1 e -c 2 (∆/δ) 1/4 . (33)
We shall thus retain the fact that Y N i a random quantity of order 3 1/4 ∆ 1/4 δ H-1/4 , with fluctuations of order ∆ -1/4 δ H+1/4 :

Y N -3 1/4 ∆ 1/4 δ H-1/4 ≈ δ H+1/4 ∆ -1/4 . ( 34
)
Step 4: Use of the interpolation inequality. Start again from equation [START_REF] Nualart | Malliavin Calculus and Related Topics[END_REF]. One would like to have an approximation of the L 1 norm of z appearing on the left hand side of this inequality, that is one would like to replace Y N by ∆. To this purpose, replace first Y N by its approximation ∆ 1/4 δ H-1/4 from the last step. This yields an inequality of the form:

∆ 1/4 δ H-1/4 1/∆ N =1 |z N ∆ | ≤ z ∞ 1/∆ N =1 Y N -3 1/4 ∆ 1/4 δ H-1/4 + δ γ-1/4 ∆ -3/4 y γ + z γ B γ ∆ γ + ζ ∞ B 2 2γ δ γ + y ♯ 3γ δ 2γ .
Rescale this inequality in order to get ∆ multiplying on the left hand side, which gives:

∆ 1/∆ N =1 |z N ∆ | ≤ z ∞ R δ,∆ + δ -(H-γ) y γ + z γ B γ ∆ γ + ζ ∞ B 2 2γ δ γ + y ♯ 3γ δ 2γ , ( 35
)
where

R δ,∆ := δ -(H-1/4) ∆ 3/4 1/∆ N =1 |Y N -3 1/4 ∆ 1/4 δ H-1/4 |. ( 36
)
Furthermore, it is well known that |∆

1/∆ N =1 |z N ∆ | -z L 1 | ≤ z γ ∆ γ , so that we can recast (35) into z L 1 ≤ z γ ∆ γ + z ∞ R δ,∆ + δ -(H-γ) y γ + z γ B γ ∆ γ + ζ ∞ B 2 2γ δ γ + y ♯ 3γ δ 2γ . ( 37
)
Recall that we take 0 < α < γ < H and set ν H := 1/(γα). According to [START_REF] Malliavin | Stochastic calculus of variation and hypoelliptic operators[END_REF] we have

z α,∞ η z γ,∞ + η -ν H z L 1 ,
for any (small enough) constant η, and plugging (37) into this last relation we obtain

z α,∞ η -ν H ∆ γ z γ,∞ + η -ν H z ∞ R δ,∆ + η z γ,∞ + η -ν H y γ,∞ δ -(H-γ) + z γ,∞ B γ δ -(H-γ) ∆ γ + ζ ∞ B 2 2γ δ 2γ-H + y ♯ 3γ δ 3γ-H .
Defining ǫ as ǫ = Hγ, we get

z α,∞ η -ν H ∆ γ z γ,∞ + η -ν H z ∞ R δ,∆ + η z γ,∞ + η -ν H y γ,∞ δ -ǫ + z γ,∞ B γ δ -ǫ ∆ γ + ζ ∞ B 2 2γ δ γ-ǫ + y ♯ 3γ δ 2γ-ǫ . ( 38
)
Step 4: Tuning the parameters. Recall that we have chosen 0 ≪ δ ≪ ∆ ≪ 1. We express this fact in terms of powers of ε, by taking δ = ε µ and ∆ = ε λ , with µ > λ > 0. We also choose η of the form η = ε τ /ν H . We shall now see how to choose λ, µ, τ conveniently: write (38) as -ǫ) . (39) In order to be able to bound z when y is assumed to satisfy y γ,∞ ≤ ε, the coefficients in the right hand side of (39) should fulfill the following conditions:

z α,∞ ε -τ +λγ z γ,∞ + ε -τ z ∞ R δ,∆ + ε τ /ν H z γ,∞ + y γ,∞ ε -τ -ǫµ + z γ,∞ B γ ε -τ -ǫµ+λγ + ζ ∞ B 2 2γ ε -τ +µ(γ-ǫ) + y ♯ 3γ ε -τ +µ(2γ
• The coefficient in front of y γ,∞ should be smaller than ε -1 .

• The other coefficients should be ≪ 1. Looking at the exponents in (39), assuming that ǫ is arbitrarily small and letting for the moment R δ,∆ apart, this imposes the following relations: λγ > τ, and 0 < τ < 1.

(

) 40
Let us go back now to the evaluation of R δ,∆ , given by expression (36), with the order of magnitude of |Y N -∆ 1/4 δ H-1/4 | given by (34). Therefore

R δ,∆ ≈ δ -(H-1/4) ∆ 3/4 ∆ -1 δ H+1/8 ∆ -1/4 = δ 1/2 ∆ -1/2 .
Expressing this in terms of powers of ε, we end up with

η -ν H R δ,∆ ≈ ε κ , with κ = µ -λ 2 -τ.
If we wish this remainder term to be small, this adds the condition

µ -λ > 2τ, (41)
which can be fulfilled easily. From now on, we shall assume that both (40) and (41) are satisfied.

Step 5: Conclusion. Recall that we wish to study the probability P( y γ,∞ < ε, and z α,∞ > ε q ). This quantity is obviously bounded by B 1 + B 2 , where

B 1 = P y γ,∞ < ε, z α,∞ > ε q , |R δ,∆ | ≤ c ∆ -1/4 δ H+1/4 (∆/δ) ε B 2 = P |R δ,∆ | ≥ c ∆ -1/4 δ H+1/4 (∆/δ) ε ,
where ε is an arbitrary small positive constant. Furthermore, inequalities [START_REF] Zähle | Integration with respect to fractal functions and stochastic calculus I[END_REF] and (36) yield, for any p ≥ 1,

B 2 ≤ c 1 e -c 2 (∆/δ) ε/2 + c 1 e -c 2 (∆/δ) 1/4 ≤ c p,λ,µ,ε ε p ,
where we have used the fact that δ/∆ = ε µ-λ .

We can now bound B 1 : notice that according to (39), if we are working on

|R δ,∆ | ≤ c ∆ -1/4 δ H+1/4 (∆/δ) ε ∩ y γ,∞ < ε ,
then there exists a ρ > 0 such that

z α,∞ ε ρ 1 + N 2 [z; Q B γ (R m )] + B 2 γ + B 2 2 2γ . Moreover, recall that N [z; Q B γ (R m )
] is assumed to be an L r random variable for all r ≥ 1, while B γ and B 2 2γ are also elements of L r , since they can be bounded by a finite chaos random variable. Thus Tchebychev inequality can be applied here, which entails

B 1 1 + E N 2l [z; Q B γ (R m )] + B 2l γ + B 2 2l 2γ ε l(ρ-q) ,
for an arbitrary l ≥ 1. It is now sufficient to choose q < ρ and l large enough so that l(ρq) = p to conclude the proof, by putting together our bounds on B 1 and B 2 .

Malliavin calculus for solutions to fractional SDEs

This section is the core of our paper, where we derive smoothness of density for the solution to [START_REF] Lyons | System control and rough paths[END_REF]. We first recall some classical notions on representations of solutions to SDEs, and then move to Malliavin calculus considerations. 4.1. Representation of solutions to SDEs. The first representations results for solutions to SDEs in terms of the driving vector fields can be traced back to the seminal work of Chen [START_REF] Chen | Integration of paths, Geometric invariants and a Generalized Baker-Hausdorff formula[END_REF]. They have then been deeply analyzed in [START_REF] Hu | Calculs formels sur les EDS de Stratonovitch[END_REF][START_REF] Strichartz | The Campbell-Baker-Hausdorff-Dynkin formula and solutions of differential equations Jour[END_REF], and also lye at the basis of the rough path theory [START_REF] Lyons | System control and rough paths[END_REF]. We have chosen here to present these formulas according to [START_REF] Baudoin | An Introduction to the Geometry of Stochastic Flows[END_REF], which is a recent and didactically useful account on the topic.

Recall that we are considering a d-dimensional fBm B with 1/3 < H < 1/2. According to Section 2.3, this allows to construct some increments B k out of B which can be seen as limits of Riemann iterated integrals over the simplex S k ([s, t]), as recalled at Proposition 2.10. Furthermore, one can solve equation [START_REF] Lyons | System control and rough paths[END_REF] under the conditions of Theorem 2.7.

Let us introduce some additional notation: let V be the space of smooth bounded vector fields over R m . If V ∈ V, the vector field exp(V ) ∈ V is defined by the relation [exp(V )](ξ) = Ψ 1 (ξ), where {Ψ t (ξ); t ≥ 0} is solution to the ordinary differential equation

∂ t Ψ t (ξ) = V (Ψ t (ξ)) , Ψ 0 (ξ) = ξ. (42)
The aim of Chen-Strichartz formula is to express the solution y t to equation [START_REF] Lyons | System control and rough paths[END_REF], for an arbitrary t ∈ [0, T ], as y t = [exp(Z t )](a) for a certain Z t ∈ V.

To this purpose, let us give some more classical notations on vector fields: if V, W ∈ V, then the vector field [V, W ] ∈ V (called Lie bracket of V and W ) is defined by

[V, W ] i = V l ∂ x l W i -W l ∂ x l V i .
Notice that this notion is usually introduced though the interpretation of V as a set of first order differential operators. A Lie bracket of order k can also be defined inductively for k ≥ 2 by setting

[U 1 • • • U k ] = [[U 1 • • • U k-1 ] , U k ] ,
for U 1 . . . U k ∈ V. With this notation in hand, our main assumption on the vector fields V 1 , . . . , V d governing equation ( 22) is the following:

Hypothesis 4.1. The vector fields V 1 , . . . , V d are n-nilpotent for some given positive integer n. Namely, for any (i 1 , . . . , i n ) ∈ {1, . . . , d} n , we have

[V i 1 • • • V in ] = 0.
We are now ready to state our formulation of Strichartz' identity, for which we need two last notations: for k ≥ 1, we call S k the set of permutations of {1, . . . , k}. Moreover, for σ ∈ S k , write e(σ) for the quantity Card({j ∈ {1, . . . , k -1}; σ(j) > σ(j + 1)}). Then the following formula is proven e.g. in [START_REF] Baudoin | An Introduction to the Geometry of Stochastic Flows[END_REF][START_REF] Hu | Calculs formels sur les EDS de Stratonovitch[END_REF][START_REF] Strichartz | The Campbell-Baker-Hausdorff-Dynkin formula and solutions of differential equations Jour[END_REF]: Proposition 4.2. Under the hypothesis of Theorem 2.7, let y be the solution to equation [START_REF] Lyons | System control and rough paths[END_REF]. Assume Hypothesis 4.1 holds true, and consider t ∈ [0, T ]. Then y t = [exp(Z t )](a), where Z t can be expressed as follows:

Z t = n-1 k=1 d i 1 ,...,i k =1 V i 1 ,...,i k ψ i 1 ,...,i k t , with ψ i 1 ,...,i k t = σ∈S k (-1) e(σ) k 2 k-1 e(σ) B k,i τ (1) ,...,i τ (k) 0t
, where we have set τ = σ -1 and V

i 1 ,...,i k = [V i 1 • • • V i k ] in the formula above.
As a warmup to the computations below, we prove now that one can extend our inequality [START_REF] Hairer | On Malliavin's proof of Hörmander's theorem[END_REF] thanks to Strichartz representation, covering the case of unbounded vector fields with bounded derivatives: Proposition 4.3. Suppose Hypothesis 4.1 holds true, and that the smooth vector fields V i , i = 1, 2, • • • , d have bounded derivative. Assume moreover that all the Lie brackets of order greater or equal to 2 are bounded vector fields. Then the solution y of equation ( 22) admits moments of any order. Namely, for any m > 1 and any T ∈ (0, ∞),

E sup 0≤t≤T |y t | m < ∞ . ( 43
)
Proof. One can restate Proposition 4.2 as follows: for any t ≤ T , the random variable y t can be expressed as y t = φ 1 , where φ t s := φ s : R d → R d satisfies (for t fixed)

∂ s φ s = n-1 k=1 d i 1 ,...,i k =1 ψ i 1 ,...,i k t V i 1 ,...,i k (φ s ) , 0 ≤ s ≤ 1 , φ 0 = a .
Let us separate the first order integrals in this equation, in order to get

∂ s φ s = d i=1 V i (φ s )B i t + n-1 k=2 d i 1 ,...,i k =1 ψ i 1 ,...,i k t V i 1 ,...,i k (φ s ) , φ 0 = a . (44)
Since V i , i = 1, 2, • • • , d, have bounded derivatives and since all the Lie brackets of order greater or equal to 2 are bounded we see that

|∂ s φ s | ≤ d i=1 |V i (φ s )| |B i t | + n-1 k=2 d i 1 ,...,i k =1 |ψ i 1 ,...,i k t ||V i 1 ,...,i k (φ s )| ≤ c 1 |φ s | d i=1 sup 0≤t≤T |B i t | + c 2 n-1 k=1 d i 1 ,...,i k =1 sup 0≤t≤T |ψ i 1 ,...,i k t | .
Thus by Gronwall's lemma, we have

|φ s | ≤ c 2 n-1 k=1 d i 1 ,...,i k =1 sup 0≤t≤T |ψ i 1 ,...,i k t | exp c 1 d i=1 sup 0≤t≤T |B i t | .
This inequality holds true for all 0 ≤ s ≤ 1. Thus More precisely, for any q > 1 and T ∈ (0, ∞),

sup 0≤t≤T |y t | ≤ c 2 n-1 k=1 d i 1 ,...,i k =1 sup 0≤t≤T |ψ i 1 ,.
E sup 0≤u≤t≤T |D u y t | q < ∞ . (45)
Proof. Go back to our representation (44), which can easily be differentiated in the Malliavin calculus sense in order to obtain

∂ s D u φ s = d i=1 ∇V i (φ s )B i t D u φ s + n-1 k=2 d i 1 ,...,i k =1 ψ i 1 ,...,i k t ∇V i 1 ,...,i k (φ s ) D u φ s + d i=1 V i (φ s )1 [i] [0,t) (u) + n-1 k=2 d i 1 ,...,i k =1 D u ψ i 1 ,...,i k t V i 1 ,...,i k (φ s ) ,
where we have set ∇V i 1 ,...,i k for the (matrix valued) gradient of V i 1 ,...,i k , and where we recall that the notation 1

[i]
[0,t) has been introduced at Section 2.3.2.

Since we assume that all the Lie brackets of order greater or equal to 2 of the vector fields V i are constant vector fields, it is easily checked that

∂ s D u φ s = d i=1 ∇V i (φ s )B i t D u φ s + d i=1 V i (φ s )1 [i] [0,t) (u) + n-1 k=2 d i 1 ,...,i k =1 D u ψ i 1 ,...,i k t V i 1 ,...,i k (φ s ) .
(46) Therefore there exist two positive constants c 1 , c 2 such that

|∂ s D u φ s | ≤ c 1 d i=1 |B i t ||D u φ s | +c 2 d i=1 [1 + |φ s |] 1 [i] [0,t) (u) + n-1 k=2 d i 1 ,...,i k =1 D u ψ i 1 ,...,i k t |V i 1 ,...,i k (φ s )| .
By Gronwall's lemma we obtain sup 0≤s≤1,0≤u≤t≤T

|D u φ s | ≤ c 2 exp c 1 d i=1 sup 0≤t≤T |B i t | × d i=1 1 + sup 0≤s≤1 |φ s | × 1 + sup 0≤t≤T n-1 k=2 d i 1 ,...,i k =1 sup 0≤s≤1,0≤u≤t≤T D u ψ i 1 ,...,i k t |V i 1 ,...,i k (φ s )| . Thus sup 0≤u≤t≤T |D u y t | ≤ c 2 exp c 1 d i=1 sup 0≤t≤T |B i t | × d i=1 1 + sup 0≤t≤T |y t | 1 + n-1 k=2 d i 1 ,...,i k =1 sup 0≤s≤1,0≤u≤t≤T D u ψ i 1 ,...,i k t ,
which ends the proof easily by boundedness of moments for y t , D u ψ i 1 ,...,i k t and B i t .

Example 4.5. A classical example of nilpotent vector fields in R 3 is due to Yamato [START_REF] Yamato | Stochastic differential equations and nilpotent Lie algebras[END_REF].

Let us check that this example fullfils our standing assumptions. Indeed, the example provided in [START_REF] Yamato | Stochastic differential equations and nilpotent Lie algebras[END_REF] is the following:

A 1 = 0 , A 2 = ∂ ∂x 1 + 2x 2 ∂ ∂x 3 , and
A 3 = ∂ ∂x 2 -2x 1 ∂ ∂x 3 . Then [A 2 , A 3 ] = -4 ∂ ∂x 3 , [[A 2 , A 3 ], A 2 ] = [[A 2 , A 3 ], A 3 ] = 0 .
It is thus readily checked that the conditions of Theorem 4.4 are met for these vector fields. Moreover, in this particular case the solution to equation ( 22) is explicit and we have y

1 1 = y 1 + B 2 t , y 2 t = y 2 + B 3 t , y 3 t = y 3 + 2 B 2,32 0t -B 2,23 0t
, if the solution starts from the initial condition (y 1 , y 2 , y 3 ). Interestingly enough, though the solution is explicit here, the smoothness of the density of y t is not immediate and we recover here the results of [START_REF] Driscoll | Smoothness of Density for the Area Process of Fractional Brownian Motion[END_REF].

Stochastic flows.

The probabilistic proof of the smoothness of density for diffusion processes originally given by Malliavin [START_REF] Malliavin | Stochastic calculus of variation and hypoelliptic operators[END_REF] heavily relies on stochastic flows methods and their relationship with stochastic derivatives. We now establish those relations for SDEs driven by a fractional Brownian motion.

To this aim, denote by y s,a the solution to equation ( 22) starting from the initial condition y s = a at time s:

dy s,a t = d i=1 V i (y s,a t ) dB i t , t ∈ [s, T ] , y s,a s = a . (47)
The above equation gives rise to a family of smooth nonlinear mappings Φ s,t : R m → R m , 0 ≤ s ≤ t ≤ T , determined by Φ s,t (a) := y s,a t , and the family {Φ s,t ; ≤ s ≤ t ≤ T } has the following flow property (we refer e.g. to [START_REF] Friz | Multidimensional stochastic processes as rough paths: theory and applications[END_REF] for the properties of flows driven by rough paths quoted below):

Φ s,t = Φ u,t • Φ s,u , ∀ 0 ≤ s ≤ u ≤ t ≤ T .
Let J s,t denote the gradient of the nonlinear mapping Φ s,t with respect to the initial condition. Then the family {J s,t ; ≤ s ≤ t ≤ T } also satisfies the relation J s,t = J u,t J s,u for 0 ≤ s ≤ u ≤ t ≤ T . In addition, the map J s,t is invertible, and we have J s,t = J 0,t J -1 0,s . The equation followed by J 0,t is obtained by differentiating formally equation (47) with respect to the initial value a, which yields dJ 0,t = d i=1 ∇V i (y t )J 0,t dB i t , J 0,0 = I .

By applying the rules of differential calculus for rough paths, we also get that J -1 0,t is solution to the following equation:

dJ -1 0,t = - d i=1 ∇V i (y t )J -1 0,t dB i t , J -1 0,0 = I . (48)
We have thus ended up with two linear equations for the derivatives of the flow. In our nilpotent case, we are able thus able to bound these derivatives along the same lines as for Theorem 4.4:

Theorem 4.6. Let the vector fields V i , 1 = 1, 2, . . . , d be smooth with all derivatives bounded, satisfying Hypothesis 4.1. Assume that all the Lie brackets of order greater or equal to 2 are constant. Then the Jacobian J 0,t and its inverse J -1 0,t have moments of any order: for any q ≥ 1 and T ∈ (0, ∞),

E sup 0≤t≤T |J 0,t | q < ∞, and E sup 0≤t≤T J -1 0,t q < ∞ . ( 49
)
Proof. As mentioned in the proof of Proposition 4.3, one can write Φ 0,t (a) = exp(Z t )(a) = φ 1 (a), where φ 1 (a) satisfies (44). Thus if we introduce Js = ∇φ s , then J 0,t = J1 where Js satisfies

∂ s Js = d i=1 B i t ∇V i (φ s ) Js + n-1 k=2 d i 1 ,...,i k =1 ψ i 1 ,...,i k t ∇V i 1 ,...,i k (φ s ) Js , J0 = I .
The first part of (49) is thus proved following the steps of Proposition 4.3.

As far as the second part of (49) is concerned, observe that J -1 0,t = J1 , where Js is inverse of Js . It is clear that Js satisfies

∂ s Js = - d i=1 ∇V i (φ s ) Js B i t - n-1 k=2 d i 1 ,...,i k =1 ψ i 1 ,...,i k t ∇V i 1 ,...,i k (φ s ) Js , J0 = I .
Once again, the methodology of Proposition 4.3 easily yields our claim.

Corollary 4.7. Under the same assumptions as in Theorem 4.6, the following holds true: (i) For any 0 < γ < H and q ≥ 1 we have

E Dy t q γ,∞ + E J 0,• q γ,∞ + E J -1 0,• q γ,∞ < c T,q , (50)
for a finite constant c T,q .

(ii) As a consequence, inequality (50) also holds true when the • γ,∞ norms are replaced by norms in H, where H has been defined at Section 2.3.1.

(iii) For any smooth bounded vector field U on R m and t ∈ [0, T ], set Z U t := J -1 0,t U(y t ), η . Then Z U is a controlled process, and satisfies the inequality E[N q [Z U ; Q γ (R m )]] ≤ c T,q for any q ≥ 1 and a finite constant c T,q .

Proof. Going back to equation (46), it is readily checked that all the terms u → D u ψ i 1 ,...,i k t are C γ 1 -Hölder continuous on [0, t] for any γ < H, since the elements ψ i 1 ,...,i k t are nice multiple integrals with respect to B. Moreover, we have

E Dψ i 1 ,...,i k t q γ,∞ < ∞,
for any m ≥ 1. This easily yields E[ Dy i 1 ,...,i k t q γ,∞ ] < ∞ by a standard application of Gronwall's lemma, as in the proof of Theorem 4.4.

Our second assertion stems from the fact that one can choose 1/2 -H < γ < H, since H > 1/3. For such γ we have C γ 1 ⊂ H, which ends the proof. Finally, our claim (iii) derives from the fact that the equation governing Z U is of the following form:

Z U t = η, U(a) + d j=1 t 0 Z [U,V j ] s dB j s . (51)
The process Z U can thus be decomposed as a controlled process as in Section 2.2, and since we already have estimates for J -1 0,t and y t , the bound on E[N q [Z U ; Q γ (R m )]] follows easily. 4.4. Proof of Theorem 1.1. As mentioned in the introduction, once we have shown the integrability of the Malliavin derivative and proved a Norris type lemma, the proof of our main theorem goes along classical lines. We have chosen to follow here the exposition of [START_REF] Hairer | On Malliavin's proof of Hörmander's theorem[END_REF], to which we refer for further details.

Step 1: Reduction to a lower bound on Hölder norms. Recall that the process Z U has been defined for any smooth vector field U in Corollary 4.7. For any p ≥ 1, our first goal is to reduce our problem to the existence of a constant c p such that

P Z V k α,∞ ≤ ε ≤ c p ε p , (52)
for 1 ≤ k ≤ d, a given α ∈ (1/3, 1/2), all ε ∈ (0, 1) and where we observe that all the norms below are understood as norms on [0, T ]. Indeed, according to [START_REF] Hairer | On Malliavin's proof of Hörmander's theorem[END_REF]Relation (4.9)] the smoothness of density can be obtained from the estimate P Z V k H ≤ ε ≤ c p ε p , where we recall that H has been defined at Section 2.3.1. Furthermore, we have

P Z V k H ≤ ε ≤ P Z V k L 2 ≤ ε ≤ P Z V k L 1 ≤ ε .
It is thus sufficient for our purposes to check

P Z V k L 1 ≤ ε ≤ c p ε p . (53)
In order to go from (53) to (52), let us use our interpolation bound [START_REF] Malliavin | Stochastic calculus of variation and hypoelliptic operators[END_REF] in the following form: for any 0 < η < 1 and 0 < α < ρ < H we have

b L 1 ≥ η 1/(ρ-α) ( b α,∞ -C α,ρ η b ρ,∞ ) .
Take now δ ∈ (0, 1) to be fixed later on and η 1(ρ-α) = ε 1-δ , that is η = ε (ρ-α) (1-δ) . Then

P Z V k L 1 ≤ ε ≤ P Z V k α,∞ ≤ 2ε δ + R, where R = P Z V k ρ,∞ ≤ 1 4c ε ν ,
(54) with ν = ρ-α-(1-(ρ-α))δ. Choose now δ small enough, so that ν > 0. Since Z V k ρ,∞ admits moments of any order according to Corollary 4.7, it is easily checked that R can be made smaller than any quantity of the form c q ε q . It is thus sufficient to prove (52) in order to get the smoothness of density for y t .

Step 2: An iterative procedure. For l ≥ 1 and x ∈ R m , let V l (x) be the vector space generated by the Lie brackets of order l of our vector fields V 1 , . . . , V d at point x:

V l (x) = Span [V k 1 • • • V k j ](x); j ≤ l, 1 ≤ k 1 , . . . , k j ≤ d .
We assume that the vector fields are ℓ-hypoelliptic for a given ℓ > 0, which can be read as V ℓ (x) = R m for any x ∈ R m . In order to start our induction procedure, we set α 1 = α, so that we have to prove P(

Z V k α 1 ,∞ ≤ ε) ≤ c p ε p .
Recall that Z V k satisfies the relation

Z V k t = η, V k (a) + d j=1 t 0 Z [V k ,V j ] s dB j s .
Thus Proposition 3.2 asserts that for any 1/3 < α 2 < α 1 < H there exists q 2 > 0 such that

P Z V k α 1 ,∞ ≤ ε ∩ ∪ d j=1 Z [V k 1 ,V k 2 ] α 2 ,∞ > ε q 2 ≤ c p ε p .
Relation (52) is thus implied by

P Z V k α 1 ,∞ ≤ ε ∩ ∩ d j=1 Z [V k 1 ,V k 2 ] α 2 ,∞ ≤ ε q 2 ≤ c p ε p .
Iterating this procedure we end up with the following claim to prove: B ℓ (ε) ≤ c p ε p for all ε ∈ (0, 1), with

B ℓ (ε) = P Z V k α 1 ,∞ ≤ ε, Z [V k 1 ,V k 2 ] α 2 ,∞ ≤ ε q 2 , . . . , Z [V k 1 •••V k ℓ ]
α ℓ ,∞ ≤ ε q ℓ , where the intersection above extends to all possible combinations 1 ≤ k 1 , . . . , k ℓ ≤ d, and where 1/3 < α ℓ < • • • < α 1 < H.

Going back now to the very definition of Z U as Z U t = J -1 0,t U(y t ), η , it is readily checked that B ℓ (ε) ≤ P ( η, V k 1 (a) ≤ ε, . . . , η,

[V k 1 • • • V k ℓ ](a) ≤ ε q ℓ ) .
Owing to the fact that V ℓ (a) = R m , we thus have B ℓ (ε) = 0 for ε small enough, which ends the proof.

  Malliavin derivative. This subsection is devoted to enhance our Proposition 4.3, and prove that the Malliavin derivative of y t has also bounded moments of any order in our particular nilpotent situation. Notice once again that the boundedness of moments of the Malliavin derivative is still an open problem for rough differential equations in the general case. We refer to Section 2.3.2 for notations on Malliavin calculus. Let the vector fields V i , 1 = 1, 2, • • • , d be smooth with all derivatives bounded, satisfying Hypothesis 4.1. Assume that all the Lie brackets of order greater or equal to 2 are constants. Then the Malliavin derivative y t has moments of any order.

				d
	t	..,i k	| exp c 1	i=1	sup 0≤t≤T	|B i t | ,
	which implies (43).
	4.2. Theorem 4.4.

Acknowledgement: We would like to thank Ivan Nourdin for pointing us out the use of Hermite polynomials in the computation of 4 th order variations (see proof of Proposition 3.2 Step 2).

ANR grant ECRU.


Rahmouni Atef
email: atef.rahmouni@fsb.rnu.tn

Uncertainty Inequalities on Laguerre hypergroup

Keywords: 2000 Mathematics Subject Classification: 42B10, 42B30, 33C45 Uncertainty principle, Heisenberg-Pauli-Weyl Inequality, Laguerre hypergroup, Laguerre Fourier transform. R n

In this paper, we give analogues of local uncertainty inequality on R n for stratified Laguerre hypergroup, connected with the spectral analysis of a given homogeneous sublaplacian L, also indicate how local uncertainty inequalities imply global uncertainty inequalities. It would be interesting to note that we deduce the local uncertainty inequalities for the radial functions on the Heisenberg group. Finally, we extend Heisenberg-Pauli-Weyl uncertainty inequality by ultracontractive properties of the semigroups generated by the differential operator and on the estimate on the heat kernel.

Introduction

The serious question of certainty in science was high-lighted by Heisenberg, in 1927, via his uncertainty principle (cf. [START_REF] Heisenberg | Über den anschaulichen Inhalt der quantentheoretischen Kinematic und Mechanik[END_REF]). He demonstrated, for instance, the impossibility of specifying simultaneously the position and the speed (or the momentum) of an electron within an atom. In 1933, according to H. Hardy (cf. [START_REF] Hardy | A theorem concerning Fourier transforms[END_REF]).

A pair of transforms cannot both be very small. [START_REF] Atef | Pseudo-differential operators associated with Laguerre hypergroups[END_REF] This aspect of the uncertainty principle was already expounded by Norbert Wiener in a lecture in Göttingen in 1925. Unfortunately, no written record of this lecture seems to have survived, apart from the nontechnical account in Wiener's autobiography (cf. [START_REF] Wiener | I am a Mathematicia[END_REF], pp. 105-107), so one can only guess at what precise versions of (1) it might have contained. Whatever influence this lecture might have had on the physicists in the audience, however, the uncertainty principle did not really sink into the minds of signal analysts until Gabor's fundamental work (cf. [START_REF] Gabor | Theory of communication[END_REF]) in 1946. Since then, it has become firmly embedded in the common culture.

On the mathematical side, there were sporadic developments relating to the uncertainty principle in the fifty years after the initial work in the 1920's, followed by a steady stream of results in the last tow decades, we refer to the survey article (cf. [START_REF] Folland | The uncertainty principle: A mathematical survey[END_REF]) for an overview of the history, and the book (cf. [START_REF] Havin | The Uncertainty Principle in Harmonic Analysis[END_REF]) of Havin and Jöricke for other forms of the uncertainty principle. These principles state that a function f and its Fourier transform f cannot be simultaneously sharply localized. The most common quantitative formulation of the uncertainty principle is the Heisenberg-Pauli-Weyl inequality. It says that,

if f ∈ L 2 (R n ) R n x 2 j | f (x)| 2 dx
where ∆ denotes the Laplacian on R n . This form of the Heisenberg-Pauli-Weyl inequality is better suited for extensions to other contexts, with the Laplacian replaced by a positive self-adjoint operator, and |x| by a distance function. The interpretation of uncertainty inequalities as spectral properties of differential operators is widely present in the literature (cf. [START_REF] Fefferman | The uncertainty principle and sharp Gårding inequality[END_REF], [START_REF] Fefferman | The uncertainty principle[END_REF], [START_REF] Thangavelu | Some uncertainty inequalities[END_REF]).

In this paper we are interested in the Laguerre hypergroup K = [0, +∞) × R which is the fundamental manifold of the radial function space for the Heisenberg group ( [START_REF] Bloom | Harmonic Analysis of Probability Measures on Hypergroups[END_REF], [START_REF] Jewett | Spaces with an abstract convolution of measures[END_REF]). Let us recall that (K, * α ) is a commutative hypergroup (cf. [START_REF] Nessibi | Inversion of the Radon Transform on the Laguerre Hypergroup by using Generalized Wavelets[END_REF], [START_REF] Trimèche | Generalized wavelets and hypergroups[END_REF] pp. 243-263 ), on which the involution and the Haar measure are respectively given by the homeomorphism (x,t) → (x,t) -= (x, -t) and the Radon positive measure dm α (x,t) = x 2α+1 πΓ(α+1) dxdt. The unity element of (K, * α ) is given by e = (0, 0), i.e. δ (x,t) * α δ (0,0) = δ (0,0) * α δ (x,t) = δ (x,t) for all (x,t) ∈ K. The convolution product * α is defined for two bounded Radon measures µ and ν on K as follows

µ * α ν, f = K×K T (α) (x,t) f (y, s)dµ(x,t)dν(y, s),
where α is a fixed nonnegative real number and {T (α) (x,t) } (x,t)∈K are the translation operators on the Laguerre hypergroup (cf. [START_REF] Bloom | Harmonic Analysis of Probability Measures on Hypergroups[END_REF], [START_REF] Nessibi | Inversion of the Radon Transform on the Laguerre Hypergroup by using Generalized Wavelets[END_REF], [START_REF] Stempak | An algebra associated with the generalized sublaplacian[END_REF], [START_REF] Trimèche | Generalized wavelets and hypergroups[END_REF]), given by

T (α) (x,t) f (y, s) = δ (x,t) * α δ (y,s) , f =    α π 1 0 2π 0 f ((ξ , η) r,θ )r(1 -r 2 ) α-1 dθ dr if α > 0, 1 2π 2π 0 f ((ξ , η) 1,θ )dθ if α = 0, where (ξ , η) r,θ = ( x 2 + y 2 + 2xyr cos θ ,t + s + xyr sin θ ).
Note that for the particular case µ = f m α and ν = gm α , f and g being two suitable functions on K, one has µ * α ν = ( f * α g)m α , where f * α g is the convolution product of f and g given by

f * α g(x,t) = K×K T (α) (-y,s) f (x,t)g(y, s)dm α (x,t).
The dual (cf. p. 46, [START_REF] Bloom | Harmonic Analysis of Probability Measures on Hypergroups[END_REF]) of Laguerre hypergroup, i.e. the space of all bounded continuous and multiplicative functions χ : K → C such that such that χ = χ, where χ(x,t) = χ(x, -t), (x,t) ∈ K, is given ( [START_REF] Nessibi | A Wiener-Tauberian and a Pompeiu type theorems on the Laguerre hypergroup[END_REF]) by

K = {ϕ λ ,m ; (λ , m) ∈ R * × N} ∪ {ϕ ρ ; ρ ≥ 0}, where ϕ λ ,m (x,t) = e -iλt L (α) m (|λ |x 2 ) and ϕ ρ = j α (ρx); L (α) m (x) = e -x 2 2 L (α) m (x)/L (α) m (0) and j α (x) = 2 α Γ(α + 1) J α (x)
x α , J α being the Bessel function of first kind and order α, and L (α) m being the Laguerre polynomial of degree m and order α (cf. [START_REF] Lebedev | Special Functions and Thier Applications[END_REF], [START_REF] Szego | Orthogonal Ploynomials[END_REF]). K which can be seen as a deformation of the hypergroup of radial functions on the Heisenberg group.

The dual of the Laguerre hypergroup K can be topologically identified with the so-called Heisenberg fan (cf. [START_REF] Faraut | Deux cours d'Analyse Harmonique[END_REF]), i.e., the subset embedded in R 2 given by m∈N

{(λ , µ) ∈ R 2 : µ = |λ |(2m + α + 1), λ = 0} {(0, µ) ∈ R 2 : µ ≥ 0} .
Moreover, the subset {(0, µ) ∈ R 2 ; µ ≥ 0} has zero Plancherel measure; therefore it will usually be disregarded. Identifying K and (R * × N) ∪ [0, +∞[, the Fourier transform of a bounded Radon measure µ on the Laguerre hypergroup is then, by

F (µ)(λ , m) = K ϕ -λ ,m (x,t)dµ(x,t) and F (µ)(ρ) = K j α (ρ, x)dµ(x,t).
The Fourier Laguerre transform of a suitable function f : K → C is given by f = F ( f dm α ), so that

f (λ , m) = K f (x,t)ϕ -λ ,m (x,t)dm α (x,t) and f (ρ) = K f (x,t) j α (ρx)dm α (x,t).
The functional analysis and Fourier analysis on K and its dual K have been extensively studied in (cf. [START_REF] Atef | Pseudo-differential operators associated with Laguerre hypergroups[END_REF], [START_REF] Nessibi | Inversion of the Radon Transform on the Laguerre Hypergroup by using Generalized Wavelets[END_REF]), and hence it is well known that the Fourier Laguerre transform given above is a topological isomorphism from the Schwartz space on K onto S( K): the Schwartz space on K. Its inverse is the operator f ∨ given by

f ∨ (x,t) = R×N ϕ λ ,m (x,t) f dγ α ,
where dγ α is the Plancherel measure on K given by dγ α (λ , m) = L α m (0)δ m ⊗ |λ | α+1 dλ . We introduce the following notations (cf. [START_REF] Nessibi | Inversion of the Radon Transform on the Laguerre Hypergroup by using Generalized Wavelets[END_REF]) L p α (K) (resp. L p α (R × N)) where 1 ≤ p ≤ ∞ the p-th Lebesgue space on K (resp. on R×N) formed by the measurable functions f :

K → C (resp. Φ : R×N → C) such that f L p α (K) < +∞ (resp. f L p α (R×N) < +∞) where f L p α (K) =    ( K | f (x,t)| p dm α (x,t)) 1/p if p ∈ [1, +∞[, ess sup (x,t)∈K | f (x,t)| if p = +∞, and
Φ L p α (R×N) =    ( R×N |Φ(λ , m)| p dγ α (λ , m)) 1/p if p ∈ [1, +∞[, ess sup (λ ,m)∈R×N |Φ(λ , m)| if p = +∞,
we have the following Plancherel formula

f L 2 α (K) = f L 2 α ( K) , f ∈ L 1 α ( K) ∩ L 2 α ( K),
and we have

f L ∞ α ≤ f L 1 α . (4)
This paper is organized as follows.

In section 2 we set some notations and collect some basic facts about the Laguerre hypergroup. In section 3 of central interest is the following slight sharpening of a local uncertainty inequality. Heisenberg's inequality says that f is highly localized, then f can not be concentrated near a single point, but it does not preclude f from being concentrated in a small neighborhood of two or more widely separated points. In fact, the latter phenomenon cannot occur either, and it is the object of local uncertainty inequality to make this precise. The first such inequalities for the Fourier transform were obtained by Faris (cf. [START_REF] Faris | Inequalities and uncertainty inequalities[END_REF]), and they were subsequently sharpened and generalized by Price (cf. [START_REF] Price | Inequalities and local uncertainty principles[END_REF], [START_REF] Price | Sharp local uncertainty principles[END_REF]). Building on the ideas of Faris, Price, and Ricci (cf. [START_REF] Ciatti | A local uncertainty inequality for step-two nilpotent Lie groups[END_REF]) we show the inequalities of the uncertainty on the local Laguerre hypergroup, it is the subject of the following results.

(a) If 0 < β < 2α+4 2 , there is a constant K = K(α, β ) such that for every f ∈ L 2 α (K), and every measurable set E ⊂ K; 0 < γ α (E) < ∞, K | f (λ , m)| 2 χ E dγ α (λ , m) ≤ Kγ α (E) β 2α+4 |(x,t)| β f 2 2 . (b) If β > 2α+4 2 , there is a constant K = K(α, β ) such that for every f ∈ L 2 α (K), and every measurable set E ⊂ K; 0 < γ α (E) < ∞, K | f (λ , m)| 2 χ E dγ α ≤ Kγ α (E) f 2-2α+4 β 2 |(x,t)| β f 2α+4 β 2 .
(c) Cases (a) and (b) fail for β = 2α+4 2 . We also indicate in the proof of the Corollary 1 how local uncertainty inequalities imply global uncertainty inequalities. It would be interesting to note that we deduce the local uncertainty inequalities for the radial functions on the Heisenberg group.

Finally (Section 4), building on the ideas of (cf. [START_REF] Ciatti | Heisenberg-Pauli-Weyl uncertainty inequalities and polynomial volume growth[END_REF]) to establish an inequality analogous to [START_REF] De Bruijn | Uncertainty principles in Fourier analysis[END_REF], and variants of it, for the Laguerre hypergroup. Our purpose is to prove a general form of the Heisenberg-Pauli-Weyl inequality in Laguerre hypergroup. More precisely, using the spectral theory, ultracontractive properties of the semigroups generated by partial differential operator and the estimate on the heat kernel on Laguerre hypergroup, that is for all

f ∈ L 2 α (K), a, b ∈ R; a, b ≥ 1 and η ∈ R such that ηa = (1 -η)b, we have K |(x,t)| 2a | f (x,t)| 2 dm α (x,t) η 2 R×N |(λ , m)| b/2 | f (λ , m)| 2 dγ α (λ , m) 1-η 2 ≥ C f 2 .
Throughout this paper, C will always represent a positive constant, not necessarily the same in each occurrence.

Preliminaries

To describe the harmonic analysis in our setting we begin with introducing the operator

L = - ∂ 2 ∂ x 2 + 2α + 1 x ∂ ∂ x + x 2 ∂ 2 ∂t 2 ,
which is positive, symmetric in L 2 α (K), we endow the space K with homogeneous of degree one norm (with respect to the family dilations (δ ρ ) ρ>0 ),

N(x,t) = |(x,t)| = (x 4 + 4t 2 ) 1/4 , (x,t) ∈ K.
For α = n -1, n being a positive integer, the operator is the radial part of the sub-Laplacian on the Heisenberg group H n (cf. [START_REF] Stempak | An algebra associated with the generalized sublaplacian[END_REF]).

Also, we introduce the operator

Λ = Λ 2 1 -(2Λ 2 + 2 ∂ ∂ λ ) 2 defined on K, where Λ = 1 |λ | (m∆ + ∆ -+ (α + 1)∆ + ) and Λ = -1
2|λ | ((m + α + 1)∆ + + m∆ -). ∆± are given for a suitable function Φ by:

∆ + Φ(λ , m) = Φ(λ , m+1)-Φ(λ , m), ∆ -Φ(λ , m) = Φ(λ , m)- Φ(λ , m -1), if m ≥ 1 and ∆ -Φ(λ , 0) = Φ(λ , 0), and the quasinorm N (λ , m) = |(λ , m)| = 4|λ |(m + α + 1 2 ), (λ , m) ∈ K.
These operators satisfy some basic properties which can be found in (cf. [START_REF] Atef | Pseudo-differential operators associated with Laguerre hypergroups[END_REF], [START_REF] Nessibi | Inversion of the Radon Transform on the Laguerre Hypergroup by using Generalized Wavelets[END_REF], [START_REF] Nessibi | A Wiener-Tauberian and a Pompeiu type theorems on the Laguerre hypergroup[END_REF]) namely one has

Lϕ λ ,m = -|(λ , m)|ϕ λ ,m and Λϕ λ ,m = |(x,t)| 4 ϕ λ ,m , by the properties of L and the symmetry one can observe that L f (λ , m) = 4|λ |(m + α+1 2 ) f (λ , m) (cf. [29]
), and we define the following operator

L b/2 f (λ , m) = |(λ , m)| b/2 f (λ , m), (λ , m) ∈ K.
We will denote by δ ρ (x,t) = (ρx, ρ 2 t), the dilated of (x,t) ∈ K and by B r (x,t) the ball centered at (x,t) of radius r, i.e., the set B r (x,t) = {(y, s)

∈ K : |(x -y,t -s)| K < r}, and ω α = Γ( α+1 2 ) 2 √ πΓ( α 2
+1)Γ(α+1) its surface area of B r = B r (0, 0) (cf. [START_REF] Guliyev | Polar coordinates in Laguerre hypergroup[END_REF]). Denote by

f ρ (x,t) = ρ (2α+4) f (δ 1 ρ (x,t))
the dilated of the function f defined on K preserving the mean of f with respect to the measure dm α , in the sense that

f ρ L 1 α (K) = f L 1 α (K) , ∀ f ∈ L 1 α (K), ρ > 0.
Throughout this paper, let us fix the notation of the norm by -L p α (K) =p , ∀p ≥ 1.

The Local Uncertainty Principle

The purpose of this section we develop a family of inequalities in their sharpest forms, which constitute the principle of local uncertainty. It is the subject of the following theorem.

Theorem 1. (a) If 0 < β < 2α+4
2 , there is a constant K = K(α, β ) such that for every f ∈ L 2 α (K), and every measur-

able set E ⊂ K; 0 < γ α (E) < ∞, K | f (λ , m)| 2 χ E dγ α (λ , m) ≤ Kγ α (E) β 2α+4 |(x,t)| β f 2 2 , ( 5
)
where

K(α, β ) = α + 2 α + 2 -β ω α β 2 β 2α+4 . ( 6
) (b) If β > 2α+4 2 , there is a constant K = K(α, β ) such that for every f ∈ L 2 α (K), and every measurable set E ⊂ K; 0 < γ α (E) < ∞, K | f (λ , m)| 2 χ E dγ α ≤ Kγ α (E) f 2-2α+4 β 2 |(x,t)| β f 2α+4 β 2 , (7)
where Proof. Let B r denote the closed unit ball in K and B c r its complement. Denote by χ B r and χ B c r the characteristic functions.

K(α, β ) = πω α (β -(α + 2)) sin( (α+2)π β ) α + 2 β -(α + 2) -α+2 β -(α+2) .
Part(a), let f ∈ L 2 α (K). By Minkowski's inequality and ( 4), for all r > 0, we have

f χ E 2 ≤ ( f χ B r )χ E 2 + ( f χ B c r )χ E 2 ≤ (γ α (E)) 1/2 ( f χ B r ) ∞ + ( f χ B c r )χ E 2 ≤ (γ α (E)) 1/2 f χ B r 1 + ( f χ B c r ) 2 . ( 8
)
On the other hand, by Hölder's inequality and hypothesis β < 2α+4 2 , we have

f χ B r 1 ≤ |(x,t)| -β χ B r 2 |(x,t)| β f 2 ≤ Γ( α+1 2 ) 2 √ πΓ( α 2 +1)Γ(α+1) α + 2 -β 1/2 r α+2-β |(x,t)| β f 2 , ≤ ω α α + 2 -β 1/2 r α+2-β |(x,t)| β f 2 . ( 9
)
By Plancherel's theorem, we have

( f χ B c r ) 2 = f χ B c r 2 ≤ |(x,t)| -β χ B c r ∞ |(x,t)| β f 2 ≤ r -β |(x,t)| β f 2 . ( 10
)
Combining the relations (8), ( 9) and [START_REF] Folland | The uncertainty principle: A mathematical survey[END_REF], by choosing r 0 > 0 to satisfy

r 2α+4 0 = β 2 γ α (E)ω α ,
we minimize the quantity on the right side to obtain

K | f (λ , m)| 2 χ E dγ α 1/2 ≤ K(α, β ) γ α (E) β 2α+4 |(x,t)| β f 2 ,
where

K(α, β ) = α + 2 α + 2 -β ω α β 2 β 2α+4 .
Part (b), from the hypothesis β > 2α+4 2 , we deduce that the function

(x,t) → (1 + |(x,t)| 2β ) -1 belongs to L 1 α (K) ∩ L 2 α (K)
and by Hölder's inequality, we have

f 2 1 ≤ (1 + |(x,t)| 2β ) 1/2 f 2 2 (1 + |(x,t)| 2β ) -1/2 2 2 ≤ f 2 2 + |(x,t)| 2β f 2 2 (1 + |(x,t)| 2β ) -1/2 2 2 . (11)
However, by standard calculus and (cf. [START_REF] Gradshteyn | Table of Integrals. Series, and Products Seventh Edition[END_REF], page 322), we have

(1 + |(x,t)| 2β ) -1/2 2 2 = πω α β sin( (α+2)π β ) . ( 12
)
For ρ > 0, we put as above f ρ (x,t) = ρ -(2α+4) f ( x ρ , t ρ 2 ), then we have

f ρ 2 2 = ρ -(2α+4) f 2 2 and |(x,t)| β f ρ 2 2 = ρ 2β -(2α+4) |(x,t)| β f 2 2 .
Replacing f by f ρ in the relation [START_REF] Gabor | Theory of communication[END_REF], and by equation ( 12) we deduce

f 2 1 ≤ πω α β sin( (α+2)π β ) ρ -2(α+2) f 2 2 + ρ 2(β -(α+2)) |(x,t)| β f 2 2 .
In particular for

ρ 2β 0 = α + 2 β -(α + 2) f 2 2 |(x,t)| β f 2 2 , we get f 2 1 ≤ πω α (β -(α + 2))β sin( (α+2)π β ) α + 2 β -(α + 2) -α+2 β -(α+2) f 2-2α+4 β 2 |(x,t)| β f 2α+4 β 2 . ( 13
)
Now, according the relations (13), the function f ∈ L 1 α (K) we have

K | f (λ , m)| 2 χ E dγ α ≤ γ α (E) f 2 ∞ ≤ γ α (E) f 2 1 (14) ≤ K(α, β )γ α (E) f 2-2α+4 β 2 |(x,t)| β f 2α+4 β 2 , (15)
where

K(α, β ) = πω α (β -(α + 2)) sin( (α+2)π β ) α + 2 β -(α + 2) -α+2 β -(α+2) .
Part (c), we collect together the counterexamples necessary to establish Theorem 1.

Counterexample1: Necessary to establish Theorem 1(a). We consider the function f 1 (x,t) = h 1 (t)g 1 (x) with

h 1 (t) =    |t| -1/4 if |t| ≤ 1, 0 elsewhere and g 1 (x) =    |x| 2( 3 2 β -3 2 -α) if |x| ≤ 1, 0 elsewhere, where 2α+4 2 ≥ β > 2α+4 6 .
The function

f 1 ∈ L 2 α (K), we shall prove that if β = 2α+4 2 then f 1 (λ , m)χ E 2 2 = +∞. Let C m = 1 πΓ(α+1)L α m (0) . Then we have f 1 (λ , m) = C m |λ | 1 2 (3β + 1 2 ) |λ | 0 u -1/4 cos(u)du |λ | 0 e -u/2 L α m (u)u 3 2 (β -1) du .
Take E = B R (0, 0) (cf. Lemma 2) and as L α 0 (0) = 1. We have

f 1 (λ , m)χ E 2 2 = ∞ ∑ m=0 L α m (0) R | f 1 (λ , m)χ E | 2 |λ | α+1 dλ . = ∞ ∑ m=0 L α m (0) R m+ α+1 2 -R m+ α+1 2 | f 1 (λ , m)| 2 |λ | α+1 dλ . ≥ 2 [πΓ(α + 1)] 2 R α+1 2 0 | g 1 (λ )| 2 |λ | α-3β + 1 2 dλ , where g 1 (λ ) = ( |λ | 0 u -1/4 cos(u)du)( |λ | 0 e -u/2 u 3 2 (β -1) du). But we have ∞ 0 u -1/4 cos(u)du = Γ( 3 4 ) cos( 3π 8 ) and ∞ 0 e -u/2 u 3 2 (β -1) du = 2 3 2 β -1 2 Γ( 3β 2 -1 2 ). So, there exist C > 0, R > 0, and r > 0, such that for R ≥ |λ | ≥ r we have | g 1 (λ )| ≥ C and f 1 (λ , m)χ E 2 2 ≥ C R α+1 2 r |λ | α-3β + 1 2 dλ → +∞ as r → 0 + , if α 3 + 1 2 < β .
In particular for β = 2α+4 2 , this provides a contradiction since the left side of ( 5) is bounded. Indeed, by [START_REF] Fefferman | The uncertainty principle and sharp Gårding inequality[END_REF] we have,

f 1 χ E 2 ≤ (γ α (E)) 1/2 f 1 χ B r 1 + ( f 1 χ B c r ) 2 . So, ( f 1 χ B c r ) 2 ≤ r 3β +1/2 4β +1/2 B( 3 8 , β +2α-1 4
) and

f 1 χ B r 1 = r 3β +1/2 3β +1/2 B( 3 8 , 3β -1 4 ), whereas f 1 χ E 2 ≤ B( 3 8 , β +2α-1 4 ) 4β +1/2 + B( 3 8 , 3β -1 4 ) 3β +1/2 r 3β +1/2 -→ 0 as r -→ 0 + .
Counterexample 2 : Necessary to establish Theorem 1(b). We consider the function f 2 (x,t) = h 1 (t)g 2 (x) with

h 1 (t) =    |t| -1/4 if |t| ≤ 1, 0 elsewhere and g 2 (x) =    |x| 2( β 2 -5 4 -α) if |x| ≤ 1, 0 elsewhere, where β ≥ 2α+4 2 > 2α+3 2 .
The function

f 2 ∈ L 2 α (K), we shall prove that if β = 2α+4 2 then f 2 (λ , m)χ E 2 2 = +∞. Let C m = 1 πΓ(α+1)L α m (0) . Then we have f 2 (λ , m) = C m |λ | β -1 2 |λ | 0 u -1/4 cos(u)du |λ | 0 e -u/2 L α m (u)u β 2 -5 2 du .
Take E = B R (0, 0) (cf. Lemma 2) and as L α 0 (0) = 1. We have

f 2 (λ , m)χ E 2 2 = ∞ ∑ m=0 L α m (0) R | f 2 (λ , m)χ E | 2 |λ | α+1 dλ . = ∞ ∑ m=0 L α m (0) R m+ α+1 2 -R m+ α+1 2 | f 2 (λ , m)| 2 |λ | α+1 dλ . ≥ 2 [πΓ(α + 1)] 2 R α+1 2 1 | g 2 (λ )| 2 |λ | α-β +2 dλ , where g 2 (λ ) = ( |λ | 1 u -1/4 cos(u)du)( |λ | 1 e -u/2 u β 2 -5 2 du). But we have ∞ 0 u -1/4 cos(u)du = Γ( 3 4 ) cos( 3π 8 ) and ∞ 0 e -u/2 u β 2 -5 2 du = 2 β 2 -3 2 Γ( β 2 -3 2 ). So, there exist C > 0, R > 1, such that for R ≥ |λ | ≥ 1 we have | g 2 (λ )| ≥ C and f 2 (λ , m)χ E 2 2 ≥ C R α+1 2 1 |λ | α-β +2 dλ → +∞ as R → ∞, if 2α+6 2 > β .
In particular for β = 2α+4 2 , this provides a contradiction since the left side of ( 7) is bounded. Indeed, by [START_REF] Hardy | A theorem concerning Fourier transforms[END_REF] we have,

f 1 χ E 2 ≤ γ α (E) f 1 1 ≤ 2C 2β + 3 B( 3 8 , β 4 - 1 8
).

Remarks 1.

1. The relations among the exponents in these inequalities are forced by homogeneity considerations Q = 2α + 4 of Laguerre hypergroup K.

Price and Sitaram ([24]

, Remark 5.2) proved the following local uncertainty inequality on the Heisenberg group

H n : Given 0 < θ < 1 2 , for each f ∈ L 1 (H n ) ∩ L 2 (H n ) and E with Lebesgue measure m(E) < ∞, E Tr(π λ ( f ) * π λ ( f ))dµ(λ ) 1/2 ≤ K θ m(E) θ | . | θ f 2 , where K θ = (2θ 2 ) -θ (1 -2θ ) -1 (1 -2θ ) θ . Using Theorem 1a), let θ = β 2α+4 which satisfies 0 < θ < 1 2 . For each f ∈ L 1 rad (H n ) ∩ L 2 rad (H n )
, we can replace their K θ by the sharper one

K θ ,rad (n -1, β ) = (2θ 2 ) -θ (1 -2θ ) -1 ω n-1 2(n + 1) 2 θ , and ω n-1 = Γ( n 2 ) 2 √ πΓ( n+1 2 )Γ(n) .
3. It would be interesting to know if this is the best possible.

We shall use the local uncertainty inequality [START_REF] Ciatti | A local uncertainty inequality for step-two nilpotent Lie groups[END_REF] to prove an analogue of Heisenberg-Pauli-Weyl uncertainty inequality, therefore we are so led to impose the most important lemma that we need for the sequel.

For this purpose we define the ball in K with center (λ , m) and radius r > 0 ( for shortness B r ) to be the set

B r = {(µ, n) ∈ K, N (λ -µ, max(n -m, 0) + α + 1 2 ) < r}.
Examples: Figs of two balls with center (0, 0) and (2, 6) respectively and radius r = 1.

λ = 0 λ = 2 m = 6 N R R N 6 6
--Lemma 1. The measure of B r with respect to the Plancherel measure dγ α is finite, in the sense that for all r > 0 and (λ , m) ∈ K γ α (B r ) < ∞.

Proof. For fixed (λ , m) ∈ K and r > 0 one has

γ α (B r ) = B r dγ α (µ, n) = ∞ ∑ n=0 L α n (0) r max(n-m,0)+ α+1 2 +λ -r max(n-m,0)+ α+1 2 +λ |µ| α+1 dµ = ∞ ∑ n=m+1 L α n (0) r n-m+ α+1 2 +λ -r n-m+ α+1 2 +λ |µ| α+1 dµ + m ∑ n=0 L α n (0) r α+1 2 +λ -r α+1 2 +λ |µ| α+1 dµ = I 1 + I 2
where

I 1 = 1 α + 2 ∞ ∑ n=m+1 L α n (0)[( r n -m + α+1 2 + λ ) α+2 -(- r n -m + α+1 2 + λ ) α+2 ]
and

I 2 = [( r α+1 2 + λ ) α+2 -(-r α+1 2 + λ ) α+2 ] α + 2 m ∑ n=0 L α n (0). Since L α n (0) ∼ n α Γ(α+1) , it follows L α n (0) α+2 [( r n+ α+1 2 +λ ) α+2 -( r n+ α+1 2 -λ ) α+2 ] ∼ C r n 2 ,
and that dγ α (B r ) < ∞, where we have computed the result above for the case N (λ , m) < r for all m ≥ 0. An analogous result follows for the complement case. Remark 1. The volume of the ball B r depends not only on its radius r, but also it is largely close to its center (λ , m) which means that the Plancherel measure dγ α is not invariant under the standard translation over R 2 .

Throughout subsequently, we take the measurable set in K by

E r = {(λ , m) ∈ K, |(λ , m)| = 4|λ |(m + α + 1 2 ) < r 2 },
that is the ball in K with center (0, 0) and radius r 2 .

Lemma 2. The measure of E r with respect to the Plancherel measure dγ α is finite, in the sense that for all r > 0 γ α (E r ) < ∞.

More precisely,

γ α (E r ) = 2 r 2(α+2) α + 2 ∑ m≥0 L α m (0) (m + α+1 2 ) α+2 . (16)
Proof. For fixed r > 0 one has

γ α (E r ) = E r dγ α (λ , m) = ∑ m≥0 L α m (0) r 2 m+ α+1 2 -r 2 m+ α+1 2 |λ | α+1 dλ = 2 r 2(α+2) α + 2 ∑ m≥0 L α m (0) (m + α+1 2 ) α+2 .
One can remark easily that the volume of the set E r is finite since L α m (0) ∼ m α Γ(α+1) , so

L α m (0) (m+ α+1 2 ) α+2 ∼ C m 2 and hence γ α (E r ) < ∞. Corollary 1. For all f ∈ L 2 α (K) and β > 0 one has K |(x,t)| 2β | f (x,t)| 2 dm α (x,t) K |(λ , m)| β /2 | f (λ , m)| 2 dγ α (λ , m) ≥ C f 4 2 ,
where C is a constant.

Proof. First, suppose that 0 < β < 2α+4 2 .

Let E r be the subset of K and E c r denotes the complement of E r in K. Then, by Plancherel's theorem, we have

f 2 2 = | f (λ , m)| 2 dγ α (λ , m) = E r | f (λ , m)| 2 dγ α + E c r | f (λ , m)| 2 dγ α . (17)
Consider

E c r | f (λ , m)| 2 dγ α = E c r | f (λ , m)| 2 |(λ , m)| β |(λ , m)| -β dγ α ≤ r -2β E c r |(λ , m)| β /2 | f (λ , m)| 2 dγ α ≤ Cr -2β K | L β /2 f (λ , m)| 2 dγ α . (18)
By the local uncertainty inequality (a), with 0 < β 2α+4 < 1 2 , we get

E r | f (λ , m)| 2 dγ α ≤ Cr 2β K |(x,t)| 2β | f (x,t)| 2 dm α . (19)
Combining the relations ( 17), ( 18) and ( 19), we obtained

f 2 2 ≤ C r 2β K |(x,t)| 2β | f (x,t)| 2 dm α + r -2β K | L β /2 f (λ , m)| 2 dγ α .
However, let g be the function defined on ]0, +∞[ by

g(r) = r 2β |(x,t)| β f 2 2 + r -2β |(λ , m)| β /2 f 2 2 ,
then, the minimum of the function g is attained of the point

r 0 = |(λ , m)| β /2 f 2 2 |(x,t)| β f 2 2 1 4β
,

and g(r 0 ) = 2 |(x,t)| β f 2 |(λ , m)| β /2 f 2 .
Then we have the desired inequality

f 4 2 ≤ C K |(x,t)| 2β | f (x,t)| 2 dm α (x,t) K | L β /2 f (λ , m)| 2 dγ α (λ , m) . (20)
Second, for β ≥ 2α+4 2 . By Hölder's inequality, we have

K | f (x,t)| 2 |(x,t)| 2β dm α (x,t) ≤ f 2 K | f (x,t)| 2 |(x,t)| 4β dm α (x,t) 1/2
, and

K |L β /2 f (λ , m)| 2 dγ α ≤ K |(λ , m)| β | f (λ , m)| | f (λ , m)|dγ α ≤ K | f (λ , m)| 2 dγ α 1/2 K |L β f (λ , m)| 2 dγ α 1/2 ≤ f 2 K |L β f (λ , m)| 2 dγ α 1/2
the value of in [START_REF] Nessibi | Inversion of the Radon Transform on the Laguerre Hypergroup by using Generalized Wavelets[END_REF] can be replaced by 2β , and doing it repeatedly, ( 20) is extended to any positive β .

Heisenberg-Pauli-Weyl inequality

In this section, we extend the inequality of the uncertainty of Heisenberg-Pauli-Weyl (Corollary 1) to the more general case. We need to use another method based on ultracontractive semigroups generated by the differential operator L and the estimation of the heat kernel, the result gives by Theorem 2. Let a, b ≥ 1 and η ∈ R such that ηa = (1η)b, then for all f ∈ L 2 α (K), we have

K |(x,t)| 2a | f (x,t)| 2 dm α (x,t) η 2 K |(λ , m)| b/2 f (λ , m)| 2 dγ α (λ , m) 1-η 2 ≥ C α f 2 (21)
where C α is a constant.

Remark 2. In the particular case when a = b = β and η = 1 2 , the previous result gives us the Heisenberg-Pauli-Weyl inequality ( cf. Corollary 1)

|(x,t)| 2β f 2 2 |(λ , m)| β /2 f 2 2 ≥ C f 4 2 .
In our proof of Theorem 2, the heat kernel h s plays an important role.

Characterization of the heat kernel.

The heat kernel on K is an analogue of the Gauss kernel p s on R n , there is associated to the operator L. Let {H s : s > 0} = {e -sL : s > 0} defines a semigroup (heat-diffusion semigroup) of operators such that for any

φ ∈ C ∞ 0 (K), H s φ is a solution of L (x,t) + ∂ ∂ s = 0
and H s φ -→ φ a.e. as s -→ 0. For every s > 0, H s is an integral operator with kernel h s , i.e. for any φ ∈ C ∞ 0 (K), H s φ = φ * h s .

Then h s , s > 0 are bi-invariant functions and h as a function of the variables s ∈ R + satisfying the properties (cf. [START_REF] Stempak | An algebra associated with the generalized sublaplacian[END_REF]).

Proposition 1. The heat kernel h s satisfies that

(1) {h s : s > 0} form a semigroup under convolution * . That is, h s * h t = h s+t for s,t > 0,

(2) h s (x, -t) = h s (x,t), K h s (x,t)dm α (x,t) = 1. (3) h s is a fundamental solution of (L + ∂ ∂ s )h s = 0, on K × (0, +∞). (4) h ρ 2 s (ρs, ρ 2 t) = ρ -(2α+4) h s (x,t). Furthermore, h s (λ , m) = e -4|λ |(m+ α+1 2 )s .
The pointwise estimate of the heat kernel h s (x,t) is given by (cf. [START_REF] Huang | A heat Kernel version of Hardy's theorem for the Laguerre Hypergroup[END_REF]).

Lemma 3. There are positive constants C and A such that

h s (x,t) = R λ 2 sinh(2λ s) α+1 e -1 2 λ coth(2λ s)x 2 e iλt dλ , (22)
and h s (x,t) ≤ Cs -(α+2) e -A s |(x,t)| 2 . ( 23
) Lemma 4. Let a > 0. Then for all f ∈ L 2 α (K), we have e -sL f 2 ≤ Cs -a/2 |(x,t)| a f 2 ( 24
)
where C is a constant.

Proof. For r > 0, let denote the characteristic function of {(x,t) : |(x,t)| < r}.

We set f r = f χ B r , and f r = f -f r . Hence, e -sL f 2 ≤ e -sL f r 2 + e -sL f r 2 ≤ Cr -a 1 + r (α+2) s -(α+1)

2

|(x,t)| a f 2 .

Choosing r = s 1/2 , we obtain (24).

Proof of Theorem 2.

Let f ∈ L 2 α (K) satisfy the hypothesis. For all s > 0, by Lemma 4 and spectral Theorem we have

  (c) Cases (a) and (b) fail for β = 2α+4 2 .

Fig 1 :

 1 Fig 1: Ball B 1 (0, 0) Fig 2: Ball B 1 (2, 6)

f 2 ≤ 2 ≤ 2 . 2 |a f η 2 | 2 . 2 | 2 , 2 ≤ 1 +/ 2 ,-b 2 |-b 2 |./b 2 .

 22222222212222 e -sL f 2 + (1 -e -sL ) f C s -a/2 |(x,t)| a f 2 + (1 -e -sL )(sL) -b/2 (sL) b/2 fThe last term is controlled, using spectral Theorem (Theorem VIII.5 p.262,[25]) bys b/2 L b/2 f 2 since (1 -e -s )s -b/2 is bounded for s ≥ 0 if b ≤ 2. Hence, we obtain f 2 ≤ C s -a/2 e -sL f 2 + s b/2 L b/2 f 2 .Then we obtain,f 2 ≤ C s -a/2 |(x,t)| a f 2 + s b/2 |(λ , m)| b/2 f (λ , m) 2 .However, let g be the function defined on ]0, +∞[ byg(s) = s -a/2 |(x,t)| a f 2 + s b/2 |(λ , m)| b/2 f 2 ,then, the minimum of the function g is attained of the points 0 = a b |(x,t)| a f (λ , m)| b/2 f 1-ηFrom which optimizing in s, we obtain the resultf 2 ≤ C |(x,t)| a f η (λ , m)| b/2 f 1-η for b ≤ 2. If b > 2, let b ≤ 2. For u ≥ 0 and b < b, u b ≤ 1 + u b which for u = |(λ ,m)| ε |(λ ,m)| ε bfor all ε > 0. It follows that |(λ , m)| b /2 f 2 ≤ ε b /2 f 2 + ε b (λ , m)| b/2 f 2 . Let g(ε) = ε b /2 f 2 + ε b (λ , m)| b/2 f 2then, the minimum of the function g is attained of the pointOptimizing in ε 0 , we get |(λ , m)| b /2 f 2 ≤ C f 1-b /b 2 |(λ , m)| b/2 fb Plugging this into[START_REF] Nessibi | A Wiener-Tauberian and a Pompeiu type theorems on the Laguerre hypergroup[END_REF] with is replaced by b, we get the result for b > 2.

  Then, since | f r (x,t)| ≤ r -a |(x,t)| a f (x,t)| and e -sL is a semigroup of contractions, e -sL f r 2 ≤ f r 2 ≤ r -a |(x,t)| a f 2 .On the other hand, we havee -sL f r 2 = f r * h s 2 ≤ f r 1 h s 2 ≤ h s 2 K |(x,t)| 2a | f r (x,t)| 2 dm α (x,t) B r dm α (x,t) = ω α r (2α+4)-2a ,which gives, e -sL f r 2 ≤ Cr -a+(α+2) |(x,t)| a f 2 h s 2 .

	1/2
	1/2
	|(x,t)| -2a χ B r dm α (x,t)
	K
	since,
	By (23), we have
	h s 2 ≤ Cs -(α+1) 2 .

K |(x,t)| -2a χ

Acknowledgements. I want to express my sincere gratitude to Professor Gerald B. Folland for useful discussions which have influenced some parts of the text for detecting and removing some misprints. Also, I would like to thank Professor Burglind Juhl-Jöricke for his advice, and his general remarks.