The linear symmetric systems associated with the modified Cherednik operators and applications

BLAISE PASCAL

Hatem Mejjaoli

The linear symmetric systems associated with the modified Cherednik operators and applications

Volume 19, n^o1 (2012), p. 213-245.

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The linear symmetric systems associated with the modified Cherednik operators and

applications

Hatem Mejjaoli

Abstract

We introduce and study the linear symmetric systems associated with the modified Cherednik operators. We prove the well-posedness results for the Cauchy problem for these systems. Eventually we describe the finite propagation speed property of it.

Les systèmes symétriques linéaires associés aux opérateurs de Cherednik modifiés et applications

Résumé

Nous présentons et étudions les systèmes symétriques linéaires associés aux opérateurs de Cherednik modifiés. Nous prouvons que le problème de Cauchy pour ces systèmes sont bien posé. Finalement nous en décrivons le principe de vitesse finie.

Dedicated to Khalifa Trimèche for his 66th birthday

1. Introduction

Leta be a real Euclidean vector space of dimensiondand letR be a root system in a. A multiplicity function is a complex-valued function k on R which is invariant with respect to the Weyl group ofR. In the mid 1990s, Ivan Cherednik associated with a triplet (a, R, k) a commutative family of first order differential-reflection operators, nowadays known as Cherednik

Keywords:Modified Cherednik operators, modified Cherednik symmetric systems, en- ergy estimates, finite speed of propagation, generalized wave equations with variable coefficients.

Math. classification:35L05, 22E30.

operators or trigonometric Dunkl operators. The original motivation for the study of these operators came from the theory of invariant differen- tial operators: if the triplet (a, R, k) arises from the structure theory of a Riemannian symmetric space of the non-compact type G/K, then it is possible to explicitly construct all radial components of the W−invariant differential operators on G/K using the Cherednik operators. The joint spectral theory of Cherednik operators is therefore naturally related to the harmonic analysis on Reimannian symmetric spaces (and to the more gen- eral theory of hypergeometric functions in several variables of Heckman and Opdam). But it is also related with the representation theory of the graded Hecke algebra of Lusztig. There are many references on the subject.

Our starting point will be the following references (cf. [3, 12, 13, 15]).

In this paper, we are interested in studying to the modified Cherednik- linear symmetric system

(S)











∂_tu(t, x)−

d

X

j=1

A_jT_ju(t, x)−A₀u(t, x) =f(t, x), (t, x)∈I×R^d u|t=0 =v,

where Tj,j= 1, . . . , d, are the modified Cherednik operators,I be an in- terval ofR, (A_p)_0≤p≤da family of functions fromI×R^dinto the space of m×mmatrices with real coefficientsap,i,j(t, x) which areW-invariant with respect tox, symmetric (i.e.a_p,i,j(t, x) =a_p,j,i(t, x)) and whose all deriva- tives inx∈R^d are bounded and continuous functions of (t, x), the initial data belongs to generalized Sobolev spaces [H_k^s(R^d)]^m andf is a continu- ous function on an intervalIwith value in [H_k^s(R^d)]^m. In the classical case, the Cauchy problem for symmetric hyperbolic systems of first order, it has been introduced and studied by Friedrichs (cf. [6]). The Cauchy problem will be solved with the aid of energy integral inequalities, developed for this purpose by Friedrichs. Such energy inequalities have been employed by Weber [18], Zaremba [19] to derive various uniqueness theorems, and by Courant-Friedrichs-Lewy [5], Friedrichs [6] to derive existence theo- rems. In all these treatments the energy inequality is used to show that the solution, at some later time, depends boundedly on the initial val- ues in an appropriate norm. To derive an existence theorem however one needs, in addition to the a priori energy estimates, auxiliary constructions.

Thus motivated by these methods we will prove by energy methods and

Friedrichs approach local well-posedness and principle of finite speed of propagation for the system (S).

Let us first summarize our well-posedness results and finite speed of propagation (Theorem 4.3 and Theorem 5.2).

Well-posedness for DLS. For all givenf ∈[C(I, H_k^s(R^d))]^m and v ∈ [H_k^s(R^d)]^m, there exists a unique solutionuof the system (S) in the space

[C(I, H_k^s(R^d))]^m\[C¹(I, H_k^s−1(R^d))]^m.

In the classical case, a similar result can be found in [2], where the authors used another method based on the symbolic calculations for the pseudo- differential operators that we can not adapt for the system (S) at the moment. Our method use some ideas inspired by the works [2, 6, 7, 8, 9, 10, 11, 5, 14].

Finite speed of propagation. Let (S) as above. We assume that f belongs to [C(I, H_k¹(R^d))]^m and v∈[H_k¹(R^d)]^m.

• There exists a positive constantC₀such that, for any positive real R satisfying

f(t, x) ≡0 for kxk< R−C₀t v(x) ≡0 for kxk< R, the unique solutionu of the system (S) verifies

u(t, x)≡0 for kxk< R−C0t.

• If the givenf and v are such that

f(t, x) ≡0 for kxk> R+C₀t v(x) ≡0 for kxk> R,

then the unique solution uof the system (S) satisfies u(t, x)≡0 for kxk> R+C₀t.

In the classical case, similar results can be found in [2, 11, 16].

A standard example of the modified Cherednik linear symmetric system is the generalized wave equations with variable coefficients defined by:

∂_t²u−div_k[A.∇_k,xu] +Q(t, x, ∂tu, Txu), t∈R, x∈R^d,

where

∇_k,xu:= (T1u, . . . , T_du), div_k(v1, . . . , v_d) :=

d

X

i=1

Tivi,

A is a real symmetric matrix which verifies some hypotheses (see subsec- tion 5.1) andQ(t, x, ∂_tu, T_xu) is differential-difference operator of degree 1 such that these coefficients areC^∞, and all derivatives are bounded. From the previous results we deduce the well-posedness of the generalized wave equations (Theorem 5.1):

Well-posedness for GDW. For all s ∈ N, u0 ∈ H_k^s+1(R^d), u1 ∈ H_k^s(R^d) and f ∈C(R, H_k^s(R^d)), there exists a unique

u∈C¹(R, H_k^s(R^d))∩C(R, H_k^s+1(R^d)) such that











∂_t²u−div_k[A.∇_k,xu] +Q(t, x, ∂tu, Txu) =f u_|t=0 =u0

∂_tu_|t=0 =u₁.

The paper is organized as follows. In Section 2 we recall the main re- sults about the harmonic analysis associated with the modified Cherednik operators. In Section 3 we introduce the generalized Sobolev spaces asso- ciated with modified Cherednik operators and we study these properties.

Section 4 is devoted to study the generalized Cauchy problem of the modi- fied Cherednik linear symmetric systems. In the last sections we give many applications. More precisely, we prove the well-posedness of the general- ized wave equations associated with the modified Cherednik operators.

Next, we prove the principle of finite speed of propagation of the linear Cherednik symetric systems.

Throughout this paper by C we always represent a positive constant not necessarily the same in each occurrence.

Acknowledgment. The author gratefully acknowledge the Deanship of Scientific Research at the University of King Faisal on material and moral support in the financing of this research project No. 130172. The author is deeply indebted to the referees for providing constructive comments and helps in improving the contents of this article.

2. Preliminaries

This section gives an introduction to the theory of modified Cherednik operators, generalized Fourier transform, and generalized convolution op- erator. Main references are [1, 4].

2.1. The eigenfunctions of the modified Cherednik operators The basic ingredient in the theory of modified Cherednik operators is finite reflection groups, acting on R^d with the standard Euclidean scalar product h., .i and kxk = ^phx, xi. On C^d, k.k denotes also the standard Hermitian norm, while hz, wi=^Pd_j=1zjwj.

Let (ej)_j=1,...,d be the Euclidean bases ofR^d, lete^∨_j = 2ej be the coroot associated to e_j and let

rej(x) =x− he^∨_j, xie_j (2.1) be the reflection in the hyperplane Hej ⊂ R^d orthogonal to ej. The re- flections r_ej, j = 1, . . . , d, generate a finite group W ⊂ O(d), called the reflection group associated with (e_j)_j=1,...,d.

Moreover, let Ak denotes the weight function

∀x∈R^d, A_k(x) = 2^γ

d

Y

j=1

|sinh(he_j , xi)|^2kjcosh^2lj(he_j , xi), (2.2) with k_j ≥l_j ≥0 andk_j 6= 0, andγ :=^Pd_j=0(k_j+l_j).

In the following we denote by

• C(R^d) the space of continuous functions on R^d.

• C0(R^d) the space of continuous functions on R^d vanishing at in- finity.

• C^p(R^d) the space of functions of class C^p onR^d.

• C_b^p(R^d) the space of bounded functions of class C^p.

• E(R^d) the space of C^∞-functions on R^d.

• S(R^d) the Schwartz space of rapidly decreasing functions on R^d.

• D(R^d) the space of C^∞-functions on R^d which are of compact support.

• S⁰(R^d) the space of temperate distributions on R^d.

The modified Cherednik operatorsTj, j= 1, . . . , d, onR^dare given by T_jf(x) := ∂

∂x_jf(x)+^hk_jcoth(he_j, xi)+l_jtanh(he_j, xi)ⁱf(x)−f(r_ej(x)). (2.3) Some properties of the T_j, j = 1, . . . , d, are given in the following: for all f and g inC¹(R^d) with at least one of them is W-invariant, we have

Tj(f g) = (Tjf)g+f(Tjg), j= 1, . . . , d. (2.4) Forf of classC¹ on R^d with compact support andgof class C¹ onR^d, we have for j = 1,2, . . . , d :

Z

R^d

T_jf(x)g(x)A_k(x)dx=− Z

R^d

f(x)T_jg(x)A_k(x)dx. (2.5) The modified Cherednik operators form a commutative system of dif- ferential-difference operators. The modified Heckman-Opdam Laplacian 4_k is defined by

4_kf(x) :=

d

X

j=1

T_j²f(x) (2.6)

=4f(x) +

d

X

j=1

h2k_jcoth(he_j, xi) + 2l_jtanh(he_j, xi)ⁱh∇f(x), e_ji

−

d

X

j=1

h k_j

(sinh(he_j, xi))² − l_j (cosh(he_j, xi))²

i

f(x)−f(rej(x)), where 4and ∇are respectively the Laplacian and the gradient onR^d.

The modified Heckman-Opdam Laplacian on W-invariant functions in denoted by 4^W_k and has the expression

4^W_k f(x) =4f(x) +

d

X

j=1

h2k_jcoth(he_j, xi) + 2l_jtanh(he_j, xi)ⁱh∇f(x), e_ji.

Example 2.1. Ford= 1, andk≥l≥0 andk6= 0, the modified Heckman- Opdam Laplacian 4^W_k is the Jacobi operator defined for even functionsf of classC² on Rby

4^W_k f(x) = d²

dx²f(x) +^h2kcoth(x) + 2ltanh(x)ⁱ d dxf(x).

We denote by G_λ the eigenfunction of the operatorsT_j,j= 1,2, . . . , d.

It is the unique analytic function on R^d which satisfies the differential- difference system

T_ju(x) =λ_ju(x), j = 1,2, . . . , d, x∈R^d u(0) = 1.

It is called the modified Opdam-Cherednik kernel.

We consider the function Fλ defined by

∀x∈R^d, Fλ(x) = 1

|W| X

w∈W

Gλ(wx).

This function is the unique analytic W-invariant function on R^d, which satisfies the differential equations

p(T)u(x) =p(λ)u(x), x∈R^d, λ∈R^d

u(0) = 1,

for all W-invariant polynomial p on R^d and p(T) =p(T₁, . . . , T_d). In par- ticular for all λ∈R^d we have

4^W_k F_λ(x) =kλk²F_λ(x).

The function Fλ is called the modified Heckman-Opdam kernel.

The functions G_λ and F_λ possess the following properties i) For allx∈R^d, the functionsG_λ and F_λ are entire onC^d. ii) There exists a positive constant M₀ such that

∀x∈R^d,∀λ∈R^d, |G_iλ(x)| ≤M0.

iii) Letp and q be polynomials of degreem and n. Then there exists a positive constant M⁰ such that for all λ ∈ C^d\{0} and for all

x∈R^d, we have

|p( ∂

∂λ)q( ∂

∂x)G_λ(x)| ≤M⁰

d

Y

j=1

(1 +|x_j|)ⁿ(1 +|λ_j|+γ)^m

|λ_j| e^{maxw∈W(Rehwλ,xi)}. (2.7) Example 2.2. When d= 1 and W =Z2, the modified Opdam-Cherednik kernel Gλ(x) is given for all λ∈Cand x∈Rby

G_λ(x) = (

ϕ^(k−

1 2,l−¹₂)

µ (x) +_λ¹_dx^dϕ^(k−

1 2,l−¹₂)

µ (x), if λ∈C\{0},

1, if λ= 0,

with λ² =µ²+ (k+l)² and ϕ^(k−

1 2,l−¹₂)

µ the Jacobi function given by ϕ^(k−

1 2,l−¹₂)

µ (x) = 2F1(k+l+iµ

2 ,k+l−iµ 2 ;k+1

2;−(sinhx)²), (2.8) where ₂F₁ is the Gauss hypergeometric function.

In this case the modified Heckman-Opdam kernel Fλ(x) is given for all λ∈Cand x∈R by

F_λ(x) =ϕ^(k−

1 2,l−¹

2)

λ (x).

2.2. The generalized Fourier transform We denote by

S₂(R^d) the space of C^∞-functions on R^d such that for all`, n ∈N, we have

sup

|µ|≤n, x∈R^d

(1 +kxk)^`

d

Y

j=1

(cosh(x_j))^2(kj+lj)|D^µf(x)|<∞, where

D^µ= ∂^|µ|

∂^µ1x₁. . . ∂^µdx_d , µ= (µ₁, . . . , µ_d)∈N^d.

PW(C^d) the space of entire functions onC^d, which are rapidly decreas- ing and of exponential type.

S₂⁰(R^d) the topological dual ofS₂(R^d).

L^p_A

k(R^d), 1≤p≤ ∞, the space of measurable functionsf onR^d satis- fying

kfk_L^p

Ak(R^d) = Z

R^d

|f(x)|^pA_k(x)dx 1/p

<∞, if 1≤p <∞ kfk_L^∞

Ak(R^d) = ess sup

x∈R^d

|f(x)|<∞.

The generalized Fourier transform of a functionf inD(R^d) is given by F_k(f)(λ) =

Z

R^d

f(x)G−iλ(x)Ak(x)dx, for allλ∈R^d. (2.9) Proposition 2.3. The transform F_k is a topological isomorphism from

(i) D(R^d) onto PW(C^d).

(ii) S₂(R^d) onto S(R^d).

The inverse transform is given by

∀x∈R^d, (F_k)⁻¹(h)(x) = Z

R^d

h(λ)Giλ(x)dνk(λ), (2.10) where dν_k(λ) :=C_k(λ)dλ is the spectral measure (cf. [1, 4]).

Remark 2.4. The functionC_k is a positive, continuous onR^dand satisfies the estimate

∀λ∈R^d, |C_k(λ)| ≤const.(1 +kλk)^b, for some b >0.

Proposition 2.5.

(i) Plancherel formula: For all f, g in D(R^d) (resp. S₂(R^d)) we have Z

R^d

f(x)g(x)A_k(x)dx= Z

R^d

F_k(f)(λ)F_k(g)(λ)dν_k(λ). (2.11) (ii) Plancherel theorem: The generalized Fourier transformF_kextends

uniquely to an isometric isomorphism of L²_A

k(R^d) into L²_νk(R^d), where L²_νk(R^d)denotes the space of measurable functions f onR^d satisfying

kfk_L2

νk(R^d)= Z

R^d

|f(x)|²dν_k(x)^1/2 <∞.

2.3. Generalized convolution operator

Definition 2.6. Let y be in R^d. The generalized translation operator f 7→τ_yf is defined onS₂(R^d) by

F_k(τ_yf)(x) =G−iy(x)F_k(f)(x), for allx∈R^d. (2.12) Using the generalized translation operator, we define the generalized convolution product of functions as follows.

Definition 2.7. The generalized convolution product offandginS₂(R^d) is the function f ∗_kg defined by

f ∗_kg(x) = Z

R^d

τxf(−y)g(y)A_k(y)dy, for allx∈R^d. (2.13) For the remainder of this subsection we collect some results proved in [1].

Proposition 2.8. Let f be in L¹_A

k(R^d) and g in L²_A

k(R^d). Then i) The function f ∗_kg defined almost everywhere onR^d by

f ∗_kg(y) = Z

R^d

τyg(−x)f(x)Ak(x)dx, belongs to L²_Ak(R^d) and we have

kf∗_kgk_L2

Ak(R^d)≤Ckfk_L1

Ak(R^d)kgk_L2 Ak(R^d). ii) We have

F_k(f ∗_kg) =F_k(f).F_k(g).

Proposition 2.9. Let ϕ be a positive function in D(R^d) such that supp ϕ⊂B(0,1) and kϕk_L1

Ak(R^d) = 1. For ε >0, we consider the function ϕε

given by

∀x∈R^d, ϕ_ε(x) = A_k(^x_ε) ε^dAk(x) ϕ(x

ε).

Then for all f in L²_A

k(R^d) we have

ε→0limkf ∗_kϕ_ε−fk_L2

Ak(R^d)= 0. (2.14) Definition 2.10. The generalized Fourier transform of a distribution τ in S₂⁰(R^d) is defined by

hF_k(τ), φi=hτ,F_k⁻¹(φ)i, for allφ∈ S(R^d). (2.15)

Proposition 2.11. The generalized Fourier transformF_k is a topological isomorphism from S⁰₂(R^d) onto S⁰(R^d).

3. The generalized Sobolev spaces

Let τ be in S₂⁰(R^d). We define the distributions Tjτ,j= 1, . . . , d,by hT_jτ, ψi = −hτ, T_jψi, for allψ ∈ S₂(R^d). (3.1) Thus we deduce

h4_kτ, ψi = hτ,4_kψi, for allψ ∈ S₂(R^d). (3.2) These distributions satisfies the following properties

F_D(Tjτ) = iyjF_D(τ), j= 1, . . . , d. (3.3) F_D(4_kτ) = −kyk²F_D(τ). (3.4) Definition 3.1. We define the generalized Sobolev space H_k^s(R^d) as

nu∈ S₂⁰(R^d) : Z

R^d

(1 +kλk²)^s|F_k(u)(λ)|²dν_k(λ)<∞^o. We provide this space with the scalar product

hu, vi_Hs

k(R^d):=

Z

R^d

(1 +kξk²)^sF_k(u)(ξ)F_k(v)(ξ)dν_k(ξ), (3.5) and the norm

kuk²_Hs

k(R^d) :=hu, ui_Hs

k(R^d). (3.6)

Proposition 3.2.

(i) The space H_k^s(R^d) provided with the scalar product h., .i_Hs k(R^d) is a Hilbert space.

(ii) Let s1, s2 in R such that s1≥s2 then H_k^s1(R^d),→H_k^s2(R^d).

(iii) Let s∈R. Then D(R^d) is dense in H_k^s(R^d).

(iv) The dual ofH_k^s(R^d) can be identified with H_k^−s(R^d). The relation of the identification is given by

hu, vi_k= Z

R^d

F_k(u)(ξ)F_k(v)(ξ)dν_k(ξ),

with u∈H_k^s(R^d) and v∈H_k^−s(R^d).

Proof. (i) It is clear thatL²(R^d,(1 +kξk²)^2sdν_k(ξ)) is complete and since F_k is an isomorphism fromS₂⁰(R^d) ontoS⁰(R^d),H_k^s(R^d) is then a Hilbert space. The result (ii) is immediately from definition of the generalized Sobolev space. As in [17], we can obtain (iii) and (iv).

Proposition 3.3. Let s₁, s, s₂ be three real numbers :s₁ < s < s₂. Then, for all ε >0, there exists a nonnegative constantCε such that for allu in H_k^s(R^d)

kuk_Hs

k(R^d)≤Cεkuk_H^s1

k (R^d)+εkuk_H^s2

k (R^d). (3.7) Proof. We consider s = (1−t)s₁+ts₂, (with t ∈ (0,1)). Moreover it is easy to see

kuk_Hs

k(R^d)≤ kuk^1−t

H_k^s1(R^d)kuk^t_H^s2

k (R^d). (3.8)

Thus

kuk_Hs

k(R^d) ≤ (ε^−1−tt kuk_H^s1

k (R^d))^1−t(εkuk_H^s2 k (R^d))^t

≤ ε^−1−tt kuk_H^s1

k (R^d)+εkuk_H^s2 k (R^d).

Hence the proof is completed for Cε =ε^−1−tt . A characterization of H_k^s(R^d), for s = m, a positive integer, is given below.

Proposition 3.4.

(i) For m∈N the space H_k^m(R^d) coincides with the space Em given by

Em=ⁿu∈L²_Ak(R^d) :T^αu∈L²_Ak(R^d), |α| ≤m^o, (3.9) where T^α = T₁^α1 ⊗ · · · ⊗T_d^αd, α = (α1, . . . , α_d) ∈ N^d and |α| = α₁+· · ·+α_d.

(ii) The norm k.k_m,k is equivalent to the norm kuk²_m,k = ^X

|ν|≤m

kT^νuk²_L2

Ak(R^d). (3.10)

For prove this proposition we need the following lemma.

Lemma 3.5. Let m ∈ N\{0}. For all α ∈ N^d with 0 < |α| ≤ m, there exists C >0 such that

∀ξ ∈R^d,

d

Y

j=1

|ξ_j|^2αj ≤(1 +kξk²)^m≤C(1 + ^X

0<|α|≤m d

Y

j=1

|ξ_j|^2αj). (3.11) Proof of Proposition 3.4. Letube inH_k^m(R^d), henceu∈L²_A

k(R^d). Using Lemma 3.5 we deduce that for all α ∈ N^d with |α| ≤ m, there exists a positive constant C such that

X

0<|α|≤m

Z

R^d

(

d

Y

j=1

|ξ_j|^2αj)|F_k(u)(ξ)|²dν_k(ξ)

≤C Z

R^d

(1 +kξk²)^m|F_k(u)(ξ)|²dν_k(ξ). (3.12) But from (3.3) we have for all ξ ∈R^d:

F_k(T^αu)(ξ) =i^|α|ξ^α₁¹. . . ξ_d^αdF_k(u)(ξ).

Thus from (3.12) we deduce that X

0<|α|≤m

Z

R^d

|T^αu(x)|²A_k(x)≤Ckuk²_Hm k(R^d). Then u belongs to Em, and

kuk²_m,k ≤Ckuk²_Hm k (R^d).

Reciprocally let u be in E_m. From Lemma 3.5 we deduce that for all α ∈N^dwith |α| ≤m, there exists a positive constantC⁰ such that

Z

R^d

(1 +kξk²)^m|F_k(u)(ξ)|²dνk(ξ)

≤C^0hkuk²_2,k+ ^X

0<|α|≤m

Z

R^d

|T^αu(x)|²A_k(x)ⁱ. Thus u belongs toH_k^m(R^d) and

kuk²_Hm

k (R^d)≤C⁰kuk²_m,k.

Proposition 3.6.

(i) Fors∈R andµ∈N^d, the Dunkl operator T^µ is continuous from H_k^s(R^d) into H_k^s−|µ|(R^d).

(ii) Letp∈N. An elementuis inH_k^s(R^d)if and only if for allµ∈N^d, with |µ| ≤p,T^µu belongs to H_k^s−p(R^d), and we have

kuk_Hs

k(R^d)∼ ^X

|µ|≤p

kT^µuk_Hs−p k (R^d). Proof. (i) Letu be inH_k^s(R^d). From (3.3) we have

Z

R^d

(1 +kξk²)^s−|µ||F_k(T^µu)(ξ)|²dνk(ξ)

= Z

R^d

(1 +kξk²)^s−|µ|kξ^µk²|F_k(u)(ξ)|²dν_k(ξ).

Thus Z

R^d

(1 +kξk²)^s−|µ||F_k(T^µu)(ξ)|²dνk(ξ)

≤ Z

R^d

(1 +kξk²)^s|F_k(u)(ξ)|²dν_k(ξ)<+∞.

Then T^µu belongs to H_k^s−|µ|(R^d), and kT^µuk

H^s−|µ|_k (R^d) ≤ kuk_Hs k(R^d).

(ii) We consider p ∈ N and u ∈ H_k^s(R^d). From (i) and Proposi- tion 3.2 (ii) for all µ∈N^d, with |µ| ≤p, we have

T^µu∈H_k^s−|µ|(R^d)⊂H_k^s−p(R^d).

Then there exists a positive constantC such that kT^µuk_H^s−p

k (R^d)≤Ckuk_H^s

k(R^d). This implies that

X

|µ|≤p

kT^µuk_Hs−p

k (R^d)≤C⁰kuk_Hs k(R^d), where C⁰ is a positive constant.

Reciprocally let p∈ Nas for all µ ∈N^d with|µ| ≤p,T^µu belongs to H_k^s−p(R^d),then

kuk²_Hs k(R^d)=

Z

R^d

(1 +kξk²)^s|F_k(u)(ξ)|²dν_k(ξ)

= Z

R^d

(1 +kξk²)^s−p(1 +kξk²)^p|F_k(u)(ξ)|²dνk(ξ).

But from Lemma 3.5, there exists a positive constant C such that (1 +kξk²)^p ≤C ^X

|µ|≤p

kξ^µk². Hence from (3.3) we deduce that

kuk²_Hs

k(R^d)≤C ^X

|µ|≤p

kT^µuk²

H_k^s−p(R^d).

This implies that u is inH_k^s(R^d).

Proposition 3.7. Let p∈N and s∈R such that s > ^b+d+p₂ , then H_k^s(R^d),→C^p(R^d),

with b the positive constant given in Remark 2.4.

Proof. Letu be in H_k^s(R^d) withs∈R such thats > ^b+d₂ . We have

Z

R^d

|F_k(u)(λ)|dν_k(λ) = Z

R^d

(1 +kλk²)^−s2(1 +kλk²)^2s|F_k(u)(λ)|dν_k(λ).

Using Hölder inequality we obtain Z

R^d

|F_k(u)(λ)|dν_k(λ)≤ Z

R^d

(1 +kλk²)^−sdν_k(λ)

1 2kuk_Hs

k(R^d). Thus from Remark 2.4, we deduce that there exists a positive constant C such that

kF_k(u)k_L1

νk(R^d)≤Ckuk_Hs

k(R^d). (3.13) Then

F_k(u)∈L¹_ν

k(R^d).

Thus from (2.10) we have u(x) =

Z

R^d

F_k(u)(λ)G_iλ(x)dν_k(λ), a.e. x∈R^d.

We identify uwith the second member, then we deduce thatu belongs to C(R^d) and using (3.13) we show that the injection ofH_k^s(R^d) intoC(R^d) is continuous.

Now letube inH_k^s(R^d) withs∈Rsuch thats > ^b+d+p₂ withpbelongs toN\{0}. From (2.7), for allx, λ∈R^d, andν ∈N^dsuch that|ν| ≤p, we have

|D^ν_xGiλ(x)| ≤Ckλk^p.

Using the same method as for p = 0, and the derivation theorem under the integral sign we deduce that

∀x ∈R^d, D^νu(x) = Z

R^d

F_k(u)(λ)D_x^νG_iλ(x)dν_k(λ).

Thus Dⁿu belongs to C(R^d), for all n ∈ N such that |ν| ≤ p. Then we show that u is in C^p(R^d) and the injection of H_k^s(R^d) into C^p(R^d) is

continuous.

4. Cherednik linear symmetric systems

Notation 4.1. For any interval I of R we define the mixed space-time spaces C(I, H_k^s(R^d)), for s ∈ R, as the spaces of functions from I into H_k^s(R^d) such that the map

t7→ ku(t, .)k_Hs k(R^d)

is continuous.

In this section, I designates the interval [0, T), T >0 and u= (u1, . . . , um), up ∈C(I, H_k^s(R^d)),

a vector with m components elements of C(I, H_k^s(R^d)). Let (A_p)_0≤p≤d a family of functions fromI×R^dinto the space ofm×mmatrices with real coefficients a_p,i,j(t, x) which areW-invariant with respect to xand whose all derivatives in x∈R^dare bounded and continuous functions of (t, x).

For a given f ∈ [C(I, H_k^s(R^d))]^m and v ∈ [H_k^s(R^d)]^m, we find u in [C(I, H_k^s(R^d))]^m satisfying the following system (S)











∂_tu(t, x)−

d

X

j=1

(A_jT_ju)(t, x)−(A₀u)(t, x) =f(t, x),(t, x)∈I×R^d u|_t=0 =v.

We shall first define the notion of symmetric systems.

Definition 4.2. The system (S) is symmetric, if and only if, for any p∈ {1, . . . , d}and any (t, x)∈I×R^dthe matricesAp(t, x) are symmetric i.e. a_p,i,j(t, x) =a_p,j,i(t, x).

In this section, we shall assumes∈Nand denote byku(t)k_s,kthe norm defined by

ku(t)k²_s,k = ^X

1≤p≤m 0≤|µ|≤s

kT_x^µu_p(t)k²_L2 Ak(R^d). The aim of this section is to prove the following theorem.

Theorem 4.3. Let (S) be a symmetric system. Assume that f is con- tained in [C(I, H_k^s(R^d))]^m andv in[H_k^s(R^d)]^m, then there exists a unique solution u of (S) in

[C(I, H_k^s(R^d))]^m\[C¹(I, H_k^s−1(R^d))]^m. The proof of this theorem will be made in several steps:

A: We prove a priori estimates for the regular solutions of the sys- tem (S).

B: We apply the Friedrichs method.

C: We pass to the limit for regular solutions and we obtain the ex- istence in all cases by the regularization of the Cauchy data.

D: We prove the uniqueness using the existence result of the adjoint system.

A: Energy estimates. The symmetric hypothesis is crucial for the en- ergy estimates which are only valid for regular solutions. More precisely we have

Lemma 4.4. (Energy Estimate in [H_k^s(R^d)]^m). For any positive integer s, there exists a positive constant λs such that, for any function u in [C¹(I, H_k^s(R^d))]^mT[C(I, H_k^s+1(R^d))]^m, we have

ku(t)k_s,k ≤e^λstku(0)k_s,k+ Z t

0

e^λs(t−t0)kf(t⁰)k_s,kdt⁰, (4.1)

for all t∈I,with

f =∂tu−

d

X

p=1

ApTpu−A0u.

To prove Lemma 4.4, we need the following Lemma.

Lemma 4.5. Let g a C¹-function on [0, T), a and b two positive contin- uous functions. We assume

d

dtg²(t)≤2a(t)g²(t) + 2b(t)|g(t)|. (4.2) Then, for t∈[0, T), we have

|g(t)| ≤ |g(0)|exp Z t

0

a(s)ds+ Z t

0

b(s) exp Z t

s

a(τ)dτds.

Proof. To prove this lemma, let us set forε >0,g_ε(t) =g²(t) +ε

1 2; the function g_ε is C¹, and we have |g(t)| ≤ g_ε(t). Thanks to the inequality (4.2), we have

d

dt(g²)(t)≤2a(t)g_ε²(t) + 2b(t)g_ε(t).

As _dt^d(g²)(t) = _dt^d(g_ε²)(t). Then d

dt(g_ε²)(t) = 2g_ε(t)dg_ε

dt (t)≤2a(t)g_ε²(t) + 2b(t)g_ε(t).

Since for all t∈[0, T) g_ε(t)>0, we deduce then dg_ε

dt (t)≤a(t)gε(t) +b(t).

Thus d dt

gε(t) exp− Z t

0

a(s)ds≤b(t) exp− Z t

0

a(s)ds. So, for t∈[0, T),

g_ε(t)≤g_ε(0) exp Z t

0

a(s)ds+ Z t

0

b(s) exp Z t

s

a(τ)dτds.

Thus, we obtain the conclusion of the lemma by tending εto zero.