Myriam Comte
email: comte@ann.jussieu.fr

Jean-Michel Coron
email: coron@ann.jussieu.fr

Sergio Guerrero
email: guerrero@ann.jussieu.fr

Pierre-Henri Tournier
email: tournier@ann.jussieu.fr

Analysis of a model of phosphorus uptake by plant roots

Keywords: quasilinear parabolic equations, regularity estimates, phosphorus uptake, plant roots, shape optimization Mathematics Subject Classification (2000) 35K20, 35K55

In this paper, we consider a model of phosphorus uptake by plant roots, governed by a quasilinear parabolic equation. We first study the well-posedness of the associated Cauchy problem. Then, we consider a shape optimization problem: how to deform the shape of the root in order to increase phosphorus uptake. Finally, we give some numerical results of the shape optimization process.

Introduction

Phosphorus (P) is an essential element for plant growth and metabolism. It is involved in many plant processes such as energy transfer, the synthesis of nucleic acids and membranes, plant respiration, photosynthesis and enzyme regulation. Adequate phosphorus nutrition stimulates early plant growth and hastens maturity. P is one of the limiting factors for plant growth in many agricultural systems. P uptake by plants is often constrained by the very low solubility of P in the soil, as P is mostly present in unavailable forms because of adsorption, precipitation, or conversion to the organic form. This leads to the application of up to four times the fertilizer necessary for crop production. This practice can result in polluted water systems, imbalanced ecosystems and degradation of the environement. Moreoever, at the current rate of usage of P fertilizer, readily available sources of phosphate rocks could be depleted in as little as 60-90 years. This brings us to consider the problem of reducing fertilizer usage of P, which suggests improved efficiency of fertilizer methods in the short term, and adaptation of genotypes to P-deficient soils in the long term.

We therefore propose to study the transfer of P in the soil as well as its uptake by plant roots, as was done by [START_REF] Doussan | Modelling of the hydraulic architecture of root systems: an integrated approach to water absorption. Model description[END_REF], [START_REF] Roose | A mathematical model of plant nutrient uptake[END_REF] and [START_REF] Mollier | A two dimensional simulation model of phosphorus uptake including crop growth and P response[END_REF]. P moves in soil through both diffusion and mass flow, although diffusion is dominant.

Let us consider a shape modeling the root surface. The exterior domain around the root is the studied section of the soil. Let us denote by Ω ⊂ R d (d = 2, 3) the soil domain, delimited by the root surface and the domain boundaries. Let Γ 1 be the boundary representing the root surface and

Γ 2 = ∂ Ω \ Γ 1 . Ω Γ 1 Γ 2 1
         ∂ t (θ c + ϕ(c)) = div A∇c -qc -R in I × Ω , αh(c) = -A∇c -qc • n on I × Γ 1 , 0 = A∇c -qc • n on I × Γ 2 = I × (∂ Ω \ Γ 1 ), c(0, x) = c 0 (x) in Ω , (1.1)
where -n is the unit outward normal to the boundary of the domain, c 0 is the initial P concentration, -A is the diffusion coefficient of P in the soil, θ is the volumetric water content, -q is the groundwater flow, ϕ is an adsorption/desorption isotherm relating the amount of adsorbed P to the equilibrium concentration of P in solution; an example is the Freundlich adsorption isotherm (McGechan and Lewis 2002), defined by:

ϕ(c) = κc b for c ∈ [0, +∞) , κ > 0 , b ∈ (0, 1),
h is a model of enzyme kinetics, relating in this case the root uptake rate of P to its concentration at the root surface; an example is the Michaelis-Menten model [START_REF] Barber | Soil nutrient bioavailability: a mechanistic approach[END_REF], given by:

h(c) = F m c K m + c for c ∈ [0, +∞) , F m > 0 , K m > 0,
-R represents additional optional source/sink terms to the system. We will only consider source terms, such as fertilizer application, α is a parameter we introduce in order to obtain sufficient regularity of the boundary condition in the case

Γ 1 ∩ Γ 2 = 0: α ∈ C 2 (∂ Ω ) such that for x ∈ ∂ Ω 0 < α(x) ≤ 1 on Γ 1 α(x) = 0 on Γ 2 .
We do not restrict ourselves to considering explicit forms for ϕ or h: we only use general properties of these functions throughout this paper. Since ϕ is defined on [0, +∞[, we consider positive solutions of problem (1.1). The paper is organized as follows: we first discuss existence and uniquess of solutions to problem (1.1). Then we introduce a shape optimization method which enables us to modify the shape of the domain in order to maximize the amount of absorbed P.

For simplicity and clarity, we only take α into account while proving existence and uniqueness of the solution, and we drop it later in the shape optimization study.

We begin by introducing some notations:

-S is the boundary of Ω , -Q is the cylinder (0, T ) × Ω , -S T is the lateral surface of Q: S T = {(t, x) | t ∈ [0, T ], x ∈ S}, -θ 0 (x) := θ (0, x) for x ∈ Ω .
Let us make the following assumptions:

hypotheses on S S ∈ C 2+β , (1.2) hypotheses on c 0 c 0 ∈ C 2+β (Ω ) , c 0 > 0, A∇c 0 -qc 0 • n + h(c 0 ) = 0 on Γ 1 , A∇c 0 -qc 0 • n = 0 on Γ 2 , (1.3)
hypotheses on ϕ ϕ ∈ C 3 ((0, +∞)) , ϕ > 0, (1.4)
hypotheses on A A ∈ C 1+β /2,2+β (Q) , A m ≥ A(t, x) ≥ A 0 > 0 in Q, (1.5) hypotheses on θ θ ∈ C 1+β /2,2+β (Q) , θ m ≥ θ (t, x) ≥ θ 0 > 0 in Q, (1.6) hypotheses on R R ∈ C 1+β /2,2+β (Q) , R < 0 in Q, (1.7)
hypotheses on q

q i ∈ C 1+β /2,2+β (Q) , i = 1, .., d, (1.8)
let us extend h to R so that

h ∈ C 2 (R) , h(0) = 0 , h L ∞ (R) + h L ∞ (R) + h L ∞ (R) ≤ C h . (1.9)
Here and in the sequel, β ∈ (0, 1) .

(1.10)

A priori estimates

In this section, we derive upper and lower bounds for the solutions of problem (1.1) in the space C 1,2 (Q). Let c be a solution of problem (1.1) in the space C 1,2 (Q), c ≥ 0 in Q.

• Estimate from below:

Let ε > 0, and let

T = max{t ∈ [0, T ] | c ≥ ε in [0, T ] × Ω }.
We now find a lower bound for c in [0, T ] × Ω .

Let us introduce the following function ĉ(t, x) := δ e -Kt ĉ0 (x), (t, x) ∈ Q, (

where δ , K > 0 are chosen below and where ĉ0 ∈ C 2 (Ω ) is a strictly positive function satisfying

A 0 ∂ ĉ0 ∂ n < -q L ∞ (Q) ĉ0 (x) -(h (0) + 1) ĉ0 (x) ∀x ∈ S, (2.2)
and 0 < ĉ0 (x) < 1 ∀x ∈ Ω .

(2.3)

Choosing now K large enough and δ small enough, ĉ satisfies

         θ ĉt -div A∇ ĉ -q ĉ + R + θ t ĉ < 0 in Q, A ∂ ĉ ∂ n < (q • n) ĉ -αh( ĉ) on S T , ĉ(0, x) < c 0 (x) in Ω .
(2.4)

Since ϕ ≥ 0, we can see that

(θ + ϕ ( ĉ)) ĉt -div A∇ ĉ -q ĉ + R + θ t ĉ < 0 in Q. (2.5)
We now apply a comparison principle which results from the following theorem:

Theorem 1 [START_REF] Friedman | Partial differential equations of parabolic type[END_REF], Theorem 17 p. 53) Let v and w be two continuous functions in Q, and let the first t-derivative and the first two x-derivatives of v, w be continuous in Q. Let F(t, x, p, p i , p i j ) (i, j = 1, ..., d) be a continuous function together with its first derivatives with respect to the p hk in a domain E containing the closure of the set of points (t, x, p, p i , p i j ) where

(t, x) ∈ Q, p ∈ (v(t, x), w(t, x)), p i ∈ ∂ v(t, x) ∂ x i , ∂ w(t, x) ∂ x i , p i j ∈ ∂ 2 v(t, x) ∂ x i ∂ x j , ∂ 2 w(t, x) ∂ x i ∂ x j ;
here (a, b) denotes the interval connecting a to b. Assume also that (∂ F/∂ p hk ) is a positive semidefinite matrix.

If                        ∂ v ∂t > F t, x, v, ∂ v ∂ x i , ∂ 2 v ∂ x i ∂ x j in Q, ∂ w ∂t ≤ F t, x, w, ∂ w ∂ x i , ∂ 2 w ∂ x i ∂ x j in Q, v(0, x) > w(0, x) on Ω , ∂ v ∂ n + γ(t, x, v) > ∂ w ∂ n + γ(t, x, w) on S T , (2.6)
for some function γ, then also v > w in Q.

We have

c t = (θ + ϕ (c)) -1 (div A∇c -qc -R) -θ t c in [0, T ] × Ω , A∇c • n = (q • n)c -αh(c) on [0, T ] × S, (2.7)
and ĉt < (θ + ϕ ( ĉ)) -1 (div A∇ ĉ -q ĉ -R) -θ t ĉ in [0, T ] × Ω , A∇ ĉ • n < (q • n) ĉ -αh( ĉ) on [0, T ] × S. (2.8)
Moreover, ĉ(0, x) < c(0, x) in Ω .

It is easy to see that in our case F satisfies the hypotheses of Theorem 1 thanks to (1.4), (1.5), (1.6), (1.7) and (1.8).

Thus, we can apply Theorem 1 to c and ĉ in

[0, T ] × Ω in order to deduce that c > ĉ ≥ δ e -KT min x∈Ω ĉ0 (x) > 0 in [0, T ] × Ω .
(2.9)

We proved that for every ε > 0, we have c > δ e -KT min

x∈Ω ĉ0 (x) in [0, T ] × Ω , with T = max{t ∈ [0, T ] | c ≥ ε in [0, T ] × Ω }.
Note that the lower bound is independent of the choice of ε. Then, it is easy to see that if we take ε small enough and we suppose that T < T , by a continuity argument (c

∈ C 1,2 (Q)) we obtain that c ≥ ε in [0, T + δt]
× Ω for some δt > 0, which leads to a contradiction. Thus T = T , and we can conclude that

c > δ e -KT min x∈Ω ĉ0 (x) > 0 in Q.
(2.10)

• Estimate from above:

Let č0 ∈ C 2 (Ω ) satisfy A 0 ∂ č0 ∂ n > q L ∞ (Q) č0 (x) ∀x ∈ S, (2.11)
and

c 0 L ∞ (Ω ) < č0 (x) < c 0 L ∞ (Ω ) + 1 ∀x ∈ Ω . (2.12) Let us define č(t, x) := e λt č0 (x), (t, x) ∈ Q, (2.13)
for λ > 0. Then, for λ large enough it is clear that

θ čt -div A∇ č -q č + R + θ t č > 0 in Q, (2.14)
which means that

(θ + ϕ ( č)) čt -div A∇ č -q č + R + θ t č > 0 in Q. (2.15)
We have

c t = (θ + ϕ (c)) -1 (div A∇c -qc -R) -θ t c in Q, A∇c • n = (q • n)c -αh(c) on S T , (2.16
)

and čt > (θ + ϕ ( č)) -1 (div A∇ č -q č -R) -θ t č in Q, A∇ č • n > (q • n) č -αh( č) on S T . (2.17) Moreover, č(0, x) > c(0, x) in Ω .
Since we know that c > δ e -KT min x∈Ω ĉ0 (x) in Q, we can now apply Theorem 1 to c and č in Q to deduce that

c(t, x) < č(t, x), (t, x) ∈ Q.
Thus, we can conclude that there exist c min , c max such that

0 < c min ≤ c(t, x) ≤ c max , (t, x) ∈ Q. (2.18) 3 Uniqueness of solutions in C 1,2 (Q)
In this section, we prove the uniqueness of solutions of problem (1.1) in the space C 1,2 (Q).

Let c 1 , c 2 be two solutions of (1.1) belonging to C 1,2 (Q). From section 2, we know that

0 < c min ≤ c 1 (t, x), c 2 (t, x) ≤ c max , (t, x) ∈ Q. (3.1)
Let us multiply the equation satisfied by c 1c 2 by c 1c 2 , integrate in Ω and integrate by parts. This yields:

1 2 d dt Ω (θ + ϕ (c 1 ))|c 1 -c 2 | 2 dx - 1 2 Ω (-θ t + ϕ (c 1 )c 1,t )(c 1 -c 2 ) 2 dx + Ω (ϕ (c 1 ) -ϕ (c 2 ))(c 1 -c 2 )c 2,t dx + Ω A|∇c 1 -∇c 2 | 2 dx - Ω q • ∇(c 1 -c 2 )(c 1 -c 2 ) dx - ∂ Ω A ∂ ∂ n (c 1 -c 2 )(c 1 -c 2 )dσ + ∂ Ω (q • n)(c 1 -c 2 ) 2 dσ = 0. (3.2)
We use that (1.4) and (3.1)) and q ∈ L ∞ (Ω × (0, T )).

θ t ∈ L ∞ (Ω × (0, T )) (see (1.6)), ϕ (c 1 ) ∈ L ∞ (Ω × (0, T )) (thanks to
We obtain: 1 2

d dt Ω (θ + ϕ (c 1 ))|c 1 -c 2 | 2 dx + Ω A|∇c 1 -∇c 2 | 2 dx ≤ K Ω (1 + |c 1,t | + |c 2,t |)|c 1 -c 2 | 2 + Γ 1 |c 1 -c 2 | 2 dσ . (3.3)
Here, we have also used that |αh(c 1 )αh(c 2 )| ≤ K|c 1c 2 | and the same property for ϕ .

• We estimate the first term in the right-hand side:

Ω |c 1 -c 2 | 2 (|c 1,t | + |c 2,t |)dx ≤ ( c 1,t L 2 (Ω ) + c 2,t L 2 (Ω ) ) c 1 -c 2 2 L 4 (Ω ) .
Using that

f L 4 (Ω ) ≤ f 1/4 L 2 (Ω ) f 3/4 L 6 (Ω ) , we deduce that Ω |c 1 -c 2 | 2 (|c 1,t | + |c 2,t |)dx ≤ ( c 1,t L 2 (Ω ) + c 2,t L 2 (Ω ) ) c 1 -c 2 1/2 L 2 (Ω ) c 1 -c 2 3/2 L 6 (Ω ) .
Using now Young's inequality (for parameters 4 and 4/3), we obtain, for every ε ∈ [0, 1] there exists K ε such that

Ω |c 1 -c 2 | 2 (|c 1,t | + |c 2,t |)dx ≤ K ε ( c 1,t 4 L 2 (Ω ) + c 2,t 4 L 2 (Ω ) ) c 1 -c 2 2 L 2 (Ω ) + ε c 1 -c 2 2 L 6 (Ω ) . From the continuous injection H 1 (Ω ) → L 6 (Ω ) (recall that d = 2, 3), we get Ω |c 1 -c 2 | 2 (1 + |c 1,t | + |c 2,t |)dx ≤ K ε (1 + c 1,t 4 L 2 (Ω ) + c 2,t 4 L 2 (Ω ) ) c 1 -c 2 2 L 2 (Ω ) + ε c 1 -c 2 2 H 1 (Ω ) , (3.4)
where K ε does not depend on K ε .

• For the second term in the right-hand side of (3.2), we consider a function ρ

∈ C 2 (Ω ) satisfying ∂ ρ ∂ n ≥ 1 on S. Then, ∂ Ω ∂ ρ ∂ n |c 1 -c 2 | 2 dσ = 2 Ω (∇ρ • ∇(c 1 -c 2 ))(c 1 -c 2 ) dx + Ω ∆ ρ (c 1 -c 2 ) 2 dx.
From Young's inequality, we easily deduce that for every ε ∈ [0, 1] there exists K ε such that

Γ 1 |c 1 -c 2 | 2 dσ ≤ K ε c 1 -c 2 2 L 2 (Ω ) + ε ∇(c 1 -c 2 ) 2 L 2 (Ω ) .
Putting this together with (3.4) and using (1.4), (1.5) and (1.6), we find 1 2

d dt Ω (θ + ϕ (c 1 ))|c 1 -c 2 | 2 dx + Ω A|∇c 1 -∇c 2 | 2 dx ≤ K ε θ -1 0 (1 + c 1,t 4
L 2 (Ω ) + c 2,t 4
L 2 (Ω ) ) Ω (θ + ϕ (c 1 ))|c 1 -c 2 | 2 dx +ε Ω A|∇c 1 -∇c 2 | 2 dx. (3.5)
Finally, taking ε > 0 small enough and using Gronwall's Lemma, we deduce that c 1 = c 2 .

4 Existence and uniqueness of solutions in

C 1+β /2,2+β (Q) Recall that β ∈ (0, 1) is fixed (1.10).
In order to apply an existence and uniqueness theorem we need to define a new problem by truncating the function ϕ:

Let us define ϕ M as follows, with

M := M 1 M 2 , 0 < M 1 < M 2 :                ϕ M (c) = ϕ(c) for M 1 ≤ c ≤ M 2 , ϕ M (c) = -ε M for c ≤ -ε M , ϕ M (c) = ϕ(M 2 ) + ε M for c ≥ M 2 + ε M , ϕ M ∈ C 3 ([-ε M , M 1 ] ∪ [M 2 , M 2 + ε M ]), ϕ M ≥ 0, for some ε M > 0. It is clear that such ϕ M exists. Note, in particular, that ϕ M (c) = ϕ M (c) = ϕ M (c) = 0 for |c| > M 2 + ε M and that ϕ M
and its derivatives of up to order three are bounded.

Let us now define the new problem: find c M such that

         ∂ t (θ c M + ϕ M (c M )) = div A∇c M -qc M -R in Q, αh(c) = -A∇c M -qc M • n on I × Γ 1 , 0 = A∇c M -qc M • n on I × Γ 2 , c M (0, x) = c 0 (x) in Ω . (4.1)
Let us make the following change of variables:

let y := θ c M + ϕ M (c M ) -θ (0, x)c 0 (x) -ϕ M (c 0 ).
Note that y is strictly monotonically increasing in the variable c M (recall that θ ≥ θ 0 > 0 in Q and ϕ M ≥ 0). Thus, we have c M = k(t, x, y) with k strictly monotonically increasing in the variable y. Moreover, k is three times continuously differentiable with respect to y.

We can write the problem as follows:

         ∂ t y -div(A∇k -qk) + R = 0 in Q, (A∇k -qk) • n + αh(k) = 0 on I × Γ 1 , (A∇k -qk) • n = 0 on I × Γ 2 , y(0, x) = 0 in Ω . (4.2)
We use the following notations for derivatives:

d dx [a(x, u(x))] = ∂ a ∂ x + ∂ a ∂ u u x = a x + a u u x .
The equation in Q gives:

∂ t y -∑ i d dx i (A d dx i k) + ∑ i d dx i (q i k) + R = 0.
Using the chain rule yields:

∂ t y -∑ i A d 2 dx 2 i k -∑ i d dx i A d dx i k + ∑ i (k d dx i q i + q i d dx i k) + R = 0.
We obtain:

∂ t y -A ∑ i k x i x i + k y y x i x i + k yx i y x i + k x i y y x i + k yy y 2 x i -∑ i A x i k x i + A x i k y y x i -kq i x i -q i k x i -q i k y y x i + R = 0.
The equation in I × ∂ (Ω ) gives:

∑ i (A d dx i k -q i k)cos(n, x i ) + αh(k) = 0.
Using the chain rule again yields:

A ∑ i k y y x i cos(n, x i ) + ∑ i (Ak x i -q i k)cos(n, x i ) + αh(k) = 0.
Finally, we can see that our problem is of the general form

    
Ly := y ta i j (x,t, y)y x i x j + b(x,t, y, y x ) = 0, L (S) y := a i j (x,t, y)y x j cos(n, x i ) + ψ(x,t, y)| S T = 0, There exists µ 1 > 0 and c 0 , c 1 , c 2 , c 3 , c 4 ≥ 0 such that for arbitrary y,

y| t=0 = 0, (4.3) with                          a i j = 0 for i = j, a ii = Ak y , b = -A ∑ i k x i x i + k yx i y x i + k x i y y x i + k yy y 2 x i -∑ i A x i k x i + A x i k y y x i -kq i x i -q i k x i -q i k y y x i + R, ψ = ∑ i (Ak x i -q i k)cos(n, x i ) + αh(k).
0 ≤ ∑ i, j a i j (x,t, y)ξ i ξ j ≤ µ 1 ξ 2 for (t, x) ∈ Q, (4.5a) -yb(x,t, y, p) ≤ c 0 p 2 + c 1 y 2 + c 2 for (t, x) ∈ Q, (4.5b) -yψ(x,t, y) ≤ c 3 y 2 + c 4 for (t, x) ∈ S T , (4.5c)
∑ i, j a i j (x,t, y)ξ i ξ j ≥ ν 1 ξ 2 for (t, x) ∈ S T . (4.5d)

There exists ν, µ > 0 such that for (t, x) ∈ Q , |y| ≤ N and for arbitrary p the functions a i j (x,t, y), b(x,t, y, p) and ψ(x,t, y) are continuous in their arguments, possess the derivatives entering into the following conditions and satisfy these conditions:

νξ 2 ≤ ∑ i, j a i j (x,t, y)ξ i ξ j ≤ µξ 2 , (4.6a) ∂ a i j (x,t, y) ∂ y , ∂ a i j ∂ x , ψ , ∂ ψ ∂ y , ∂ ψ ∂ x ≤ µ, (4.6b) |b(x,t, y, p)| ≤ µ(1 + p 2 ), (4.6c
)

|ψ yy (x,t, y) , ψ yx , ψ yt , a i j t , ψ t | ≤ µ, (4.7a
)

|b p |(1 + |p|) + |b y | + |b t | ≤ µ(1 + p 2 ), (4.7b)
|a i j yy , a i j yt , a i j yx j , a i j x j t | ≤ µ.

For (t, x) ∈ Q , |y| ≤ N and |p| ≤ N, the functions a i jx (x,t, y) are Hölder continuous in the variables x with exponent β , ψ x (x,t, y) is Hölder continuous in x and t with exponent β and β /2 respectively, and b(x,t, y, p) is Hölder continuous in x with exponent β . Let us verify hypotheses (4.5a) to (4.9b) for our problem:

Proof of (4.5a) (4.5d) (4.6a). We have Proof of (4.5b) (4.5c). We have

k y = 1
y k • k = 1 θ + ϕ M (k) ,
and since 0 < A 0 ≤ A(t, x) ≤ A m in Q and 0 < θ 0 ≤ θ (t, x) + ϕ M (k) < +∞ in Q × R,
-yb(x,t, y, p) = ∑ i Ak x i x i y + A x i k x i y -kq i x i y -q i k x i y -Ry + A ∑ i k yx i p i y + k x i y p i y + k yy p 2 i y + ∑ i [A x i k y p i y -q i k y p i y] .
-Terms not involving p, for example Ak x i x i y, are handled in this way:

k x i x j = - θ x i x j k θ + ϕ M (k) + 2θ x i θ x j k (θ + ϕ M (k)) 2 - θ x i θ x j ϕ M (k)k 2 (θ + ϕ M (k)) 3 + θ 0 x i x j c 0 + θ 0 x i c 0 x j + (θ 0 + ϕ M (c 0 ))c 0 x i x j + (θ 0 x j + ϕ M (c 0 )c 0 x j )c 0 x i θ + ϕ M (k) -θ 0 x i c 0 + (θ 0 + ϕ M (c 0 ))c 0 x i θ x j (θ + ϕ M (k)) 2 + ϕ M (k)(θ 0 x i c 0 + (θ 0 + ϕ M (c 0 ))c 0 x i ) (θ + ϕ M (k)) 3 - ϕ M (k)θ x i k (θ + ϕ M (k)) 3 .
We have θ 0 ≤ θ (t, x)+ϕ M (k) < +∞. Moreover, for |y| large enough there holds k = θ -1 y+c t (with c t bounded and independent of y) and ϕ M (k) = 0. Taking into account θ , A, θ 0 , c 0 and their space derivatives of up to order two are bounded , it follows that there exists c 3a , c 3b ≥ 0 such that for arbitrary y

|Ak x i x i y| ≤ c 3a y 2 + c 3b for (t, x) ∈ Q.
Remaining terms not involving p are handled similarly.

k y , k yx i and k x i y , the derivatives appearing in terms involving p, can be bounded independently of y. In the case of the term Ak yx i p i y, we have

k yx i = -θ x i (θ + ϕ M (k)) 2 + θ x i ϕ M (k)k (θ + ϕ M (k)) 3 - ϕ M (k) θ 0 x i c 0 + (θ 0 + ϕ M (c 0 ))c 0 x i (θ + ϕ M (k)) 3 ,
which leads to the fact that there exists c t 2 , c 5a , c 5b ≥ 0 such that for arbitrary y

|Ak yx i p i y| ≤ c t 2 |p i ||y| ≤ c 5a p 2 + c 5b y 2 for (t, x) ∈ Q.
-The term Ak yy p 2 i y remains. We have

k yy = -ϕ M (k)k y (θ + ϕ M (k)) 2 = -ϕ M (k) (θ + ϕ M (k)) 3 ,
which vanishes for |y| large enough. It follows that there exists c 6 ≥ 0 such that for arbitrary y |Ak yy p 2 i y| ≤ c 6 p 2 for (t, x) ∈ Q. Consequently, (4.5b) holds. (4.5c) can be verified in the same way as (4.5b).

Proof of (4.6b) (4.6c) (4.7a) (4.7b) (4.7c). It is easy to see that under the assumptions we made, all appearing quantities in these hypotheses are defined and bounded, thus it is clear that these conditions are verified.

Proof of (4.8). We have

∂ a ii (x,t, y) ∂ x j = A x j k y + Ak yx j , ∂ ψ ∂ x j = ∑ i [(A x j k x i + Ak x i x j -q i x j k -q i k x j )n i + (Ak x i -q i k)n i x j ] + αh (k)k x j + α x j h(k),
and one can easily verify that under the assumptions we made, the Hölder continuity hypotheses on a i jx (x,t, y), ψ x (x,t, y) and b(x,t, y, p) required by (4.8) hold true.

Proof of (4.9a). We made the assumption that S ∈ C 2+β .

Using this change of variable in (5.2) leads to

T 0 Ω -(θ c s + ϕ(c s ))∂ t Φ s V | det(DT s )| + T 0 Ω (A( * D(T s )) -1 ∇c s -qc s • ( * D(T s )) -1 ∇Φ s V | det(DT s )| + T 0 Ω R • T s Φ s V | det(DT s )| = Ω (θ z • T s + ϕ(z • T s ))Φ s V (0, x)| det(DT s )| - T 0 Γ 1 Φ s V h(c s )| det(DT s )| ( * D(T s )) -1 (n) , (5.3)
with

Φ(T, •) = 0.
As Φ is any element in H 1 (Ω s ), Φ s V describes H 1 (Ω ). Then using (5.3) leads to

T 0 Ω -(θ c s + ϕ(c s ))∂ t Φ| det(DT s )| + T 0 Ω (A( * D(T s )) -1 ∇c s -qc s • ( * D(T s )) -1 ∇Φ| det(DT s )| + T 0 Ω R • T s Φ| det(DT s )| = Ω (θ z • T s + ϕ(z • T s ))Φ(0, x)| det(DT s )| - T 0 Γ 1 Φh(c s )| det(DT s )| ( * D(T s )) -1 (n) . (5.4)
Furthermore, using (1.1) gives

T 0 Ω -(θ c + ϕ(c))∂ t Φ + (A∇c -qc • ∇Φ + T 0 Ω RΦ = Ω (θ z + ϕ(z)Φ(0, x) - T 0 Γ 1
Φh(c).

(5.5) Subtracting (5.3) to (5.4), dividing by s and letting s go to 0 leads to

T 0 Ω -(θ ċ + ϕ (c) ċ + θ c div V + ϕ(c) div V)∂ t Φ + T 0 Ω A(∇ ċ + (div V - * DV -DV )∇c) • ∇Φ + T 0 Ω -q • ċ -c * DV + c div V ∇Φ + T 0 Ω (∇R • V + R div V)Φ = Ω (θ ∇z • V + θ z div V + ∇(ϕ • z) • ∇V + ϕ(z) div V)Φ(0, x) - T 0 Γ 1 (h (c) ċ + h(c)(div V -DV n • n))Φ. (5.6)
Integrating by parts the first term gives

T 0 Ω ∂ ∂t (θ ċ + ϕ (c) ċ + θ c div V + ϕ(c) div V)Φ + T 0 Ω A(∇ ċ + (div V - * DV -DV )∇c) • ∇Φ + T 0 Ω -q • ċ -c * DV + c div V ∇Φ + T 0 Ω (∇R • V + R div V)Φ = Ω θ ∇z • V -θ + ϕ (c(0, x)) ċ(0, x) + ∇(ϕ • z) • ∇V Φ(0, x) - T 0 Γ 1 (h (c) ċ + h(c)(div V -DV n • n))Φ.
Integrating by parts the following two terms leads to

T 0 Ω ∂ ∂t θ ċ + ϕ (c) ċ + θ c div V + ϕ(c) div V Φ - T 0 Ω A ∆ ċ + div(div V - * DV -DV )∇c Φ + T 0 ∂ Ω A ∂ ċ ∂ n + (div V - * DV -DV )∇c • n Φ + T 0 Ω q • ∇ ċ -c∇(div V) + div V∇c Φ+ T 0 Ω -DV q • ∇c + c div(DV q) Φ+ + T 0 ∂ Ω -ċq • n + DV (cq) • n -div Vcq • n Φ + T 0 Ω (∇R • V + R div V)Φ = Ω θ ∇z • V -θ + ϕ (c(0, x)) ċ(0, x) + ∇(ϕ • z) • ∇V Φ(0, x)+ - T 0 Γ 1 (h (c) ċ + h(c)(div V -DV n • n))Φ.
Finally, the equation verified by ċ in Q is

∂ ∂t θ ċ + ϕ (c) ċ + θ c div V + ϕ(c) div V -A ∆ ċ + div(div V - * DV -DV )∇c +q • ∇ ċ -c∇(div V) + div V∇c - -DV q • ∇c + c div(DV q) = -∇R • V -R div V, (5.7)
with boundary conditions

         A ∂ ċ ∂ n + (div V - * DV -DV )∇c • n -ċq • n + DV (cq) • n -div Vcq • n = -h (c) ċ -h(c)(div V -DV n • n) on I × Γ 1 , A ∂ ċ ∂ n + (div V - * DV -DV )∇c • n -ċq • n + DV (cq) • n- -div Vcq • n = 0 on I × Γ 2 ,
(5.8) and initial value

(θ + ϕ (c(0, x))) ċ(0, x) = θ ∇z • V + ∇(ϕ • z) • ∇V.
(5.9)

Using the original equation satisfied by c, we obtain

∂ ∂t θ ċ + ϕ (c) ċ -A∆ ċ + q • ∇ ċ -c∇(div V) -A∇(div V) • ∇c -DV q • ∇c + c div(DV q) +A div( * DV + DV )∇c = -∇R • V in Q, (5.10)
with boundary conditions

         A ∂ ċ ∂ n -( * DV + DV )∇c • n -ċq • n +DV (cq) • n = -h (c) ċ + h(c)DV n • n on I × Γ 1 , A ∂ ċ ∂ n -( * DV + DV )∇c • n -ċq • n + DV (cq) • n = 0 on I × Γ 2 ,
(5.11) and initial value

(θ + ϕ (c(0, x))) ċ(0, x) = θ ∇z • V + ∇(ϕ • z) • ∇V.
(5.12)

Let us now denote by c the derivative of c with respect to the domain:

c = ċ -∇c • V.
The equation satisfied by c is

∂ ∂t θ c + ϕ (c)c -A∆ c + q • ∇c = 0 in Q, (5.13)
with boundary conditions

         A ∂ c ∂ n -c q • n + h (c)c = -A ∂ 2 c ∂ n 2 -∂ h(c) ∂ n + ∇c • q (V • n) +A∇ T (V • n) • ∇ T c -∇ T (cV • n) • q on I × Γ 1 , A ∂ c ∂ n -c q • n = -A ∂ 2 c ∂ n 2 + ∇c • q (V • n) +A∇ T (V • n) • ∇ T c -∇ T (cV • n) • q on I × Γ 2 ,
(5.14) and initial value c (0, x) = 0, (5.15) where ∇ T is the tangential part of the gradient. Now, consider the derivative of the functional J at Ω in the direction V:

dJ(c, V) = T 0 Γ 1 h (c)c + Γ 1 Hh(c)(V • n) + Γ 1 h (c) ∂ c ∂ n (V • n) , (5.16)
where H is the mean curvature of the boundary of the domain.

In order to get rid of the shape derivative c (which we would have to compute for every choice of V) in the expression of dJ, we use the adjoint state technique. Let us introduce p the solution to the following adjoint state problem:

         -(θ + ϕ (c))∂ t p -div(A∇p) -q • ∇p = 0 in Q, (A∇p) • n + h (c)p = h (c) on I × Γ 1 , (A∇p) • n = 0 on I × Γ 2 , p(T, x) = 0 in Ω ?
(5.17)

Multiplying by c and integrating over Q yields:

- T 0 Ω (θ + ϕ (c))∂ t pc - T 0 Ω -div(A∇p)c - T 0 Ω q • ∇pc = 0. ( 5

.18)

Successive integrations by parts lead to

T 0 Ω ∂ t (θ c + ϕ (c)c )p - Ω (θ + ϕ (c(x, T )))c (x, T )p(x, T ) + Ω (θ + ϕ (c(x, 0)))c (x, 0)p(x, 0) + T 0 Ω A∇c • ∇p - T 0 Γ Ac ∂ p ∂ n + T 0 Ω p div(qc ) - T 0 Γ c pq • n = 0, (5.19)
and finally

T 0 Ω ∂ t (θ c + ϕ (c)c )p - T 0 Ω div(A∇c )p + T 0 Γ A ∂ c ∂ n p - T 0 Γ Ac ∂ p ∂ n + T 0 Ω p div(qc ) - T 0 Γ c pq • n = 0.
(5.20)

On another hand, recall that

∂ t (θ c + ϕ (c)c ) -div(A∇c -qc ) = 0 in Q. ( 5
T 0 Γ A ∂ c ∂ n p - T 0 Γ Ac ∂ p ∂ n - T 0 Γ c pq • n = 0.
Considering the boundary conditions satisfied by p, it follows that

T 0 Γ A ∂ c ∂ n p + T 0 Γ 1 h (c)c p - T 0 Γ c pq • n = T 0 Γ 1 h (c)c . (5.22)
We can now use (5.22) in equation (5.16) verified by dJ in order to get rid of c :

dJ(c, V) = T 0 Γ 1 h (c)c + Γ 1 Hh(c)(V • n) + Γ 1 h (c) ∂ c ∂ n (V • n) = T 0 Γ A ∂ c ∂ n p + Γ 1 h (c)c p - T 0 Γ c pq • n + Γ 1 Hh(c)(V • n) + Γ 1 h (c) ∂ c ∂ n (V • n) = T 0 Γ A∇ T c • ∇ T (V • n)p -A ∂ 2 c ∂ n 2 (V • n)p -∇ T (cV • n) • qp + q • ∇c(V • n)p - T 0 Γ 1 h (c) ∂ c ∂ n (V • n)p -(Hh(c) + h (c) ∂ c ∂ n )(V • n) , (5.23)
where the last equality comes from using the boundary conditions satisfied by c . Now, note that

Γ A∇ T c • ∇ T (V • n)p = - Γ div T (Ap∇ T c)(V • n) = - Γ A∇ T p • ∇ T c(V • n) - Γ p div T (A∇ T c)(V • n).
(5.24)

On the other hand, we have

div T (A∇ T c) = A∆ T c = A∆ c -AH ∂ c ∂ n -A ∂ 2 c ∂ n 2 = ∂ t (θ c + ϕ(c)) + div(qc) -AH ∂ c ∂ n -A ∂ 2 c ∂ n 2 .
(5.25)

It follows that

Γ A∇ T c • ∇ T (V • n)p = - Γ A∇ T p • ∇ T c(V • n) - Γ ∂ t (θ c + ϕ(c))p(V • n) - Γ div(qc)p(V • n) + Γ AH ∂ c ∂ n p(V • n) + Γ A ∂ 2 c ∂ n 2 p(V • n).
Finally,

dJ(c, V) = - T 0 Γ A∇ Γ p • ∇ Γ c(V • n) - Γ c∇ T p • q(V • n) + Γ ∂ t (θ c + ϕ(c))p(V • n) + T 0 Γ AH ∂ c ∂ n p(V • n) - Γ 1 h (c) ∂ c ∂ n (V • n)p + Γ 1 Hh(c)(V • n) + Γ 1 h (c) ∂ c ∂ n (V • n) .
Note that the shape gradient dJ is now expressed in the following convenient way:

dJ(c, V) = Γ j(V • n),
where j does not depend on V.

With that in mind, a simple yet effective approach to maximize J consists in choosing V such that V • n = j, i.e. V = jn. this brings dJ(c, V) = Γ j 2 > 0, which ensures that J increases as the domain is iteratively deformed.

Note that this method restricts the choice of the deformation, as V is taken colinear with n.

Numerical resolution of the state and adjoint equations in two spatial dimensions is carried out using the free finite element software FreeFEM++. Spatial discretization is done using Lagrange P2 finite elements. The backward Euler method is applied for the discretization in time. Nonlinearities in the state equation are handled by Newton's method. A built-in adaptive anisotropic mesh refinement algorithm is used in order to improve accuracy near the boundary while preserving an acceptable computational cost. The constant volume constraint is enforced by a lagrange multiplier. Additionally, a minimum diameter constraint is put on the shape in order to prevent unsuitable deformations of the domain.

Numerical values used in this example are as follows:

-F m = 0.282 µmol cm -2 d -1 , K m = 5.8 × 10 -3 µmol cm -3 , κ = 6.15 , b = 0.72, θ = 0.35 cm 3 cm -3 , -A = 0.102 cm 2 d -1 , c 0 = 2.9 × 10 -3 µmol cm -3 , -q = 0, R = 0.

-The initial shape is an ellipse of diameters 1.33 cm and 2.66 cm. As expected, this example shows that maximizing root surface area to volume ratio is indeed an essential component of root uptake efficiency. Future work could consist in coupling a similar shape sensitivity analysis with an explicit geometry root system growth model, in order to study and improve the efficiency of different types of root systems regarding nutrient and water uptake in heterogeneous soils.

Fig. 1

 1 Fig. 1 Configuration of the domain

(4. 4 )Theorem 2 (

 42 We can now apply the following theorem to problem (4.3): Ladyzhenskaya et al. (1968), Theorem 7.4 p. 491) Suppose the following conditions are fulfilled:

(4. 8 )

 8 S ∈ C 2+β . (4.9a) ψ(x, 0, 0)| x∈S = 0. (4.9b)Then problem (4.3) has a unique solution y(t, x) in the class C 1+β /2,2+β (Q).

Fig. 2

 2 Fig. 2 Snapshots of the domain and P concentration at different steps of the shape optimization process

Fig. 3 Fig. 4

 34 Fig. 3 Evolution of the total amount of absorbed P during the optimization process

Acknowledgements This work was supported by Agropolis Fondation, CONVENTION DE SOUTIEN n o 0902-008. JMC was partially supported by the ERC advanced grant 266907 (CPDENL) of the 7th Research Framework Programme (FP7).

Proof of (4.9b). We have the following initial and boundary compatibility condition:

(A∇c 0 -qc 0 ) • n + αh(c 0 ) = 0 on S, which, since k| t=0 = c 0 , leads to ψ(x, 0, 0) = ∑ i (Ac 0

x iq i c 0 )cos(n, x i ) + αh(c 0 ) = 0 on S, and (4.9b) holds.

We can then choose M so that M 1 < c min and M 2 > c max . It follows that the unique solution c M of problem (4.1) is also a solution of the original problem (1.1). Thus, using the fact that problem (1.1) has at most one solution in the space C 1,2 (Q) which was proven in section 3, we can deduce that problem (1.1) has a unique solution in the space C 1+β /2,2+β (Q).

Shape optimization

In this section, we use the tools of shape optimization presented by Sokolowski and Zolesio (1992), [START_REF] Haslinger | Introduction to Shape Optimization: Theory, Approximation, and Computation[END_REF] and [START_REF] Henrot | Variation et optimisation de formes: une analyse géométrique[END_REF] to find root shapes that increase the amount of absorbed P. More specifically, we want to deform Ω so as to maximize the shape functional

where |Ω | = a given constant. It is assumed that A and q are constants for simplicity. We first calculate the material and shape derivatives of c with respect to the domain:

We introduce a vector field V ∈ C 2 (R d , R d ) and we consider Ω s = (Id + sV)(Ω ) where s is a small parameter. Let T s (V) = Id + sV and let c s be the unique solution of

We first determine the form of ċ, the material derivative of c, which is the derivative of s → c s • T s at s = 0. For s small enough, c s satisfies

for all Φ ∈ H 1 (Ω s ) such that Φ(T, •) = 0. We use the change of variable x = T s (V)(X) and introduce c s = c s • T s (V) and Φ s V = Φ • T s (V). The Jacobian of the transformation is then det(D(T s (V)) = det(DT s ).


Isabelle Gallagher
email: gallagher@math.jussieu.fr

Yannick

Yannick Sire
email: sire@cmi.univ-mrs.fr

BESOV ALGEBRAS ON LIE GROUPS OF POLYNOMIAL GROWTH

published or not. The documents may come

Besov algebras on Lie groups of polynomial growth

1. Introduction 1.1. Lie groups of polynomial growth. In this paper G is an unimodular connected Lie group endowed with the Haar measure. By "unimodular" we mean that the Haar measure is left and right-invariant. Denoting by G the Lie algebra of G, we consider a family X = {X 1 , ..., X k } of left-invariant vector fields on G satisfying the Hörmander condition, i.e. G is the Lie algebra generated by the X ′ i s. In the following, although not stated, all the functional spaces depend on the field X. A standard metric on G, called the Carnot-Caratheodory metric, is naturally associated with X and is defined as follows: let ℓ : [0, 1] → G be an absolutely continuous path. We say that ℓ is admissible if there exist measurable functions c 1 , ..., c k : [0, 1] → C such that, for almost every t ∈ [0, 1], one has ℓ ′ (t) = . For all x, y ∈ G, define d(x, y) as the infimum of the lengths of all admissible paths joining x to y (such a curve exists by the Hörmander condition). This distance is left-invariant. For short, we denote by |x| the distance between e, the neutral element of the group, and x so that the distance from x to y is equal to |y -1 x|. For all r > 0, denote by B(x, r) the open ball in G with respect to the Carnot-Caratheodory distance and by V (r) the Haar measure of any ball. There exists d ∈ N * (called the local dimension of (G, X)) and 0 < c < C such that, for all r ∈]0, 1[, cr d ≤ V (r) ≤ Cr d , see [START_REF] Nagel | Balls and metrics defined by vector fields i: Basic properties[END_REF]. When r > 1, two situations may occur (see [START_REF] Guivarc | h. Croissance polynomiale et période des fonctions harmoniques[END_REF]):

• Either there exist c, C, D > 0 such that, for all r > 1, cr D ≤ V (r) ≤ Cr D where D is called the dimension at infinity of the group (note that, contrary to d, D does not depend on X). The group is said to have polynomial volume growth.

The first author is partially supported by the ANR project ANR-08-BLAN-0301-01 "Mathocéan", as well as by the Institut Universitaire de France. The second author is supported by the ANR project "PREFERED".

• Or there exist c 1 , c 2 , C 1 , C 2 > 0 such that, for all r > 1, c 1 e c 2 r ≤ V (r) ≤ C 1 e C 2 r and the group is said to have exponential volume growth. When G has polynomial volume growth, it is plain to see that there exists a constant C > 0 such that for all r > 0, V (2r) ≤ CV (r). In turn this implies that there exist C > 0 and κ > 0 such that for all r > 0 and all θ > 1, V (θr) ≤ Cθ κ V (r).

We denote ∆

G = k i=1 X 2
i the sub-laplacian on G.

1.2. Nilpotent Lie groups. A Lie group is said to be nilpotent if its Lie algebra G is nilpotent: more precisely writing G 1 = G and defining inductively

G k+1 = [G k , G k ],
then there is n such that G n = {0}. It can be shown that such groups are always of polynomial growth (see for instance [START_REF] Dungey | Analysis on Lie groups with polynomial growth[END_REF]).

1.3. Stratified (Carnot) and H-type groups. Stratified groups are a particular version of nilpotent groups, which admit a stratified structure and for which V (r) ∼ r Q for some positive Q, for all r > 0. One advantage of this additional structure is that such groups admit dilations. Important examples of such groups are H-type groups, a particular example being the Heisenberg group. More precisely, a stratified (or Carnot) Lie group G is simply connected and its Lie algebra admits a stratification, i.e. there exist linear subspaces

V 1 , ..., V r of G such that G = V 1 ⊕ ... ⊕ V r which satisfy [V 1 , V i ] = V i+1 for i = 1, ..., r -1 and [V 1 , V r ] = 0. By [V 1 , V i ] we mean the subspace of G generated by the elements [X, Y ] where X ∈ V 1 and Y ∈ V i .
Carnot groups are nilpotent. Furthermore, via the exponential map, G and G can be identified as manifolds. The dilations γ δ (δ > 0) are then defined (on the Lie algebra level) by

γ δ (x 1 + ... + x r ) = δx 1 + δ 2 x 2 + ... + δ r x r , x i ∈ V i .
We define the homogeneous dimension

Q = dimV 1 + 2dimV 2 + • • • + rdimV r .
If G is a Carnot group, we have for all r > 0, V (r) ∼ r Q (see [START_REF] Folland | Hardy spaces on homogeneous groups[END_REF]). We shall say that the Q-dimensional Carnot group is of step r: for instance the Heisenberg group H d is a Carnot group and

Q = 2d + 2.
The previous abstract definition of Carnot groups is not always very practical. It is however possible to prove (see [START_REF] Bonfiglioli | Nonlinear Liouville theorems for some critical problems on H-type groups[END_REF]) that any N-dimensional Carnot group of step 2 with m generators is isomorphic to (R N , •) with the law given by ( 1) , x (2) ) • (y (1) , y (2) ) = x

N = m + n, x (1) ∈ R m , x (2) ∈ R n ) (x (
(1) 1) , U (j) y (1) , j = 1, ..., n , where U (j) are m × m linearly independent skew-symmetric matrices. With this at hand, one can give the definition of a group of Heisenberg-type (Htype henceforth). These groups are two-step stratified nilpotent Lie groups whose Lie algebra carries a suitably compatible inner product, see [START_REF] Kaplan | Fundamental solutions for a class of hypoelliptic PDE generated by composition of quadratic forms[END_REF]. One of these groups is the nilpotent Iwasawa subgroup of semi-simple Lie groups of split rank one (see [START_REF] Korányi | Geometric properties of Heisenberg-type groups[END_REF]). More precisely, an H-type group is a Carnot group of step 2 with the following property: the Lie algebra G of G is endowed with an inner product •, • such that if Z is the center of G, then [Z ⊥ , Z ⊥ ] = Z and moreover for every z ∈ Z, the map J z : Z ⊥ → Z ⊥ defined by J z (v), w = z, [v, w] for every w ∈ Z ⊥ is an orthogonal map whenever z, z = 1. If m = dim (Z ⊥ ) and n = dim (Z), then any H-type group is canonically isomorphic to R m+n with the above group law, where the matrices U (j) satisfy the additional property U (r) U (s) + U (s) U (r) = 0 for every r, s ∈ {1, ..., n} with r = s. Whenever the center of the group is one-dimensional, the group is canonically isomorphic to the Heisenberg group on R m+1 . We shall always identify Z ⊥ with C ℓ with 2ℓ = m and Z to R n thanks to the discussion above. Note that the homogeneous dimension of a Htype group so defined is Q = 2ℓ + n. On an H-type group G, the vector-fields in the algebra G are given by

j + y (1) j , j = 1, ..., m x (2) j + y (2) j + 1 2 x (
X j = ∂ ∂x j + 1 2 n k=1 2ℓ l=1 z l U (k) l,j ∂ ∂t k and Y j = ∂ ∂y j + 1 2 n k=1 2ℓ l=1 z l U (k) l,j+ℓ ∂ ∂t k for j = 1, ..., ℓ, z = (x, y) ∈ R 2ℓ and t ∈ R n .
In the following we shall denote by X any element of the family (X 1 , . . . , X ℓ , Y 1 , . . . , Y ℓ ). The hypo-elliptic Kohn Laplacian on H-type groups writes

∆ G = m j=1 ∂ 2 ∂x 2 j + 1 4 |x| 2 n s=1 ∂ 2 ∂t 2 s + n s=1 m i,j=1 x i U (s) ij ∂ 2 ∂t s ∂x j • 1.4.
Main results and structure of the paper. In [START_REF] Coulhon | Sobolev algebras on Lie groups and Riemannian manifolds[END_REF], the authors investigate the algebra properties of the Bessel space

L p α (G) = f ∈ L p (G), (-∆ G ) α 2 f ∈ L p (G)
and their homogeneous counterpart, where G is any unimodular Lie group.

Our first theorem considers Besov spaces in the general setting of groups with polynomial volume growth. The case s ∈ (0, 1) is obtained, both for inhomogeneous and homogeneous spaces, by using an equivalent definition in terms of differences (see [START_REF] Saloff-Coste | Analyse sur les groupes de Lie à croissance polynômiale[END_REF]). The general case is only proved in the case of inhomogeneous spaces and uses the fact that local Riesz transforms are continuous in L p for 1 < p < ∞ (whence the restriction on p below), along with an interpolation argument to obtain all values of s.

Theorem 1. Let G be a Lie group with polynomial volume growth.

For every s ∈ (0, 1) and 1 ≤ p, q ≤ ∞, the spaces B s p,q (G)∩L ∞ (G) and Ḃs p,q (G)∩L ∞ (G) are algebras under pointwise multiplication. The same property holds if s ≥ 1 for B s p,q (G) ∩ L ∞ (G), with the additional restriction that 1 < p < ∞.

Remark 1.1. We shall give a generalization of Theorem 1 to the case when the space L ∞ (G) is replaced by L r (G) (see Propositions 3.3 and 3.4).

One can recover the full range of indexes p, as well as homogeneous Besov spaces, in the context of H-type groups thanks to the paraproduct algorithm. Before stating the result let us give an intermediate statement in the case of nilpotent groups. Its proof requires the continuity of Riesz transforms, as well as a result which is to our knowledge new even in the context of the Heisenberg group (see Proposition 4.1) and which links Ḃs p,q (G) and Ḃs+1 p,q (G) in terms of the action of X i and not only powers of the sublaplacian.

Theorem 2. Let G be a nilpotent Lie group.

For every 1 ≤ s < d, the space Ḃs d s ,1 (G) is embedded in L ∞ (G) and is an algebra. Moreover for every 1 < s and every 1 < p < ∞, if f and g belong to the space Ḃs

p, s-1 s ∩ L ∞ (G) then f g belongs to Ḃs p,1 ∩ L ∞ (G). Finally if 1 < p 1 , p 2 < ∞ with 1/p = 1/p 1 + 1/p 2 , if 1 ≤ q ≤ ∞
, and if f belongs to Ḃs p 1 ,q ∩ L p 1 (G) and g belongs to Ḃs p 2 ,q ∩ L p 2 (G) then f g ∈ Ḃs p,q ∩ L p (G), for any s > 0.

Remark 1.2. Unfortunately we are unable to recover, in the case of nilpotent groups, the full algebra property due to the (technical) fact that Besov spaces do not interpolate very well when the integrability indexes are different. The second property in Theorem 2 is almost an algebra property, except for a loss in the third (summation) index. As to the last property this time the integrability index is changed in the product. The reason for those losses will appear clearly in the proof of the theorem.

Finally, in the context of H-type groups, thanks to paraproduct techniques, one can enlarge the range of admissible spaces and prove the following result.

Theorem 3. Let G be an H-type group. For every s > 0 and 1 ≤ p, q ≤ ∞, the spaces B s p,q (G) ∩ L ∞ (G) and Ḃs p,q (G) ∩ L ∞ (G) are algebras under pointwise multiplication.

Besov spaces are defined in the coming section, and Theorems 1 and 2 are proved in Sections 3 and 4 respectively. We present the proof of Theorem 3 in Section 5.

We shall write A B if there is a universal constant C such that A ≤ CB. Similarly we shall write A ∼ B if A B and B A.

Acknowledgements. The authors are very grateful to L. Saloff-Coste for comments on a previous version of this text. They also thank the anonymous referee for questions and suggestions which improved the presentation.

Littlewood-Paley decomposition on groups of polynomial growth, and Besov spaces

This section is devoted to a presentation of the Littlewood-Paley decomposition on groups of polynomial growth, together with some standard applications. A general approach to the Littlewood-Paley decomposition on Lie groups with polynomial growth is investigated in [START_REF] Furioli | Littlewood-Paley decompositions and Besov spaces on Lie groups of polynomial growth[END_REF]. We also refer to [START_REF] Bahouri | Estimations de Strichartz généralisées sur le groupe de Heisenberg[END_REF] or [START_REF] Bahouri | Paraproduit sur le groupe de Heisenberg et applications[END_REF] for the case of the Heisenberg group. We recall here the construction of the homogeneous and inhomogeneous decompositions. For details and proofs of the results presented in this section we refer to [START_REF] Chamorro | Inégalités de Gagliardo-Nirenberg précisées sur le groupe de Heisenberg[END_REF], [START_REF] Furioli | Littlewood-Paley decompositions and Besov spaces on Lie groups of polynomial growth[END_REF] and [START_REF] Hulanicki | A functional calculus for Rockland operators on nilpotent Lie groups estimates[END_REF].

2.1. Littlewood-Paley decomposition. We first review the dyadic decomposition constructed in [START_REF] Furioli | Littlewood-Paley decompositions and Besov spaces on Lie groups of polynomial growth[END_REF]. Let χ ∈ C ∞ (R) be an even function such that 0 ≤ χ ≤ 1 and χ

= 1 on [0, 1/4], χ = 0 on [1, ∞[. Define ψ(x) = χ(x/4) -χ(x), so that the support of ψ is included in [-4, -1/4] ∪ [1/4, 4]. The following holds: ∀τ ∈ R * , j∈Z ψ(2 -2j τ ) = 1 and χ(τ ) + j≥0 ψ(2 -2j τ ) = 1, ∀τ ∈ R.
Introduce the spectral decomposition of the hypo-elliptic Laplacian

-∆ G = ∞ 0 λdE λ .
Then we have

χ(-∆ G ) = ∞ 0 χ(λ)dE λ and ψ(-2 -2j ∆ G ) = ∞ 0 ψ(2 -2j λ)dE λ .
We then define for j ∈ N the operators [START_REF] Furioli | Littlewood-Paley decompositions and Besov spaces on Lie groups of polynomial growth[END_REF]). Let G be a Lie group with polynomial growth and p ∈ (1, ∞). Then u belongs to L p (G) if and only if S 0 u and

S 0 f = χ(-∆ G )f and ∆ j f = ψ(-2 -2j ∆ G )f. The homogeneous Littlewood-Paley decomposition of f in S ′ (G) is f = j∈Z ∆ j f, while the inhomogeneous one is f = S 0 f + ∞ j=0 ∆ j f. Theorem 4 ([
∞ j=0 |∆ j u| 2 belong to L p (G). More- over, we have u L p (G) ∼ S 0 u L p (G) + ∞ j=0 |∆ j u| 2 1 2 L p (G)
.

In the following we shall denote by Ψ j the kernel of the operator ψ(-2 -2j ∆ G ). One can show that Ψ j is mean free (see Corollary 5.1 of [START_REF] Chamorro | Improved Sobolev Inequalities and Muckenhoupt weights on stratified Lie groups[END_REF] for Carnot groups, and Theorem 7.1.2 of [START_REF] Chamorro | Inégalités de Gagliardo-Nirenberg précisées sur le groupe de Heisenberg[END_REF] for an extension to groups of polynomial growth). In the context of Carnot groups, Ψ j satisfies the dilation property

Ψ j (x) = 2 Qj Ψ 0 (2 j x).
In the more general context of groups of polynomial growth, this does not hold but one has nevertheless the following important estimates: let α ∈ N and I ∈ β∈N {1, . . . , k} β be given, as well as p ∈ [1, ∞]. The following result is due to [START_REF] Furioli | Littlewood-Paley decompositions and Besov spaces on Lie groups of polynomial growth[END_REF]:

(2.1) ∀j ≥ 0, (1 + | • |) α X I Ψ j L p (G) 2 j( d p ′ +|I|) , where 1/p + 1/p ′ = 1. We have denoted X I = X i 1 . . . X i β and |I| = β. Moreover as proved in [10], Theorem 7.1.2, one has (2.2) ∀j ∈ Z, X i Ψ j L 1 (G) 2 j .
Finally putting together classical estimates on the heat kernel (see [START_REF] Coulhon | Sobolev algebras on Lie groups and Riemannian manifolds[END_REF] or [START_REF] Th | Analysis and geometry on groups[END_REF] for instance) and the methods of [START_REF] Furioli | Littlewood-Paley decompositions and Besov spaces on Lie groups of polynomial growth[END_REF] allows to write that for any α ≥ 0,

(2.3) ∀j ∈ Z, | • | α Ψ j L 1 (G) 2 jα .
2.2. Besov spaces. As a standard application of the Littlewood-Paley decomposition, one can define (inhomogeneous) Besov spaces on Lie groups with polynomial volume growth in the following way: let s ∈ R and 1 ≤ p ≤ +∞ and 0 < q ≤ ∞, then B s p,q (G) is the space

f ∈ S ′ (G), f B s p,q (G) = S 0 f L p (G) + ∞ j=0 (2 js ∆ j f L p (G) ) q 1/q < ∞
with the obvious adaptation if q = ∞. When p = q = 2 one recovers the usual Sobolev spaces (see for instance [START_REF] Bahouri | Paraproduit sur le groupe de Heisenberg et applications[END_REF] for a proof in the case of the Heisenberg group). Note that when s > 0 one sees easily that S 0 f L p (G) may be replaced by f L p (G) . Using Bernstein inequalities (Proposition 4.2 of [START_REF] Furioli | Littlewood-Paley decompositions and Besov spaces on Lie groups of polynomial growth[END_REF]) one gets immediately that if 0 < s then (2.4)

p 1 ≤ p 2 =⇒ B s+ d p 1 -d p 2 p 1 ,q ∩ L p 2 ֒→ B s p 2 ,q ∩ L p 1
where recall that d is the local dimension of G.

One can also define the homogeneous counterpart of the above norm:

f Ḃs p,q (G) = j∈Z (2 js ∆ j f L p (G) ) q 1/q
but proving that this does provide a (quasi)-Banach space is not an easy matter, and is actually not true in general, even in the euclidean case (see for instance [2, Chapter 2] for comments on that subject). To recover a Banach space in the context of Carnot groups, the homogeneous space Ḃs p,q (G) can be defined as the set of functions in S ′ (G) modulo polynomials, such that the above norm is finite (see [START_REF] Fuhr | Homogeneous Besov spaces on stratified Lie groups and their wavelet characterization[END_REF]). In the present study however this will not be an issue, even if the group is not stratified: we define Ḃs p,q (G) as the completion for the above norm of the set of smooth functions such that ∆ j f → 0 as j → -∞, and we shall always be considering the intersection of Ḃs p,q (G) with a Banach space (such as L ∞ ).

Note that Bernstein inequalities imply as in (2.4) that

p 1 ≤ p 2 =⇒ Ḃs+ d p 1 -d p 2 p 1 ,q ֒→ Ḃs p 2 ,q .
Besov spaces are often rather defined using the heat flow (the advantage being that it does not require the Littlewood-Paley machinery). In [START_REF] Furioli | Littlewood-Paley decompositions and Besov spaces on Lie groups of polynomial growth[END_REF], the authors prove that if s ∈ R, then f ∈ B s p,q (G) is equivalent to: for all t > 0, the function e t∆ G f belongs to L p (G) and (2.5)

1 0 t -sq/2 (t(-∆ G )) m/2 e t∆ G f q L p (G) dt t 1/q < ∞
for m ≥ 0 greater than s. We shall not be using this characterization here.

3. Proof of Theorem 1 3.1. The case s ∈ (0, 1). We start by dealing with the case s ∈ (0, 1), and use an idea of [START_REF] Coulhon | Sobolev algebras on Lie groups and Riemannian manifolds[END_REF] which consists in representing the norm on Besov spaces by suitable functionals. More precisely, we introduce the following functional (note that it differs slightly from that used in [START_REF] Coulhon | Sobolev algebras on Lie groups and Riemannian manifolds[END_REF]), writing τ w f (w ′ ) = f (w ′ w):

S s,p f (w) = τ w f -f L p (G) |w| s • Proposition 3.1.
Let G be a Lie group with polynomial growth. Then for any s ∈ (0, 1) and p, q ∈ [1, +∞], we have

f B s p,q (G) ∼ f L p + S s,p f L q (G, 1 |y|≤1 dy V (|y|) )
and

f Ḃs p,q (G) ∼ S s,p f L q (G, dy V (|y|) )
. Once Proposition 3.1 is proved, the algebra property follows immediately in the case when s ∈ (0, 1). Indeed, let f, g belong to the space (B s p,q ∩ L ∞ )(G) for s ∈ (0, 1). It is easy to see that

(3.1) S s,p (f g) ≤ f L ∞ S s,p g + g L ∞ S s,p f,
hence the result using the equivalence in Proposition 3.1. The same holds in the homogeneous case.

Remark 3.2. One can extend (3.1) to the following, with 1/a i + 1/b i = 1/p:

S s,p (f g) ≤ f L a 1 S s,b 1 g + g L a 2 S s,b 2 f .
We now prove Proposition 3.1. Note that this result was already proved in [START_REF] Saloff-Coste | Analyse sur les groupes de Lie à croissance polynômiale[END_REF] using the characterization (2.5). We choose to present a proof using the Littlewood-Paley definition here, which is inspired by the proof of the euclidean case in [START_REF] Bahouri | Fourier Analysis and Nonlinear PDEs[END_REF] for instance. We need to prove that for s ∈ (0, 1)

j∈Z (2 js ∆ j f L p (G) ) q ∼ f q L p + G 1 |w|≤1 τ w f -f q L p (G) V (|w|)|w| sq dw
with the obvious modification if q = ∞. Compared to the euclidean case, one is missing the usual dilation property, which will be replaced by estimate (2.1). The classical proof also uses a Taylor expansion at order one, which we must adapt to our context in order to use only horizontal vector fields (which alone appear in (2.1)). Let us start by noticing that

f q L p ≤ j≤0 (2 js ∆ j f L p (G) ) q .
Next let us bound the quantity

τ w ∆ j f -∆ j f L p (G) . Recalling that ∆ j = |j ′ -j|≤1 ∆ j ∆ j ′ , we have τ w ∆ j f -∆ j f = |j ′ -j|≤1 ∆ j ′ f ⋆ τ w Ψ j -Ψ j ,
where Ψ j is the kernel associated with ψ(2 -2j ∆ G ). It follows by Young's inequality that

τ w ∆ j f -∆ j f L p (G) ≤ |j ′ -j|≤1 ∆ j ′ f L p τ w Ψ j -Ψ j L 1 .
Now let us estimate τ w Ψ j -Ψ j L 1 . We have

(τ w Ψ j -Ψ j )(x) = 1 0 d ds Ψ j (xϕ(s)) ds = k ℓ=1 1 0 c ℓ (s)(X ℓ (xϕ(s))Ψ j )(xϕ(s)) ds,
where ϕ is an admissible path linking e to w. It follows that

τ w Ψ j -Ψ j L 1 ≤ G k ℓ=1 1 0 |c ℓ (s)| (X ℓ (xϕ(s))Ψ j )(xϕ(s)) dsdx ≤ k ℓ=1 1 0 |c ℓ (s)| ds X ℓ Ψ j L 1
τ w Ψ j -Ψ j L 1 2 j |w|.
This implies that there is a sequence (c j ) in the unit ball of ℓ q such that

(3.2) ∀j ∈ N, τ w ∆ j f -∆ j f L p (G) c j |w|2 j(1-s) f B s p,q (
G) . On the other hand one has of course

(3.3) τ w ∆ j f -∆ j f L p (G) c j 2 -js f B s p,q (G) . Now let us define j w ∈ Z such that 1 |w| ≤ 2 jw ≤ 2 |w| • Then using (3.
2) for low frequencies and (3.3) for high frequencies allows to write

τ w f -f L p (G) f B s p,q (G) j≤jw c j 2 j(1-s) |w| + j>jw c j 2 -js .
Let us first consider the case q = ∞. Then one finds directly that

τ w f -f L p (G) |w| s f B s p,q (G)
which proves one side of the equivalence. The case q < ∞ is slightly more technical but is very close to the euclidean case. We include it here for sake of completeness.

We have that

τ w f -f L p |w| s q L q (G, 1 |w|≤1 V (|w|) )
2 q f q Ḃs p,q

(I 1 + I 2 )
where

I 1 = G 1 |w|≤1
j≤jw c j 2 j(1-s) q |w| q(1-s) dw V (|w|) and

I 2 = G 1 |w|≤1 j>jw c j 2 -js q |w| -qs dw V (|w|) •
Hölder's inequality with the weight 2 j(1-s) and the definition of j w imply j≤jw c j 2 j(1-s) q |w| -(1-s)(q-1) j≤jw c q j 2 j(1-s) .

By Fubini's theorem, we deduce that

I 1 j∈N B(0,2 -j+1 ) |w| 1-s dw V (|w|) 2 j(1-s) c q j 1,
since (c j ) ℓ q ≤ 1. The estimate on I 2 is very similar. Note that it is crucial here that s ∈ (0, 1). The converse inequality is easy to prove and only depends on the fact that the mean value of Ψ j is zero. We write indeed

∆ j f (w) = τ v f (w)Ψ j (v) dv = (τ v f (w) -f (w))Ψ j (v) dv so that 2 js ∆ j f L p ≤ sup v∈G τ v f -f L p |v| s 2 js |v| s |Ψ j (v)| dv.
Then (2.3) implies that

2 js ∆ j f L p ≤ sup v∈G τ v f -f L p |v| s • Since sup |v|≥1 τ v f -f L p |v| s ≤ 2 f L p , we get finally sup j∈Z 2 js ∆ j f L p ≤ sup |v|≤1 τ v f -f L p |v| s + 2 f L p ,
the result follows in the case q = ∞. The case q < ∞ is similar though a little more technical, as above.

The homogeneous case is dealt with in a similar fashion. We leave the details to the reader. This proves Proposition 3.1.

Using Remark 3.2, the same proof provides the following result, which will be useful in the next section.

Proposition 3.3. Let G be a Lie group with polynomial volume growth. For every 0 < s < 1 and 1 ≤ p, q ≤ ∞ one has, writing

1/p = 1/a i + 1/b i f g B s p,q ≤ f L a 1 g B s b 1 ,q + g L a 2 f B s b 2 ,q and f g Ḃs p,q ≤ f L a 1 g Ḃs b 1 ,q + g L a 2 f Ḃs b 2 ,q .
3.2. The case s ≥ 1 (inhomogeneous spaces). We shall first deal with the case when s is not an integer. We use the well-known fact that the "local Riesz transforms" (Id

-∆ G ) m-1 2 X i (Id -∆ G ) -m
2 are bounded over L p (G) for 1 < p < ∞ (see for instance [START_REF] Dungey | Analysis on Lie groups with polynomial growth[END_REF]). This implies easily (see the next section where the same result is proved in the more difficult homogeneous case) that

f ∈ B s+1 p,q ⇐⇒ f ∈ B s p,q
and X i f ∈ B s p,q ∀i = 1, . . . , k.

We can then follow the lines of [START_REF] Coulhon | Sobolev algebras on Lie groups and Riemannian manifolds[END_REF], by writing f g

B s+1 p,q ∼ f g B s p,q + k i=1 X i (f g) B s p,q
and by arguing by induction: let us detail the case s = 1 + s ′ with 0 < s ′ < 1. On the one hand we know that for all 1 ≤ p, q ≤ ∞ and if 1/a i + 1/b i = 1/p,

f g B s ′ p,q f L a 1 g B s ′ b 1 ,q + g L a 2 f B s ′ b 2 ,q
.

Then we write, by the Leibniz rule,

X i (f g) B s ′ p,q ≤ f X i g B s ′ p,q + gX i f B s ′ p,q
and we have, by Proposition 3.3,

(3.4) f X i g B s ′ p,q f L a 1 X i g B s ′ b 1 ,q + f B s ′ a 2 ,q X i g L b 2 .
The estimate on gX i f in B s ′ p,q is similar so we shall not write the details for that term. The first term on the right-hand side of (3.4) is very easy to estimate since

X i g B s ′ b 1 ,q g B s b 1 ,q
. So let us turn to the second term. Let us first estimate f in B s ′ a 2 ,q . We have clearly, since

s ′ ≤ s, f B s ′ a 2 ,q f B s a 2 ,q . Now let us turn to the estimate of X i g in L b 2 , choosing 1 < b 2 < ∞. We use the fact that X i g L b 2 X i (Id -∆) -1 2 (Id -∆) 1 2 g L b 2 (Id -∆) 1 2 g L b 2 by the continuity of the local Riesz transforms. Since [∆ j , ∆ G ] = 0, Bernstein's lemma (see Proposition 4.3 of [18]) implies ∆ j (Id -∆) 1 2 g L b 2 2 j ∆ j g L b 2 .
This implies that (Id -∆)

1 2 g L b 2 S 0 (Id -∆) 1 2 g L b 2 + j≥0 2 j ∆ j g L b 2 ≤ g L b 2 + j≥0 2 js ∆ j g L b 2 2 j(1-s) g B s b 2 ,q
since s > 1. This gives the required estimate for the second term in (3.4) and that allows to conclude the proof of Theorem 1 in the case when s ∈ R + \ N. The general case s > 0 is then obtained by interpolation: we recall indeed that the following complex interpolation is true (see [START_REF] Bergh | Interpolation Spaces, an Introduction[END_REF]Theorem 6.4.5], whose proof only relies on the dyadic decomposition and may be easily adapted to our situation)

(3.5) [B 1-ε p,q ; B 1+ε p,q ] 1 2 = B 1 p,q .
The multilinear interpolation result of [START_REF] Bergh | Interpolation Spaces, an Introduction[END_REF]Theorem 4.4.1] provides the case s = 1 and the other integer cases are obtained similarly. Note that the above proof actually gives the following result. Proposition 3.4. Let G be a Lie group with polynomial volume growth. For every s ≥ 1, 1 ≤ q ≤ ∞ and 1 < p < ∞ one has, writing

1/p = 1/a i + 1/b i and choosing 1 < a i , b i < ∞, f g B s p,q ≤ f L a 1 g B s b 1 ,q + g L a 2 f B s b 2 ,q .

Proof of Theorem 2

As in the previous case the idea is to argue by induction for the noninteger values of s, and then by interpolation. To do so, we need the following result, which is new to our knowledge, even in the context of the Heisenberg group. Proposition 4.1. Let G be a nilpotent Lie group and let s > 0 and p ∈ (1, ∞) be given. Then f ∈ Ḃs+1 p,q (G) if and only if for all i = 1, ..., k, we have X i f ∈ Ḃs p,q (G). Proof. On the one hand we need to prove that for all i = 1, . . . , k and j ∈ N,

∆ j X k f L p 2 j ∆ j f L p .
Using Bernstein's inequality and by density of polynomials in the space of continuous functions it is then actually enough to prove that for all integers m,

(4.6) (-∆ G ) m 2 (-∆ G ) -1 2 X k f L p (-∆ G ) m 2 f L p .
Indeed if (4.6) holds, then one also has, multiplying both sides of the inequality by 2 jm , (-

2 2j ∆ G ) m 2 (-∆ G ) -1 2 X k f L p (-2 2j ∆ G )
m 2 f L p so for smooth compactly supported function ϕ, we get by functional calculus (4.7). So let us prove (4.6). Actually according to [START_REF] Lohoué | Remarques sur les transformées de Riesz sur les groupes de Lie nilpotents[END_REF] the operator

(4.7) ϕ(-2 2j ∆ G )(-∆ G ) -1 2 X k f L p ϕ(-2 2j ∆ G )f L p But recalling that ∆ j = ψ(-2 2j ∆ G ) we have ∆ j X k f L p = ψ(-2 2j ∆ G )X k f L p = ψ(-2 2j ∆ G )(-∆ G ) 1 2 (-∆ G ) -1 2 X k f L p . Then we can write ψ(-2 2j ∆ G )(-∆ G ) 1 2 (-∆ G ) -1 2 X k f L p 2 j ψ(-2 2j ∆ G )(-∆ G ) -1 2 X k f L p 2 j ∆ j f L p due to Bernstein's inequality ∆ j (-∆ G ) 1 2 f L p 2 j ∆ j f L p along with
L k m = (-∆ G ) m-1 2 X k (-∆ G ) -m 2 is bounded over L p (G) for 1 < p < ∞.
That property is false if the group is not nilpotent (see for instance [START_REF] Sire | An application of homogenization theory to harmonic analysis: Harnack inequalities and Riesz transforms on Lie groups of polynomial growth[END_REF]) so it is here that the assumption that G is nilpotent is used. So writing

(-∆ G ) m 2 (-∆ G ) -1 2 X k f = (-∆ G ) m 2 (-∆ G ) -1 2 X k (-∆ G ) -m 2 (-∆ G ) m 2 f = L k m (-∆ G ) m 2
f, the result follows. On the other hand, using again the fact that polynomials are dense in the space of continuous functions, we also need to check that for all f ,

(-∆ G ) m+1 2 f L p sup k (-∆ G ) m 2 X k f L p .
To prove that we simply use again the fact that L k m is bounded over L p (G) for every index 1 < p < ∞. Indeed we can write

(-∆ G ) m+1 2 f L p ≤ k (-∆ G ) m-1 2 X 2 k f L p = k (-∆ G ) m-1 2 X k (-∆ G ) -m 2 (-∆ G ) m 2 X k f L p = k L k m (-∆ G ) m 2 X k f L p
whence the result. Proposition 4.1 is proved.

Proposition 4.1 allows to obtain rather easily Theorem 2 when s ∈ R + \ N, using also Proposition 3.3. Let us give the details.

The fact that Ḃs

d s ,1 (G) is embedded in L ∞ (G)
simply follows from the easy calculations:

f L ∞ (G) ≤ j∈Z ∆ j f L ∞ (G) j∈Z 2 js ∆ j f L d s (G)
by the Bernstein inequality (Proposition 4.2 of [START_REF] Furioli | Littlewood-Paley decompositions and Besov spaces on Lie groups of polynomial growth[END_REF]). Now let us prove that Ḃs d s ,1 (G) is an algebra, and then let us prove that for every 1 < s and every 1 < p < ∞, if f and g belong to Ḃs

p, s-1 s ∩ L ∞ (G) then f g belongs to Ḃs p,1 ∩ L ∞ (G).
We follow the lines of the inhomogeneous case treated above, but we need to be careful that the norms are now homogeneous. Let us define s = 1 + s ′ with s ′ ∈ (0, 1). We write as in the inhomogeneous case, by the Leibniz rule,

X i (f g) Ḃs ′ p,q ≤ f X i g Ḃs ′ p,q + gX i f Ḃs ′ p,q
and study more particularly the first term on the right-hand side, which satisfies due to Proposition 3.3, for 1/a i + 1/b i = 1/p (and choosing from now on 1 < a i , b i < ∞),

(4.8) f X i g Ḃs ′ p,q f L a 1 X i g Ḃs ′ b 1 ,q + f Ḃs ′ a 2 ,q X i g L b 2 . On the one hand X i g Ḃs ′ b 1 ,q g Ḃs b 1 ,q
, so it suffices to estimate f Ḃs ′ a 2 ,q X i g L b 2 . In the case when q = 1 and p = d/s we choose a 2 = d/(s -1) = d/s ′ and b 2 = d and use Bernstein's inequality which states that

f Ḃs ′ a 2 ,1 f Ḃs d s ,1 .
Then according to [START_REF] Coulhon | Sobolev algebras on Lie groups and Riemannian manifolds[END_REF] we have

X i g L d (-∆ G ) s 2 g 1 s L d s g 1-1 s L ∞ and moreover (-∆ G ) s 2 g L d s ≤ j∈Z ∆ j (-∆ G ) s 2 g L d s j∈Z 2 js ∆ j g L d s g Ḃs d s ,1
by Bernstein's inequality, so we infer that

X i g L d g 1 s Ḃs d s ,1 g 1-1 s L ∞
and the result follows in the case 1 < s < 2. The other noninteger cases are obtained by induction. To prove the result in the integer case we use a nonlinear interpolation argument as in the inhomogeneous case above: let us detail the case s = 1 for instance.

We have indeed (as pointed out in [START_REF] Bergh | Interpolation Spaces, an Introduction[END_REF], the interpolation results hold in the homogeneous case):

[ Ḃ1-ε d 1-ε ,1 , Ḃ1+ε d 1+ε ,1 ] 1 2 = Ḃ1 d,1
so the result follows. The other cases are obtained similarly.

In the case when f and g belong to Ḃs p, s-1 s then we use as above the fact that

f L a 1 X i g Ḃs ′ b 1 ,q f L a 1 g Ḃs b 1 ,q
, and in particular we can take a 1 = ∞ and b 1 = p, and we choose a 2 = ps/(s -1) and b 2 = ps. Then Hölder's inequality gives

2 js ′ ∆ j f L ps s-1 2 js ′ ∆ j f s-1 s L p ∆ j f 1 s L ∞ 2 js ∆ j f L p s-1 s f 1 s L ∞ . Since as above X i g L ps (-∆ G ) s 2 g 1 s L p g 1-1 s L ∞ g 1 s Ḃs p,1 g 1-1 s L ∞
the result follows in the case 1 < s with s non integer. Let us detail for instance how to recover the case s = 2: we write for instance

[ Ḃ 9 4 p, 5 9 , Ḃ 9 5 p, 4 9 ] 5 9

= Ḃ2

p, 1 2 so the result follows by bilinear interpolation. The other cases are obtained similarly.

Finally let us turn to the last statement of the theorem, namely the fact that if 1 < p 1 , p 2 < ∞ with 1/p = 1/p 1 + 1/p 2 , if 1 ≤ q ≤ ∞, and if f belongs to Ḃs p 1 ,q ∩ L p 1 (G) and g belongs to Ḃs p 2 ,q ∩L p 2 (G) then f g ∈ Ḃs p,q ∩L p (G). We recall the real interpolation result [7, Theorem 6.4.5], which holds also in the homogeneous case as indicated in [START_REF] Bergh | Interpolation Spaces, an Introduction[END_REF], according to which (4.9)

[ Ḃs 1 p,q 1 ; Ḃs 2 p,q 2 ] θ,r = Ḃs p,r , s = θs 1 + (1 -θ)s 2 , s 1 = s 2 so in particular [ Ḃ0 a 2 ,∞ ; Ḃs a 2 ,q ] s-1 s ,1 = Ḃs ′ a 2 ,1 ֒→ Ḃs ′ a 2 ,q .

We infer that f belongs to Ḃs ′ a 2 ,q as soon as f belongs to

L a 2 ∩ Ḃs a 2 ,q ֒→ Ḃ0 a 2 ,∞ ∩ Ḃs a 2 ,q . Similarly [ Ḃ0 b 2 ,∞ ; Ḃs b 2 ,q ] 1 s ,1 = Ḃ1 b 2 ,1
so using the fact that

X i g L b 2 X i g Ḃ0 b 2 ,1 g Ḃ1 b 2 ,1
we infer that X i g belongs to L b 2 as soon as

g ∈ L b 2 ∩ Ḃs b 2 ,q ⊂ Ḃ0 b 2 ,∞ ∩ Ḃs b 2 ,q
. The result follows for 1 < s < 2, and the theorem is proved by an easy induction, and interpolation as in the inhomogeneous case.

Paradifferential calculus on H-type groups

In this section, we describe several topics related to harmonic analysis on H-type groups, which we recall are particular cases of Carnot groups where it turns out that an explicit Fourier transform is available.

5.1. Fourier transforms. In order to construct para-differential and pseudo-differential calculus on H-type groups, one needs to introduce a suitable Fourier transform. This is classically done through the infinite-dimensional unitary irreducible representations on a suitable Hilbert space since H-type groups are non commutative. Two representations are available: the Bargmann representation (see [START_REF] Kaplan | Harmonic analysis on groups of Heisenberg type[END_REF] for instance) and the Schrödinger representation (see [START_REF] Corwin | Representations of nilpotent Lie groups and their applications[END_REF] for instance). 5.1.1. General definitions. Let us define generally what a Fourier transform is on non commutative groups. Consider a Hilbert space H λ (K ℓ ) of functions defined on some field K (which can be R or C). The irreducible unitary representations π λ : G → H λ (K ℓ ) are parametrized by λ ∈ R n \ {0} where n is the dimension of the center of the group. We have then the following definition. Definition 5.1. We define the Fourier transform on G by the following formula: let f ∈ L 1 (G). Then the Fourier transform of f is the operator on H λ (K ℓ ) parametrized by λ ∈ R n \ {0} defined by

F (f )(λ) = G f (z, t)π λ (z, t) dz dt. Note that one has F (f ⋆ g)(λ) = F (f )(λ) • F (g)(λ). Let F α,λ , α ∈ N ℓ be a Hilbert basis of H λ (K ℓ ). We recall that an operator A(λ) of H λ such that α∈N ℓ |(A(λ)F α,λ , F α,λ ) H λ | < +∞
α∈N d R n F (f )(λ)F α,λ H λ |λ| ℓ dλ < ∞ then we have for almost every w, f (w) = 2 ℓ-1 π ℓ+1 R n tr π λ (w -1 )F (f )(λ) |λ| ℓ dλ.
Following [START_REF] Yang | The heat kernel on H-type groups[END_REF], the representation π λ on G determines a representation π * λ on its Lie algebra G on the space of C ∞ vectors. The representation π * λ is defined by π * λ (X)f = d dt π λ (exp(tX))f | t=0 for every X in the Lie algebra G. We can extend π * λ to the universal enveloping algebra of left-invariant differential operators on G. Let K be a left-invariant operator on G, then we have

K(π λ f, g) = (π λ π * λ (K)f, g)
where (•, •) stands for the H λ inner product. [START_REF] Dooley | Heisenberg-type groups and intertwining operators[END_REF])

(π λ (z, t)F )(ξ) = F (ξ -z)e i λ,t -|λ|(|z| 2 + z,ξ ) .
It is a well-known fact that the Fock space admits an orthonormal basis given by the

monomials F α,λ (ξ) = ( 2|λ| ξ) √ α ! α , α ∈ N ℓ .
A very important property for us is the following classical diagonalization result (see [START_REF] Kaplan | Harmonic analysis on groups of Heisenberg type[END_REF], or [START_REF] Bahouri | Phase-space analysis and pseudodifferential calculus on the Heisenberg group[END_REF] and the references therein).

Proposition 5.2. Let F B be the Fourier transform associated to the Bargmann representation π.The following diagonalization property holds: for every f ∈ S(G),

F B (∆ G f )(λ)F α,λ = -4|λ|(2|α| + ℓ)F B (f )(λ)F α,λ .
This allows to define the following formula, for every ρ ∈ R:

F B ((-∆ G ) ρ f )(λ)F α,λ = (4|λ|(2|α| + ℓ)) ρ F B (f )(λ)F α,λ .
5.1.3. The L 2 representation on H-type groups. Another useful representation is the so-called Schrödinger, or L 2 representation. In this case, the unitary irreducible representations are given on L 2 (R ℓ ) by, for λ ∈ R n (and writing z = (x, y)): (π λ (z, t)F )(ξ) = e i λ,t +|λ|i( ℓ j=1 x j ξ j + 1 2 x j y j ) F (ξ +y). The intertwining operator between the Bargmann and the L 2 representations is the Hermite-Weber transform K λ :

H λ → L 2 (R ℓ ) given by (K λ φ)(ξ) = C ℓ |λ| ℓ/4 e |λ| |ξ| 2 2 φ - 1 2|λ| ∂ ∂ξ e -|λ| |ξ| 2 ,
which is unitary and satisfies K λ π λ (z, t) = πλ (z, t)K λ . Following [START_REF] Yang | The heat kernel on H-type groups[END_REF] and the previous description, we have π * λ (X j ) = i|λ|ξ j and π * λ (Y j ) = ∂ ∂ξ j for j = 1, ..., ℓ, and similarly

for k = 1, ..., n, π * λ (∂ t k ) = iλ k . Therefore, we have π * λ (-∆ G ) = - n j=1 ∂ 2 ∂ξ 2 j + |λ| 2 |ξ| 2 .
Notice that this is a Hermite operator and the eigenfunctions of

π * λ (-∆ G ) are Φ λ α (ξ) = |λ| n/4 Φ α ( |λ|ξ), α = (α 1 , ..., α ℓ ) where Φ α (ξ) is the product ψ α 1 (ξ 1 )...ψ α ℓ (ξ ℓ ) and ψ α j (ξ j ) is the eigenfunction of -∂ 2 ∂ξ 2 j + ξ 2
j with eigenvalue 2α j + 1. This leads to the following formula, where |α| = α 1 + ...

+ α ℓ : π * λ (-∆ G )Φ λ α = (2|α| + ℓ)|λ|Φ λ α .
As a consequence, one has the following lemma.

Lemma 5.3. Let F S be the Fourier transform associated to the Schrödinger representation π.The following diagonalization property holds: for every f in S(G),

F S ((-∆ G )f )Φ λ α = (2|α| + ℓ)|λ|F S (f )Φ λ α . Proof. We have by definition F S ((-∆ G )f )Φ λ α = G (-∆ G )f (z, t)π λ (z, t)Φ λ α = G f (z, t)(-∆ G )π λ (z, t)Φ λ α .
Using the definition of the dual representation, we have

F S ((-∆ G )f )Φ λ α = G f (z, t)π λ (z, t)π * λ (-∆ G )Φ λ α
and using the properties of the Hermite operator, this gives the result.

A localization lemma.

As in [START_REF] Bahouri | Paraproduit sur le groupe de Heisenberg et applications[END_REF], one can prove a localization lemma (also called Bernstein Lemma), which we state here in the context of the Bargmann representation. The proof is omitted as it is identical to the Heisenberg situation treated in [START_REF] Bahouri | Paraproduit sur le groupe de Heisenberg et applications[END_REF]. Note that using Proposition 4.1, the last statement of the lemma could be extended to iterated vector fields X I . We denote C 0 the ring {τ ∈ R 1/2 ≤ |τ | ≤ 4} and by B 0 the ball {τ ∈ R |τ | ≤ 2}.

Lemma 5.4. Let p and q be two elements of [1, ∞], with p ≤ q, and let u ∈ S(G) satisfy for all α ∈ N ℓ , F B (u)(λ)F α,λ = 1 λ∈(2|α|+ℓ) -1 2 2j B 0 F B (u)(λ)F α,λ . Then we have

∀k ∈ N, sup |β|=k X β u L q (G) ≤ C k 2 N j( 1 p -1 q )+kj u L p (G) .
On the other hand, if F B (u)(λ)F α,λ = 1 λ∈(2|α|+ℓ) -1 2 2j C 0 , then for all ρ ∈ R,

C -1 ρ 2 -jρ (-∆ G ) ρ 2 u L p (G) ≤ u L p (G) ≤ C ρ 2 -jρ (-∆ G ) ρ 2 u L p (G) .

5.3.

Paraproduct on H-type groups. In order to develop a paraproduct on H-type groups, one has to prove that the product of two functions is localized in frequencies whenever the functions are localized. This is the object of the next lemma, whose proof is the same as that of Proposition 4.2 of [START_REF] Bahouri | Paraproduit sur le groupe de Heisenberg et applications[END_REF]. ∆ j u ∆ j ′ v.

Remark 5.7. It is clear that formally uv = T u v + T v u + R(u, v).

One of the classical consequences of Lemma 5.5 is the following result, which is obtained using the previous decomposition as well as localization properties of the paraproduct and remainder terms.

Corollary 5.8. Let ρ > 0 and (p, r) ∈ [1, +∞] 2 be three real numbers. Then

f g B ρ p,r (G) ≤ C( f L ∞ g B ρ p,r (G) + g L ∞ f B ρ p,r (G)
). If ρ 1 + ρ 2 > 0 and if p 1 is such that ρ 1 < Q/p 1 , then for all (p 2 , r 2 ) ∈ [1, +∞] 2 writing ρ = ρ 1 + ρ 2 -Q/p 1 ,

f g B ρ p 2 ,r 2 (G) ≤ C( f B ρ 1 p 1 ,∞ g B ρ 2 p 2 ,r 2 + g B ρ 1 p 1 ,∞ f B ρ 2 p 2 ,r 2
).

Moreover, if ρ 1 + ρ 2 ≥ 0, ρ 1 < Q/p 1 and

1 r 1 + 1 r 2 = 1, then f g B ρ p,∞ (G) ≤ C( f B ρ 1 p 1 ,r 1 g B ρ 2 p 2 ,r 2
+ g B ρ 1 p 1 ,r 1 f B ρ 2 p 2 ,r 2
).

Finally if ρ 1 + ρ 2 > 0, ρ j < Q/p j and p ≥ max(p 1 , p 2 ), then for all (r 1 , r 2 ), .

f g B ρ 12 p,r (G) ≤ C f B ρ 1 p 1 ,
The same results hold in the case of homogeneous Besov spaces. Once the paraproduct algorithm is in place, one can obtain (refined) Sobolev and Hardy inequalities (see [START_REF] Chemin | Inclusions de Sobolev en calcul de Weyl-Hörmander et champs de vecteurs sous-elliptiques[END_REF] and [START_REF] Bahouri | Paraproduit sur le groupe de Heisenberg et applications[END_REF] for Sobolev embeddings in the euclidean case and for the Heisenberg group, and [START_REF] Bahouri | Refined Hardy inequalities[END_REF] for the Hardy inequalities -see also [START_REF] Chamorro | Improved Sobolev Inequalities and Muckenhoupt weights on stratified Lie groups[END_REF] for recent extensions). One can also construct, in the context of H-type groups, an algebra of pseudo-differential operators exactly as on the Heisenberg group. We refer to [START_REF] Bahouri | Phase-space analysis and pseudodifferential calculus on the Heisenberg group[END_REF] for details.

c

  i (t)X i (ℓ(t)). If ℓ is admissible, its length is defined by |ℓ| =

j k ℓ=1 1 0

 1 by the Fubini theorem and a change of variables. Using (2.2) we get ∀j ∈ N, τ w Ψ j -Ψ j L 1 2 |c ℓ (s)| ds so by definition of |w| and by the Cauchy-Schwarz inequality we find ∀j ∈ N,

Theorem 5 .

 5 is said to be of trace-class. One then sets tr (A(λ)) = α∈N ℓ (A(λ)F α,λ , F α,λ ) H λ , and the following inversion theorem holds. If a function f satisfies

Lemma 5 . 5 .Definition 5 . 6 .

 5556 There is a constant M 1 ∈ N such that the following holds. Consider f and g two functions of S(G) such thatF B (f )(λ)F α,λ = 1 λ∈(2|α|+ℓ)2 2m C 0 (λ)F B (f )(λ)F α,λ and F B (g)(λ)F α,λ = 1 λ∈(2|α|+ℓ)2 2m ′ C 0 (λ)F B (g)(λ)F α,λ for m and m ′ integers. If m ′ -m > M 1 , then there exists a ring C such that F B (f g)(λ)F α,λ = 1 λ∈(2|α|+ℓ)2 2m ′ C (λ)F B (f g)(λ)F α,λ .On the other hand, if |m ′ -m| ≤ M 1 , then there exists a ball B such thatF B (f g)(λ)F α,λ = 1 λ∈(2|α|+ℓ)2 2m ′ B (λ)F B (f g)(λ)F α,λ .We shall call paraproduct of v by u and shall denote by T u v the bilinear operator T u v = j S j-1 u ∆ j v. We shall call remainder of u and v and shall denote by R(u, v) the bilinear operator R(u, v) = |j-j ′ |≤1

  5.1.2. Bargmann representations on H-type groups. Given λ ∈ R n \ {0}, consider the Hilbert space (called the Fock space) H λ (C ℓ ) of all entire holomorphic functions F |F (ξ)| 2 e -|λ||ξ| 2 dξ is finite. The corresponding irreducible unitary representation π λ of the group G is realized on H λ (C ℓ ) by (recall that t ∈ R n and z, ξ ∈ C ℓ ) (see

	on C ℓ such that F 2 H λ =	2|λ| π	ℓ	C ℓ

  = max(r 1 , r 2 ), and if ρ 1 + ρ 2 ≥ 0,

	1 p 1 ) and r with ρ j < Q/p j and with + 1 p 2 -1 p 1 r 1 1 + r 2

r 1 g B ρ 2 p 2 ,r 2 , with ρ 12 = ρ 1 + ρ 2 -Q( = 1, then for all p ≥ max(p 1 , p 2 ), f g B ρ 12 p,∞ (G) ≤ C f B ρ 1 p 1 ,r 1 g B ρ 2 p 2 ,r

2