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

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


Introduction
Let a be a real Euclidean vector space of dimension d and let R 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 of R. 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 operators or trigonometric Dunkl operators. The original motivation for the study of these operators came from the theory of invariant differential 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 general 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 Cheredniklinear symmetric system where T j , j = 1, . . . , d, are the modified Cherednik operators, I be an interval of R, (A p ) 0≤p≤d a family of functions from I × R d into the space of m×m matrices with real coefficients a p,i,j (t, x) which are W -invariant with respect to x, symmetric (i.e. a p,i,j (t, x) = a p,j,i (t, x)) and whose all derivatives in x ∈ R d are bounded and continuous functions of (t, x), the initial data belongs to generalized Sobolev spaces [H s k (R d )] m and f is a continuous function on an interval I with value in [H s k (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 theorems. In all these treatments the energy inequality is used to show that the solution, at some later time, depends boundedly on the initial values 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 given
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 pseudodifferential 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, • There exists a positive constant C 0 such that, for any positive real R satisfying the unique solution u of the system (S) verifies • If the given f and v are such that then the unique solution u of the system (S) satisfies 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: A is a real symmetric matrix which verifies some hypotheses (see subsection 5.1) and Q(t, x, ∂ t u, T x u) is differential-difference operator of degree 1 such that these coefficients are C ∞ , and all derivatives are bounded. From the previous results we deduce the well-posedness of the generalized wave equations (Theorem 5.1): The paper is organized as follows. In Section 2 we recall the main results about the harmonic analysis associated with the modified Cherednik operators. In Section 3 we introduce the generalized Sobolev spaces associated with modified Cherednik operators and we study these properties. Section 4 is devoted to study the generalized Cauchy problem of the modified Cherednik linear symmetric systems. In the last sections we give many applications. More precisely, we prove the well-posedness of the generalized 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.

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

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 ., . and x = x, x . On C d , . denotes also the standard Hermitian norm, while z, w = d j=1 z j w j . Let (e j ) j=1,...,d be the Euclidean bases of R d , let e ∨ j = 2e j be the coroot associated to e j and let with k j ≥ l j ≥ 0 and k j = 0, and γ := d j=0 (k j + l j ). In the following we denote by • C(R d ) the space of continuous functions on R d .
• C 0 (R d ) the space of continuous functions on R d vanishing at infinity.
• C p (R d ) the space of functions of class C p on R d .
• S(R d ) the Schwartz space of rapidly decreasing functions on R d .

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• D(R d ) the space of C ∞ -functions on R d which are of compact support.
• S (R d ) the space of temperate distributions on R d .
The modified Cherednik operators T j , j = 1, . . . , d, on R d are given by (2.3) Some properties of the T j , j = 1, . . . , d, are given in the following: for all f and g in C 1 (R d ) with at least one of them is W -invariant, we have For f of class C 1 on R d with compact support and g of class C 1 on R d , we have for j = 1, 2, . . . , d : (2.5) The modified Cherednik operators form a commutative system of differential-difference operators. The modified Heckman-Opdam Laplacian k is defined by and ∇ are respectively the Laplacian and the gradient on R d . The modified Heckman-Opdam Laplacian on W -invariant functions in denoted by W k and has the expression 218

Symmetric systems and applications
Example 2.1. For d = 1, and k ≥ l ≥ 0 and k = 0, the modified Heckman-Opdam Laplacian W k is the Jacobi operator defined for even functions f of class C 2 on R by We denote by G λ the eigenfunction of the operators T j , j = 1, 2, . . . , d. It is the unique analytic function on R d which satisfies the differentialdifference system It is called the modified Opdam-Cherednik kernel. We consider the function F λ defined by This function is the unique analytic W -invariant function on R d , which satisfies the differential equations The functions G λ and F λ possess the following properties i) For all x ∈ R d , the functions G λ and F λ are entire on C d .
ii) There exists a positive constant M 0 such that iii) Let p and q be polynomials of degree m and n. Then there exists a positive constant M such that for all λ ∈ C d \{0} and for all 219 In this case the modified Heckman-Opdam kernel F λ (x) is given for all λ ∈ C and x ∈ R by

The generalized Fourier transform
We denote by PW(C d ) the space of entire functions on C d , which are rapidly decreasing and of exponential type.
The generalized Fourier transform of a function f in D(R d ) is given by (2.9) The inverse transform is given by where dν k (λ) := C k (λ)dλ is the spectral measure (cf. [1,4]).
Remark 2.4. The function C k is a positive, continuous on R d and satisfies the estimate

Generalized convolution operator
(2.12) Using the generalized translation operator, we define the generalized convolution product of functions as follows.
For the remainder of this subsection we collect some results proved in [1].
For ε > 0, we consider the function ϕ ε given by .

The generalized Sobolev spaces
Thus we deduce These distributions satisfies the following properties We provide this space with the scalar product (3.5) and the norm The relation of the identification is given by 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). (3.7) Proof. We consider s = (1 − t)s 1 + ts 2 , (with t ∈ (0, 1)). Moreover it is easy to see (3.8) Thus A characterization of H s k (R d ), for s = m, a positive integer, is given below.

Proposition 3.4.
(i) For m ∈ N the space H m k (R d ) coincides with the space E m given by (3.10) For prove this proposition we need the following lemma.
Using Lemma 3.5 we deduce that for all α ∈ N d with |α| ≤ m, there exists a positive constant C such that Then u belongs to E m , and Reciprocally let u be in E m . From Lemma 3.5 we deduce that for all α ∈ N d with |α| ≤ m, there exists a positive constant C such that Thus u belongs to H m k (R d ) and
(i) For s ∈ R and µ ∈ N d , the Dunkl operator T µ is continuous from Then T µ u belongs to H s−|µ| k (R d ), and (ii) We consider p ∈ N and u ∈ H s k (R d ). From (i) and Proposition 3.2 (ii) for all µ ∈ N d , with |µ| ≤ p, we have . Then there exists a positive constant C such that where C is a positive constant.
Reciprocally let p ∈ N as for all µ ∈ N d with |µ| ≤ p, T µ u belongs to But from Lemma 3.5, there exists a positive constant C such that Hence from (3.3) we deduce that This implies that u is in H s k (R d ). Proposition 3.7. Let p ∈ N and s ∈ R such that s > b+d+p 2 , then Using Hölder inequality we obtain . Thus from Remark 2.4, we deduce that there exists a positive constant C such that . Thus from (2.10) we have We identify u with the second member, then we deduce that u belongs to C(R d ) and using (3.13) we show that the injection of H s with p belongs to N\{0}. From (2.7), for all x, λ ∈ R d , and ν ∈ N d such that |ν| ≤ p, we have |D ν x G iλ (x)| ≤ C λ p . Using the same method as for p = 0, and the derivation theorem under the integral sign we deduce that Thus D n u 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 s k (R d ) into C p (R d ) is continuous. For a given f ∈

Cherednik linear symmetric systems
] m satisfying the following system (S) We shall first define the notion of symmetric systems. In this section, we shall assume s ∈ N and denote by u(t) s,k the norm defined by The aim of this section is to prove the following theorem.
The proof of this theorem will be made in several steps:

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

B:
We apply the Friedrichs method.
C: We pass to the limit for regular solutions and we obtain the existence in all cases by the regularization of the Cauchy data.

D:
We prove the uniqueness using the existence result of the adjoint system.
To prove Lemma 4.4, we need the following Lemma.
Since for all t ∈ [0, T ) g ε (t) > 0, we deduce then So, for t ∈ [0, T ), Thus, we obtain the conclusion of the lemma by tending ε to zero.
Proof. of Lemma 4.4. We prove this estimate by induction on s. We firstly assume that u belongs to [C 1 (I, We will estimate the third term of the sum above by using the symmetric hypothesis of the matrix A p . In fact from (2.4) and (2.5) we have The matrix A p being symmetric, we have Thus Since the coefficients of the matrix A p , as well as their derivatives are bounded on I × R d , there exists a positive constant λ 0 such that To complete the proof of Lemma 4.4 in the case s = 0 it suffices to apply Lemma 4.5. We assume now that Lemma 4.4 is proved for s. T 1 u, . . . , T d u).
for any j ∈ {1, . . . , d}, applying the operator T j on the last equation and using (2.4), we obtain We can then write B: Estimate about the approximated solution. We notice that the necessary hypothesis to the proof of the inequalities of Lemma 4.4 require exactly one more derivative than the regularity which appears in the statement of the theorem that we have to prove. We then have ] m . Moreover, as J 2 n = J n , it is obvious that J n u n is also a solution of (S n ). We apply Proposition 4.6 we deduce that J n u n = u n . The function u n is then belongs to [C 1 (I, H s k (R d ))] m for any integer s and so (S n ) can be written as Now, let us estimate the evolution of u n (t) s,k .
Proof. The proof uses the same ideas as in Lemma 4.4.

C: Construction of solution.
This step consists on the proof of the following existence and uniqueness result: For s ≥ 0, we consider the symmetric system There exists a unique solution u belonging to the space )] m and satisfying the energy estimate : for all σ ≤ s + 3 and t ∈ I.
Proof. Us consider the sequence (u n ) n defined by the Friedrichs method and let us prove that this sequence is a Cauchy one in [L ∞ (I, H s+1 k (R d ))] m .
We put V n,p = u n+p − u n , we have Moreover, by a simple calculation we find Similarly, we have By Lemma 4.7 we deduce that Finally we will prove the inequality (4.5). From Lemma 4.7 we have

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Since for any t ∈ I, the sequence (u n (t)) n∈N tends to

then there exists a solution of a symmetric system (S) in the space
. Proof. We consider the sequence (ũ n ) n∈N of solutions of From Proposition 4.8 (ũ n ) n is in [C 1 (I, H s k (R d ))] m . We will prove that (ũ n ) n is a Cauchy sequence in [L ∞ (I, H s k (R d ))] m . We putṼ n,p =ũ n+p −ũ n . By difference, we find By Lemma 4.7 we deduce that with ., . k defined by ] m for any t in I and the fact that A j is symmetric we obtain As u is not very regular, we have to justify the integration by parts in time on the quantity I u(t, .), ∂ t ϕ(t, .) k dt. Since J n u(., x), J n ϕ(., x) are C 1 functions on I, then by integration by parts, we obtain, for any x ∈ R d , Since u(0, .) = ϕ(T, .) = 0, we have A is a real symmetric matrix such that there exists m > 0 satisfying and Q(t, x, ∂ t u, T x u) is differential-difference operator of degree 1, and we assume also that the matrix A is W -invariant with respect to x; the coefficients of A and Q are C ∞ and all derivatives are bounded. If we put B = √ A it is easy to see that the coefficients of B are C ∞ and all derivatives are bounded.
We introduce the vector U with d + 2 components Then, the equation P (u) = f can be written as and B = (b ij ). Thus the system (5.4) is symmetric and from Theorem 4.3 we deduce the following.

Finite speed of propagation
Theorem 5.2. Let (S) be a symmetric system. There exists a positive constant C 0 such that, for any positive real R, any function where the function ψ ∈ E(R d ) will be chosen later. By a simple calculation we see that There exists a positive constant K such that if T j ψ L ∞ k (R d ) ≤ K for any j = 1, . . . , d, we have for any (t, x) Re(B τ y),ȳ ≤ Re(A 0 y),ȳ for all τ ≥ 1 and y ∈ C m .
However if (t 0 , x 0 ) verifies x 0 < R − C 0 t 0 , we can find a function ψ of precedent type such that t 0 < ψ(x 0 ). Thus the theorem is proved.
the unique solution u of system (S) belongs to [C(I, L 2 A k (R d ))] m with u(t, x) ≡ 0 for x < R − C 0 t.
Proof. If f ε ∈ [C(I, H 1 k (R d ))] m , v ε ∈ [H 1 k (R d ))] m are given such that f ε → f in [C(I, L 2 A k (R d ))] m and v ε → v in [L 2 A k (R d )] m , we know by Section 4 that the solution u ε belongs to [C(I, H 1 k (R d ))] m and verifies u ε → u in [C(I, L 2 A k (R d ))] m . Therefore, if we construct f ε and v ε satisfying (5.8) and (5.9) with R replaced by R − ε, we obtain the result by applying Theorem 5.2. To this end let us consider χ ∈ D(R d ) and radial such that suppχ ⊂ B(0, 1) and For ε > 0, we put The hypothesis (5.8) and (5.9) are then satisfied by f ε and u 0,ε if we replace R by R − ε. On the other hand the solution u ε associated with f ε and u 0,ε is [C 1 (I, H s k (R d ))] m for any integer s. Finally applying Proposition 2.9 and Theorem 5.2 we obtain the result.
Theorem 5.4. Let (S) be a symmetric system. We assume that the func- Then the unique solution u of system (S) belongs to [C(I, H 1 k (R d ))] m with u(t, x) ≡ 0 for x > R + C 0 t.
Proof. The proof uses the same ideas as in Theorem 5.2.
As above we obtain the following result.
Corollary 5.5. Let (S) be a symmetric system. We assume that the func- Then the unique solution u of system (S) belongs to [C(I, L 2 A k (R d ))] m with u(t, x) ≡ 0 for x > R + C 0 t.