Guy Métivier
email: guy.metivier@math.u-bordeaux.fr

Remarks on Bronštein's root theorem

Introduction

In [START_REF] Bronštein | Smoothness of the roots of polynomials depending on parameters[END_REF], M.D.Bronštein proved that the roots of hyperbolic polynomials

(1.1) p(t, τ ) = τ m + m k=1 p k (t)τ m-k .
which depend on t varying in some interval I ⊂ R, are locally Lipschitz functions of t, if the coefficients p k are C r functions of t and the multiplicity of the roots does not exceed r; more importantly, he also proved that the roots are locally uniformly Lipschitz if p depends also continuously of parameters varying in a compact set. His proof is based on delicate inequalities between the coefficients p k implied by hyperbolicity. The goal of this note is to give a different and simpler proof, based on an induction on the degree. We will also drop compactness, to keep only uniform bounds of coefficients in W r,∞ .

This result is crucial in the proof of other estimates that M.D.Bronštein proved in his other paper [Br2], on the well posedness in Gevrey spaces of hyperbolic equations. We will also quickly discuss these estimates in the last section of this note, as they are indeed equivalent to the Lispchitz smoothness of the roots.

To fix the notations, we make the following assumption.

Assumption 1.1. For t ∈ I := [-T, T ], p(t, •) is a monic hyperbolic polynomial of degree m, which means that it has only real roots which denote by λ 1 (t) ≤ . . . ≤ λ m (t). We assume that their multiplicity is less than or equal to r ∈ {1, . . . , m}. We further assume that all the coefficients p k belong to the space W r,∞ (I) and that ∀k ∈ {1, . . . , m}, p k W r,∞ (I) ≤ R, (1.2)

∀t ∈ I, ∀τ, , r j=0 |∂ j τ p(t, τ )| ≥ δ > 0. (1.3)
Here W r,∞ (I) denotes the space of functions in L ∞ with derivatives up to order r in L ∞ . This is the space of C r-1 functions such that the (r -1)-th derivative satisfies a Lipschitz condition. The condition (1.3) is a quantitive way to express that the multiplicities of the roots do not exceed r.

Theorem 1.2. For all T 1 < T , there is a constant K which depends on T ,T 1 , R, δ and m, such that for all family of polynomials p(t, •) which satisfies the Assumption 1.1 one has for all t ∈ [-T 1 , T 1 ]:

(1.4) p(t, τ ) = 0 ⇒ |∂ t p(t, τ )| ≤ K|∂ τ p(t, τ )|.
Moreover, for all j = 1, . . . , m and t and t in [-T 1 , T 1 ]:

(1.5)

|λ j (t) -λ j (t )| ≤ K|t -t |.
The property (1.4) is stated in Lemma 4 of [START_REF] Bronštein | Smoothness of the roots of polynomials depending on parameters[END_REF], as an intermediate step in the proof of the Lipschitz estimate. The equivalence of (1.4) and (1.5) for strictly hyperbolic polynomials is immediate, since the implicit function theorem implies that the roots roots λ j and that

∂ t λ j (t) = -∂ t p(t, λ j )/∂ τ p(t, λ j ).
Approximating p by strictly hyperbolic polynomials, immediately implies the equivalence of (1.4) and (1.5) (see Lemma 2.5 below an the discussion which follows).

To prove (1.4) at t 0 , we will make several reductions. When r < m, we will factorize p, on an interval of length which depends only on R and δ, into products of polynomials of degree at most r, and thus the general case follows from the case where we take r = m. Note that in this case, the condition (1.3) is trivial since ∂ m τ p(t, τ ) = m!. Moreover, by translation and dilatation in t, the key estimate is the following: Proposition 1.3. For all R, there is a constant K which depends on R and m, such that for all family of polynomials p(t, •) on [-1, 1] which satisfies

(1.6) ∀k ∈ {1, . . . , m}, p k W m,∞ ([-1,1]) ≤ R, then (1.7) p(0, τ ) = 0 ⇒ |∂ t p(0, τ )| ≤ K|∂ τ p(0, τ )|.
This proposition is proved by induction on m in Section 3 and the proof of Theorem 1.2 is completed in Section 4. Additional remarks are given in the last Section 5. Before starting the proof we collect in Section 2 several preliminary lemmas which will be needed later on.

Remark 1.4. The Lipschitz regularity of the root stated in Theorem 1.2 is local in ] -T, T [ and in general, does not extend to the boundary, as shown by the example p(t, τ ) = τ 2 -t on [0, 1]. The estimates are not valid at t = 0.

Remark 1.5. The condition on the W r,∞ norm of the coefficients is sharp, in the sense that it cannot be replaced by W s,∞ with s < r. For r = m = 2, this is seen on the example p(t, τ ) = τ 2 -a(t) where one can choose a ≥ 0 in W s,∞ with s < 2 and √ a not Lipschitz. This can be extended in the following way.

Proposition 1.6. For all m ≥ 2 and > 0, there are hyperbolic polynomials (1.1) with coefficients p k in C m-([-1, 1]) such that there is a root λ j (t) which is not Lipschitz continuous near the origin.

The proof is given in Section 4 below.

Preliminaries

In this section, we gather several elementary lemmas which will be useful during the proof.

Sets of hyperbolic polynomials

Denote by P m the set of monic polynomials (2.1)

p(τ ) = τ m + m-1 k=0 p m-k τ k = m j=1 (τ -λ j )
and we identify p with (p 1 , . . . , p m ) ∈ C m . In particular we use the norm |p| = max |p k | on P m . It is also natural to introduce the quantity

(2.2) p = max |p k | 1/k , since for all ρ > 0, (2.3) p ρ (τ ) = ρ m p(τ /ρ) ⇒ p ρ = ρ p .
Moreover, the roots satisfy

(2.4) ∀j, |λ j | ≤ 2 p since 0 = |p(λ j )| ≥ |λ j | m 1 - m-1 k=0 p /|λ j | m-k
and the parenthesis is positive when p /|λ j | < 1/2.

We denote by H m the set of hyperbolic polynomials, that is polynomials with only reals roots λ j . We label them in nondecreasing order. For p ∈ H m , the coefficients p k are real. Let H m 0 denote the subset of polynomials p ∈ H m such that the coefficient p 1 vanishes. In this case, the roots λ j satisfy (2.5)

p 1 = λ j = 0, p 2 = - 1 2 λ 2 j ≤ 0. This implies that forall k > 2, |p k | ≤ m k |2p 2 | k/2 , or in short (2.6) p ∈ H m 0 ⇒ p ≤ m|2p 2 | 1 2 .

Smooth factorisation

Our proof relies on the factorization of polynomials and a good control of the factors. It is convenient to introduce the following terminology.

Definition 2.1. We say that there is an holomorphic factorization on a a neighborhood O of p ∈ P m if there are holomorphic mappings from φ j from O to P m j , for j = 1, . . . , l, with 1 ≤ m j < m, such that for all p ∈ O,

(2.7) p(τ ) = p 1 (τ ) . . . p l (τ ), p j = φ j (p).

Recall first the following local result.

Lemma 2.2. Suppose that p ∈ P m can be factorized as p 1 . . . p l with p j ∈ P m j with 1 ≤ m j < m with non common roots. Then, there is a neighborhood O and a holomorphic factorization on O with φ j (p) = p j . Moreover, if p is hyperbolic, all the factors φ j (p) are hyperbolic.

Proof. Consider sets D j which contains of all the roots of a j and such that

D j ∩ D k = ∅ for j = k.
For p in a neighborhood O of p, p does not vanish on ∪∂D j and by Rouché's theorem p has m j roots in D j , which yields the decomposition p = p 1 . . . p l . Moreover, the symmetric functions of the roots of p j are given by

σ k (p j ) = 1 2iπ ∂D j ∂ τ p(τ ) p(τ ) τ k dτ.
Thus they are holomorphic functions of the coefficients of p. Thus the coefficients of p j , which are polynomials of the σ k (p j ), are also holomorphic functions of the coefficients of p, proving the first part of the lemma.

Because the roots of p j are roots of p, it is clear that if p is hyperbolic then all the p j are hyperbolic.

Corollary 2.3. Suppose that K is a compact subset of P m of polynomials with roots of multiplicity at mots r < m. Then there is ρ > 0 such that for all p ∈ K, there is an analytic factorization on the ball of center p and radius ρ in polynomials p j of degree m j ≤ r.

Proof. By assumption and the lemma, there is an analytic factorization on a neighborhood of all p ∈ K. By compactness, there is ρ > 0 such that one can cover K by balls B j of radius ρ such that there is analytic factorization on the ball Bj of radius 2ρ with the same center. All p is contained in one of the B j and the ball of center ρ and radius ρ is contained in Bj .

In particular, we will use the following corollary.

Corollary 2.4. For R > 0, there is ρ > 0 such that for all p ∈ H m 0 such that

(2.8) |p| ≤ R, |p 2 | ≥ 1.
there is an analytic factorization p = p 1 p 2 on the ball of center p and radius ρ.

Proof. Denote by K the set of polynomials in H m 0 which satisfy (2.8). The roots of p ∈ K satisfy λ j = 0 and λ 2 j ≥ 2. Therefore there is at least one positive root and at least one negative root, which shows that the multiplicities of the roots is strictly less than m.

Approximation by strictly hyperbolic polynomials

Recall the following result proved by W.Nuij [Nuij] .

Lemma 2.5. If p ∈ H m and ε = 0, then p ε = (1 + ε∂ τ ) m p has m distinct real roots.

For the sake of completeness we sketch the proof. Let p = (τ -µ j ) r j where the µ 1 < . . . < µ l are distinct. The roots of p + ε∂ τ p are the multiple roots µ j of p, with multiplicity decreased by one, and the zeros of f (τ ) = 1+ ε r j /(τ -µ j ), which vanishes in each interval ]µ j , µ j+1 [ and in -] -∞, µ 1 [ or in ]µ l , +∞[ depending on the sign of ε. Hence p + ε∂ τ p has m real roots counted with their multiplicity and the maximal multiplicity is decreased by one. Iterating m times, all the roots are real and simple.

Estimates for functions of one real variable

On intervals I ⊂ R and for integers r ≥ 1, we consider the spaces W r,∞ (I) of functions of class C r-1 such that the (r -1)-th derivative is Lipschitz continuous on I. We give below two elementary lemmas.

Lemma 2.6. There is a consant C which depends only on m such that for all a ∈ W m,∞ ([-1, 1]]) and j ≤ m:

(2.9)

∂ j t a L ∞ ≤ C a L ∞ + ∂ m t a L ∞ .
Proof. Intregrating m times from 0 to t we see that the function b

∈ W m,∞ ([0, 1]) such that ∂ m t b = ∂ m t a, ∂ j t b(0) = 0 for j < m, is bounded and ∂ j t b L ∞ ≤ ∂ m t a L ∞ .
Moreover, the function a -b is a polynomial of degree less than m and therefore its coefficients are bounded by its L ∞ norm, and there is a constant C which depends only on m such that

∂ j t (a -b) L ∞ ≤ C a -b L ∞ ≤ C a L ∞ + ∂ m t a L ∞ .
The lemma follows.

Lemma 2.7. For all nonnegative a ∈ W 2,∞ ([-1, 1]) one has

(2.10) |t| ≤ θ a(0)/M ⇒ 1 2 a(0) ≤ a(t) ≤ 2a(0), where M = a W 2,∞ ([-1,1]) and θ = ( √ 5 -2).
Proof. Recall first that for nonnegative functions a ∈ W 2,∞ ([-1, 1]):

(2.11)

|∂ t a(0)| ≤ max (2a(0) ∂ 2 t a L ∞ ) 1 2 , 2a(0) ≤ 2M 1 2 a(0) 1 2 .
The first estimate follows from the inequality which is valid for all s ∈ [-1, 1]:

0 ≤ a(s) ≤ a(0) + s∂ t a(0) + 1 2 s 2 M 2 with M 2 = ∂ 2 t a L ∞ . If |∂ t a(0)| ≤ M 2 we can choose s = -∂ t a(0)/M 2 and we find that |∂ t a(0)| 2 ≤ 2M 2 |a(0)|. If |∂ t a(0)| ≥ M 2 , we take s = ±1 to obtain that 0 ≤ a(0) -|∂ t a(0)| + 1 2 M 2 ≤ 0 ≤ a(0) - 1 2 |∂ t a(0)|
and thus |∂ t a(0)| ≤ 2a(0). Hence, with δ = a(0), one has

|a(t) -δ 2 | ≤ 2|t| √ M δ + 1 2 t 2 M ≤ 1 2 δ 2 if |t| ≤ ( √ 5 -2)δ/ √ M proving the lemma.

The main induction

In this section we prove Proposition 1.3 by induction on m, the case m = 1 being trivial. So we assume that m ≥ 2 and that the proposition is proved for polynomials of degree less than m. More precisely, we assume that for all R 1 , there is K 1 such that for all family of monic hyperbolic polynomials

{q(t, •), t ∈ [-1, 1]} of degree ≤ m -1 with W m-1,∞ coefficients and such that (3.1) ∀k, q k W m-1,∞ ([-1,1]) ≤ R 1 , one has (3.2) q(0, τ ) = 0 ⇒ |∂ t q(0, τ )| ≤ K 1 |∂ τ q(0, τ )|.
We fix R > 0 and we look for K such that (1.7) is satisfied for all family of polynomials p(t, •) ∈ H m which satisfy (1.6).

We first consider families of hyperbolic polynomials in H m 0 . Proposition 3.1. Given R, there is K such that for all family of hyperbolic polynomials {p(t, •) ∈ H m 0 , t ∈ [-1, 1]}, which satisfies the bound (1.6), the property (1.7) is satisfied.

Proof. a) Because p(t, •) ∈ H m 0 , we know that p 2 (t) ≤ 0. If p 2 (0) = 0 then, because m ≥ 2, ∂ t p 2 (0) = 0 and |p 2 (t)| ≤ 1 2 R|t| 2 .
Hence by (2.6) all the coefficients satisfy p k (t) = O(|t| k ). In particular, τ = 0 is the unique root of p(0, •) and therefore the estimate is trivially true since ∂ τ p(0, 0) = ∂ t p(0, 0) = 0.

From now on we assume that p 2 (0) < 0 and we set δ = -p 2 (0)/2 > 0.

b) Let ρ = ( √ 5 -2) (2/R). By Lemma (2.7), for |t| ≤ ρδ one has δ 2 ≤ -p 2 (t) ≤ 4δ 2 . Note that the interval [-ρδ, ρδ] is contained in [-1, 1] since δ ≤ R/2. By (2.6), the coefficients of p satisfy (3.3) ∀t ∈ [-ρδ, ρδ], |p k | ≤ Cδ k
where C depends only on m. For s ∈ [-1, 1], introduce the polynomial

(3.4) q(s, τ ) = δ -m p(sρδ, δτ ) = τ m + m k=2 q k (s)τ m-k
the coefficients of which are (3.5) q k (s) = δ -k p k (ρδs).

The estimate

(3.6) q k L ∞ ([-1,1] ≤ C follow from (3.3). Moreover, (3.7) |∂ m s q k (s)| ≤ ρ m δ m-k ∂ m t p k L ∞ ≤ R
, where R depends only on R. Here we crucially use that the order of derivation is larger than or equal to k, so that the exponent of δ is nonnegative.

By Lemma 2.6 one can interpolate between (3.6) and (3.7) and there is R which depends only on R and m such that

(3.8) ∀j ≤ m, ∀s ∈ [-1, 1], |∂ j t q k (s)| ≤ R . Moreover, by construction (3.9) -q 2 (s) = -δ -2 p 2 (sρδ) ≥ 1.
Summing up, we have proved that the q(s, •) are hyperbolic polynomials in H m 0 for s ∈ [-1, 1], which satisfy uniform bounds (3.8), and the additional condition -q 2 (s) ≥ 1. c) By Corollary 2.4 there is ρ > 0, which depends only on R , and an analytic factorization q = q 1 q 2 on the ball centered at q(0, •) of radius ρ. The condition |q(s, •) -q(0, •)| = max k |q k (s) -q k (0)| ≤ ρ is satisfied on the interval |s| ≤ ρ/R , and hence, there is T > 0, T ≤ 1, depending only on R and m, such that for s ∈ [-T, T ] (3.10) q(s, τ ) = q 1 (s, τ )q 2 (s, τ ).

Moreover, the coefficients of q 1 and q 2 are analytic functions of the coefficients of q and there is R such that their coefficients satisfy

∀j ≤ m, ∀s ∈ [-T, T ], , |∂ j t q l k (s)| ≤ R .
Consider then ql (s, τ ) = q(sT, τ ) defined for |s| ≤ 1. Then, there is R 1 , depending only on R and m, such that the ql are of degree less than m and satisfy the estimates (3.1) of the induction hypothesis. Therefore, there is K 1 which depends only on R 1 and m such that the polynomials q1 and q2 satisfy (3.2). Therefore, the q l satisfy (3.2) with the constant K 1 /T . At the roots of q 1 one has ∂ t q = ∂ t q 1 q 2 and ∂ τ q = ∂ τ q 1 q 2 . There is a similar and symmetric result at the roots of q 2 . Because q 1 and q 2 have no root in common, this implies that (3.11) q(0, τ ) = 0 ⇒ |∂ t q(0, τ

)| ≤ K 1 /T |∂ τ q(0, τ )|.
d) From the definition (3.4) of q, we see that q(0, τ ) = 0 is equivalent to p(0, δτ ) = 0 and that ∂ s q(0, τ ) = δ 1-m ρ∂ t p(0, δτ ), ∂ τ q(0, τ ) = δ 1-m p(0, δτ ).

Hence (3.11) implies that at roots of p(0, τ ) = 0 (3.12)

|∂ t p(t 0 , τ )| ≤ K 1 /T ρ |∂ τ p(t 0 , τ )|.
This shows that (1.7) is satisfied with the constant K = K 1 /T ρ which depends only on R and m.

We now relax the condition p 1 = 0 and finish the proof of Proposition 1.3.

Proposition 3.2. Given R, there is K such that for all family of hyperbolic polynomials {p(t, •) ∈ H m , t ∈ [-1, 1]}, which satisfies the bound (1.6), the property (1.7) is satisfied.

Proof. Consider (3.13) p(t, τ ) = p(τ + p 1 (t)/m).

The polynomials p(t, •) all belong to H m 0 : one has p1 = 0 and the other coefficients are

pk = k l=0 m-l m-k (-p 1 /m) k-l p l .
Thus, by (1.2), |∂ j t pk (t)| ≤ R 1 where R 1 depends only on R and m. Thus, by Proposition 3.1, there is K 1 , depending only on R 1 , such that

p(0, τ ) = 0 ⇒ |∂ t p0, τ )| ≤ K 1 |∂ τ p(0, τ )|. Because ∂ t p(t, τ ) = ∂ t p(t, τ + p 1 /m) + 1 m ∂ t p 1 (t) ∂ τ p(t, τ + p 1 /m) the estimate (1.7) follows with K = K 1 + R/m.
The proof of Proposition 1.3 is now complete.

4 Proof of Theorem 1.2

4.1 Proof of (1.4)

Recall the notations. The polynomials are defined for |t| ≤ T and we want to prove (1.4) for |t 0 | ≤ T 1 , where T 1 < T are given. By translation, we can always assume that t 0 = 0 and that the polynomials are defined on the fixed interval |t| ≤ T -T 1 . By dilation, we can further assume that T -T 1 ≥ 1, and therefore, the estimate (1.4) immediately follows from Proposition 3.2 when r = m. We now consider the case r < m. In addition to the uniform bound (1.2) of the W r,∞ norm of the coefficients, we assume (1.3). Again it is sufficient to prove the estimate for t 0 = 0 and T > 0, R and δ are fixed. Consider the compact set K ⊂ H m of polynomials with coefficients bounded by R and satisfying (4.1) min

τ ∈R r j=0 |∂ j τ p(τ )| ≥ δ.
This is a compact set of polynomials with roots of mulitplicity ≤ r. By Corollary 2.3, there is ρ which depends only on K, that is on R and δ, such that there is an holomorphic factorisation p = p 1 . . . p l for |p -p(0, •)| ≤ ρ.

Hence, for |t| ≤ ρ/R, there is a factorisation

p(t, τ ) = p 1 (t, τ ) . . . p l (t, τ )
where the coefficients of p j are holomorphic functions of the coefficients of p. Therefore, there is R 1 , which depends only on R and δ, such that the W r,∞ norms of the coefficients of the p j are bounded by R 1 . Moreover, the p j are hyperbolic of degree at most r. Hence, by the first step, there is

K 1 such that (4.2) p j (0, τ ) = 0 ⇒ |∂ t p j (0, τ )| ≤ K 1 |∂ τ p j (0, τ )|. This implies (1.4) with K = K 1 R l-1 1 .
4.2 Proof of (1.5)

As already said, (1.4) and (1.5) are equivalent when p is strictly hyperbolic. By Lemma 2.5, p ε = (1 + ε∂ τ ) m ∈ H m and the W r,∞ norm of its coefficients is uniformly bounded if ε ∈ [0, 1[. Moreover, decreasing δ if necessary, the property (1.3) remains satisfied uniformly in ε for ε small. Hence, there is a constant K such that the property (1.4) is satisfied for all p ε . Hence, for ε > 0 small, the roots of p ε are simple and satisfy on [-T 1 , T 1 ]

|λ ε,j (t) -λ ε,j (t )| ≤ K|t -t |.
The roots are continuous functions of ε and hence (1.5) follows.

Proof of Proposition 1.6

Consider a strictly hyperbolic polynomial q(τ ) with 0 as a root:

(4.3) q(τ ) = τ m + m-1 k=1 q m-k τ k .
In particular, q m-1 = ∂ τ q(0) = 0. There is δ > 0 such that for all real a ∈ [-δ, δ], q(τ ) + a is strictly hyperbolic, with roots µ 1 (a) < . . . < µ m (a) which are analytic in a. One of them, say µ j , vanishes at a = 0 and ∂ a µ j (0) = -1/q m-1 = 0. Therefore, decreasing δ if necessary, there is c > 0 such that

(4.4) |a| ≤ δ ⇒ |∂ a µ j (a)| ≥ c. For t ∈ [-1, 1], consider the polynomial p(t, a, τ ) = t m (q(τ /t) + a) = τ m + m-1 k=1 t m-k q m-k τ k + t m a.
It is hyperbolic and its roots are tµ l (a). In particular, for t > 0, one has λ j (t, a) = tµ j (a). Now we chose a = a(t) = δ cos(t -α ) with 0 < α < 1/m. The coefficients p 1 , . . . , p m-1 of p(t, a(t), •) are smooth, and the last one p m (t) = δt m cos(t -1-α ) belongs to the Hölder class C m-mα . On the other hand, the t derivative of its j-th root is

µ j (a(t)) + αδt -α sin(t -α ) ∂ a µ j (a(t)),
which by (4.4) is not bounded as t → 0.

Bronštein estimates for hyperbolic polynomials

In [Br2], M.D Bronštein proved the well posed-ness of the Cauchy problem in Gevrey spaces for a hyperbolic operator P (t, x, D t , D x ) , using the construction of a parametrix for e -tγ(Dx) P e tγ(Dx) , with principal symbol q = 1/P (t, x, τ -iγ(ξ), ξ). The key step is to prove that q is a good symbol, and this relies on the following estimates.

Theorem 5.1. Suppose that {p(t, •), t ∈ [-T, T ]} is a family of hyperbolic polynomials which satisfies Assumption 1.1. Then, for all T 1 < T there is a constant C such that for k ≤ m and l ≤ r, one has for t ∈ [T 1 , T 1 ] and

τ ∈ C with |Im τ | ≤ 1 (5.1) |Im τ | k+l ∂ k τ ∂ l t p(t, τ ) ≤ C |p(τ, a)|.
This theorem essentially rephrases the Proposition 2 of [Br2]. What we want to emphasize here is that these estimates are equivalent to (1.4) (1.5) so that Theorems 1.2 and 5.1 are somehow equivalent.

The estimates of the τ derivatives are classical, using that

(5.2) |p(t, τ )| ≥ c|Im τ | r
and that m! k! ∂ k τ p is monic and hyperbolic with maximal multiplicity at most r -1.

The estimates of the derivatives with respect to the parameter t are proved by induction on the degree m using the following lemma which follows from (1.4) (1.5). This lemma is implicit in [Br2] at the bottom of page 92, but it is sufficiently important to be stated on its own. Lemma 5.2. Under Assumption 1.1, given T 1 < T , there is ε 1 > 0 such that for |ε| ≤ ε 1 q = ∂ τ p+ε∂ t p is hyperbolic with roots of multiplicity ≤ r-1.

Proof. For the convenience of the reader we recall the essence of Bronštein's proof. Write

(5.3) p(τ, a) = τ m + m k=1 p k (a)τ m-k = m j=1 (τ -λ j (a)).
Because p is monic, q is of degree ≤ m -1 and the coefficient of τ m-1 is

(5.4) a(t) = m + ε∂ a p 1 (t)
Thus there is ε 1 > 0 depending only on R, such that for ε ≤ ε 1 , a ≥ 1 2 m and q is of degree m -1.

Suppose that µ 1 < . . . < µ n are the distinct roots of p(t 0 , •) with multiplicity r j . For k ≤ r j -1, µ j is a root of ∂ k-1 τ p(t 0 , •) and thus (1.4) implies that there is K, which depends only on T 1 and on the bounds R and δ such that (5.5)

|∂ t ∂ k-1 τ p(t 0 , µ j )| ≤ K|∂ k τ p(t 0 , µ j )|.
In particular, if r j > 1, µ j is a root of ∂ τ p(t 0 , •), thus of ∂ t p(t 0 , •) and q(t 0 , •), of multiplicity ≥ r j -1. Moreover (5.5) implies that when τ → µ j ,

|∂ t p(t 0 , τ )| ≤ K + O |τ -µ j |) |∂ τ p(t 0 , τ )|.
Therefore, if |ε|K < 1, q(t 0 , τ )/∂ τ p(t 0 , τ ) > 0 for τ close to µ j . On ]µ j , µ j+1 [, p(t 0 , •) has a constant sign, and ∂ τ p(t 0 , •) has opposite sign near the ends µ j and µ j+1 . By the remark above, near the ends, q(t 0 , •) has the same sign as ∂ τ p(t 0 , •), provided that |ε|K < 1. Thus q(t 0 , •) has opposite sign near the ends and q has also a root in the interval ]µ j , µ j+1 [. Because q is of degree m -1, this implies that q(t 0 , •) has only real roots: the µ j with multiplicity r j -1 and one simple root in each of the interval ]µ j , µ j+1 [. Proof of Theorem 5.1. We proceed by induction on l.

a) If l = 0, denoting by λ j the roots of p, one has

∂ k τ p/p = |I|=k j∈I (τ -λ j ) -1 ,
where I denotes subsets of {1, . . . , m} and |I| is the number of elements in it. The estimate follows since |τ -

λ j | ≥ |Im τ |. b) Similarly, if l = 1, (∂ k τ ∂ t p)/p is a sum of terms ∇λ j j∈I (τ -λ j ) -1
with |I| = k + 1. The estimate follows, knowing the estimate of ∇λ j from Theorem 1.2. c) Suppose now that the estimate has been proved up to derivatives of order l -1. Consider the polynomial q = 1 a ∂ τ p + ε∂ a p , a = m + ε∂ t p 1 .

By Lemma 5.2, q is monic and hyperbolic, and for T 2 ∈]T 1 , T [ and ε small, depending only on R and δ, l the family {q(t, •), t ∈ [-T 2 , T 2 ]} satisfies the Assumption 1.1 for some constants R 1 and δ 1 which depend only on R and δ. Hence, one can apply the induction hypothesis to q and conclude that and the estimate is proved at the order l.

∂ k τ ∂ l-1 t q(τ,
Remark 5.3. Theorems 1.2 and (5.1) are closely related. We have used the former to prove the latter, but the converse is also true : if the estimates (5.1) are satisfied, then the main estimate of (1.4) is satisfied. Indeed, it is sufficient to assume that

γ |∂ t p(t 0 , τ -iγ)| ≤ K 1 |p(t 0 , τ -iγ)|.
At a root τ 0 of p(t 0 , τ 0 ) = 0, the right hand side is γ |∂ τ p(t 0 , τ 0 | + O(γ 2 ). Thus dividing out by γ and letting γ tend to 0 implies that |∂ t p(t 0 , τ 0 )| ≤ K 1 |∂ τ p(t 0 , τ 0 )|.

Remark 5.4. When p(τ, ξ) is polynomial in (τ, ξ) the estimates (5.1) are proved in a completely different way in [Hör] Theorem 12.4.6, using real algebraic geometry. Bronštein's theorem is much more general since it applies when the coefficient are only C r functions of the parameters. Note that (5.7) is indeed equivalent to (5.1) for k = 0, since there is a constant C such that for all polynomial and all γ > 0, q(s) = l≤r q l s l one has

C -1 sup γ l |q l | ≤ sup |s|≤γ |q(s)| ≤ C sup γ l |q l |
A direct proof of (5.7) and (5.8) is given in [CNR].

  a) ≤ C|Im τ | -k-l+1 |q(τ, a)| ≤ C|Im τ | -k-|α| |p(τ, a)|, since, by step b), |q| ≤ C|Im τ | -1 |p|.One has similar estimates for aq, since a is smooth, and for ∂ τ p by the induction hypothesis, thus ε∂ k τ ∂ l t p(τ, a) |Im τ | -k-|α| |p(τ, a)|

  implies that for |s| |Im τ |, (5.7) |p(t, s, τ )| |p(t, τ )| and that there is δ > 0 such that for |s| ≤ δ|Im τ |, , τ )| ≤ |p(t, s, τ )| ≤ 2|p(t, τ )|.


Dr Josef Finsterer
email: fifigs1@yahoo.de

Correspondence

To the Editor: With interest, we read the article by Kwon et al. [START_REF] Kwon | Peripheral nerve block is safely administered in a patient with Kearns-Sayre syndrome[END_REF] about a single patient with Kearns-Sayre syndrome (KSS), who successfully and without complications underwent peripheral nerve block for multiple muscle biopsies. We have the following comments and concerns.

The main disadvantage of this study is that the diagnosis of KSS was not genetically confirmed. Although the patient underwent muscle biopsy, the authors did not report the results of genetic studies of the mitochondrial DNA (mtDNA) extracted from the muscle. [START_REF] Kwon | Peripheral nerve block is safely administered in a patient with Kearns-Sayre syndrome[END_REF] The cause of KSS is usually a single mtDNA deletion; [START_REF] Mancuso | Redefining phenotypes associated with mitochondrial DNA single deletion[END_REF] however, in rare cases, single mtDNA point mutations might be causative. [START_REF] Zhang | Phenotype heterogeneity associated with mitochondrial DNA A3243G mutation (In Chinese)[END_REF] KSS should not be diagnosed upon a muscle biopsy alone.

The authors should also provide information if the family history was positive or negative for a mitochondrial disorder (MID). Did any of the first-degree relatives present with a history of a neuromuscular disorder, in particular, an MID, or did any of them have a history of complications during anesthesia? It is not conceivable why the patient required multiple muscle biopsies. [START_REF] Kwon | Peripheral nerve block is safely administered in a patient with Kearns-Sayre syndrome[END_REF] In general, a single biopsy is sufficient to show typical histological, immune-histological, or ultrastructural abnormalities, such as ragged-red fibers, cytochrome-C-oxidase-negative fibers, abnormal mitochondria with both unusual cristae and paracrystalline inclusions and biochemical abnormalities, such as complex-I-, complex-II-, complex-III-, or complex-IV-deficiency, alone or in combination.

It should be mentioned that general anesthesia or even regional anesthesia might deteriorate myopathy and, thus, weakness in MID patients. [START_REF] Finsterer | Deterioration of Kearns-Sayre syndrome following articaine administration for local anesthesia[END_REF][START_REF] Kawagoe | A case of intraoperative cardiac arrest in a patient with mitochondrial encephalomyopathy undergoing lung resection[END_REF] Did the patient experience previous complications during anesthesia? Did he ever receive long-duration or high-dose propofol, which should be avoided in MID patients? Since KSS might be associated with cardiomyopathy, we should be informed about the results of long-term electrocardiogram (ECG) recordings, echocardiography, and cardiac magnetic resonance imaging. KSS patients might present with hypertrophic cardiomyopathy, dilated cardiomyopathy, or Takotsubo syndrome (TTS). It should be discussed that particularly general anesthesia and surgery in MID patients might trigger a TTS. [START_REF] Kawagoe | A case of intraoperative cardiac arrest in a patient with mitochondrial encephalomyopathy undergoing lung resection[END_REF] Long-term ECG recordings are important since KSS patients may die from sudden cardiac death (SCD). Since SCD might be even the initial manifestation of cardiac involvement in KSS, it is essential to monitor KSS patients cardiologically as soon as the neurological diagnosis is established. Cardiac conduction defects or arrhythmias preceding SCD in KSS patients include QT-prolongation, Torsade de pointes, or polymorphic ventricular tachycardia. KSS patients rather require an implantable cardioverter defibrillator than a pacemaker.

It is also important to mention that KSS patients usually present with short stature. Which was the height of the presented patient, his parents, and his siblings?

Overall, this interesting case requires genetic confirmation of the diagnosis since MIDs frequently show a wide range of phenotypic mimicry and a work-up of first-degree family members. To profit most from the description of a KSS patient, all aspects of the syndrome should be addressed. Particularly, addressed should be cardiac involvement since these patients might die from SCD.

Financial support and sponsorship

Nil.

Conflicts of interest

There are no conflicts of interest.

Diagnosing Kearns-Sayre Syndrome Requires Genetic Confirmation

Josef Finsterer 1 , Sinda Zarrouk-Mahjoub 2

1 Krankenanstalt Rudolfstiftung, 1030 Vienna, Austria 2 Genomics Platform, Pasteur Institute of Tunis, 1002 Tunis, Tunisia


F Colombini

Guy Métivier
email: guy.metivier@math.u-bordeaux.fr.

The Cauchy problem for weakly hyperbolic systems

Keywords: MSC Classification : 35 L 50, 35 L 45, 35 L 40 hyperbolic systems, Cauchy problem, symmetrizers, well posedness, Gevrey spaces

. The main new ingredient is the construction, for all hyperbolic matrix A, of a family of approximate symmetrizers, S ε , the coefficients of which are polynomials of ε and the coefficients of A and A * .

Introduction

It is now well known that the Cauchy problem is not necessarily well posed in C ∞ for general weakly hyperbolic equations or systems (see e.g. [CDGS] 1 [CS] for counterexamples and [IvPe] for necessary conditions). On the other hand, in [Br], M.D.Bronštein has proved that the Cauchy problem for weakly hyperbolic equations and systems is well posed in Gevrey spaces G s for s ≤ 1+1/(m-1) where m is the maximum multiplicity of the characteristics, provided that the coefficients themselves have the Gevrey regularity G s with respect to the space variables and are sufficiently differentiable in time, see also [START_REF] Kajitani | The Cauchy problem for nonstrictly hyperbolic systems in Gevrey Classes[END_REF] for systems. A different proof, based on symmetrizer technics, is given in [CNR]. The result has also been extended to nonlinear systems, [START_REF] Kajitani | The Cauchy problem for nonlinear hyperbolic systems[END_REF].

For systems, not only the algebraic multiplicities are important, but also the diagonalizability properties play a role. In particular, for uniformly diagonalizable systems, it is shown in [START_REF] Kajitani | The Cauchy problem for uniformly diagonalizable hyperbolic systems in Gevrey classes[END_REF] that the Cauchy problem is well posed in G s , for s < 2, independently of the multiplicities.

It has been noticed that the general result can be improved in some special cases, in particular when the coefficients depend only on time. For the wave equation

(1.1) ∂ 2 t u - a j,l (t)∂ x j ∂ x l u = f,
with a j,l (t)ξ j ξ l ≥ 0 for all t and ξ, it is proved in [CJS] that the Cauchy problem is well posed in Gevrey spaces G s for s < 1 + k/2 if the coefficients a j,k are C k . In the same paper, the authors prove that the threshold index 1 + k/2 is sharp. This result has been extended to 2 × 2 uniformly diagonalizable systems in [CoNi]. However, S.Tarama has proved in [START_REF] Tarama | Une note sur les Systèmes Hyperboliques Uniformément Diagonalisables[END_REF] that the Cauchy problem is well posed in Gevrey classes G s for s < 1 + k for a class of 2 × 2 uniformly diagonalizable systems, and, by counterexamples, that this index is sharp. We will show that Tarama's result extends to all 2 × 2 uniformly diagonalizable weakly hyperbolic systems.

More generally, this paper is concerned with the well posedness of the Cauchy problem in Gevrey spaces for N × N weakly hyperbolic systems (1.2)

Lu := ∂ t u + d j=1 A j (t)∂ x j u = ∂ t u + A(t, ∂ x )u.
The matrices A j are defined for t in some interval [0, T ]. Below, we always assume that L is weakly hyperbolic, that is that for all t and ξ ∈ R d , the eigenvalues of A(t, ξ) = ξ j A j (t) are real. Before stating our results, let us recall some definitions. For s > 1 and Ω ⊂ R d , the Gevrey space G s (Ω) is the set of C ∞ functions u on Ω such that for all compact set K ⊂ Ω there is a constant C such that

(1.3) ∀α ∈ N d , ∂ α x u L ∞ (K) ≤ C |α|+1 (|α|!) s .
We denote by G s 0 (Ω) the subset of functions which are compactly supported in Ω. For functions depending also on time, u(t, x) with (t, x) ∈ Ω a relatively open subset of [0, T [×R d , we say that u ∈ C 0 G s ( Ω) if all the derivatives ∂ α

x u are continuous on Ω and for all compact set K ⊂ Ω there is a constant C such that

(1.4) ∀α ∈ N d , ∂ α x u L ∞ ( K) ≤ C |α|+1 (|α|!) s .
Note that it is required that these estimates are valid up to t = 0 (when there are such points in Ω).

We use the following terminology:

Definition 1.1. We say that the Cauchy problem for (1.2) is locally wellposed in G s if for all neighborhood Ω ⊂ R d of the origin, there is a neighborhood Ω of 0 in [0, T [×R d such that for all f ∈ C 0 G s ( Ω)) and h ∈ G s (Ω) the problem

(1.5) Lu = f, u |t=0 = h, has a unique solution u ∈ C 0 G s ( Ω).
We say that the problem is globally well posed in

G s if for Ω = R d one can take Ω = [0, T [×R d .
Our main question is to know wether the smoothness in time of the coefficients improves Bronštein's threshold index m/(m -1). In general the answer is no: Theorem 1.2. There are weakly hyperbolic systems (1.2) with analytic coefficients such that the Cauchy Problem is not locally well posed in G s for s > N/(N -1). This result is quite elementary, but not in the literature to the knowledge of the authors. It is proved in Section 2. The idea is that, in general, the variation in time of the coefficients have the same effect as adding a general zero-th order term B to L, and even in the constant coefficient case, the well posedness is stable by such perturbations only if s ≤ m/(m -1).

This shows the importance of the uniform diagonalizability assumptions in the papers [START_REF] Kajitani | The Cauchy problem for uniformly diagonalizable hyperbolic systems in Gevrey classes[END_REF][START_REF] Colombini | Two by two strongly hyperbolic systems and Gevrey classes[END_REF][START_REF] Tarama | Une note sur les Systèmes Hyperboliques Uniformément Diagonalisables[END_REF] cited above, because in sharp contrast, the well-posedness of the Cauchy problem remains valid for all bounded perturbation B(t). Recall the definition: Definition 1.3. The system (1.2) is said to be uniformly diagonalizable if for all t and ξ the matrix A(t, ξ) is diagonalizable and the eigenprojectors are bounded uniformly with respect to t and ξ.

There are several equivalent conditions (see e.g. [Me] where this condition is called strong hyperbolicity of the symbol). One of them is that there exists a bounded family of symmetrizers, that is a family of self adjoint and positive matrices S(t, ξ) such that S(t, ξ)A(t, ξ) is self adjoint and S and S -1 are uniformly bounded.

Under this condition, the Gevrey index can be improved when the coefficients are smooth in time. For k ∈ N, µ ∈]0, 1], we denote by C k,µ ([0, T ]) the space of C k functions on [0, T ] such that their k-th derivative satisfies a Hölder condition of order µ (Lipschitz if µ = 1).

When k + µ > m(m -1), the following result improves the general result of K. Kajitani [Ka2] who proved the well posedness for s < 2.

Theorem 1.4. Consider a uniformly diagonalizable weakly hyperbolic system (1.2), with C k,µ coefficients. Then, for all bounded matrix B(t), the Cauchy problem for L + B is locally and globally well posed in Gevrey spaces G s with s < 1 + (k + µ)/m(m -1) where m is the maximal multiplicity of the eigenvalues.

Corollary 1.5. Consider a uniformly diagonalizable weakly hyperbolic system (1.2), with C ∞ coefficients. Then, for all bounded matrix B(t), the Cauchy problem for L + B is locally and globally well posed in all Gevrey spaces G s with s ∈ [1, ∞[. When m = 2 the computations are more explicit and we are able to get a better control of the symmetrizers. We will obtain in Section 5 the well posedness for s < 1 + k + µ for coefficients in the class C k,µ . This is sharp by S.Tarama's result [START_REF] Tarama | Une note sur les Systèmes Hyperboliques Uniformément Diagonalisables[END_REF].

Theorem 1.6. Consider a uniformly diagonalizable weakly hyperbolic system (1.2) with coefficients in C k,µ ([0, T ]), k ∈ N, µ ∈]0, 1]. If the multiplicity of the eigenvalues is at most 2, then , for all bounded matrix B(t), the Cauchy problem is locally and globally well posed in Gevrey spaces G s with s < 1 + k + µ.

In particular, this extends Tarama's result [START_REF] Tarama | Une note sur les Systèmes Hyperboliques Uniformément Diagonalisables[END_REF] to general 2 × 2 weakly hyperbolic systems. We will also show that the threshold index 1 + k + µ is also valid for a special class of N × N systems (see Theorem 7.4 below). For general m, the index 1 + (k + µ)/m(m -1) is likely to be not optimal, but it is sufficient to imply Corollary 1.5.

The paper is organized as follows. We prove Theorem 1.2 in Section 2. Next, in Section 3, we reduce the proof of the well-posedness in G s to the construction of approximate symmetrizers S(t, ξ) for the matrices A(t, ξ) = ξ j A j . In practice, for |ω| = 1, one constructs families of approximate symmetrizers S ε (t, ω) of A(t, ω) depending on the parameter ε > 0 and next one chooses

(1.6) S(t, ξ) = S ε (t, ω), ω = ξ |ξ| , ε = |ξ| -γ .
Two conditions are in competitions in the proof of the energy estimate:

(1.7) Im S ε (t, ω)A(t, ξ) ε|ξ|S ε (t, ω), ∂ t S ε (t, ω) ϕ ε (t, ω)ε -β S ε (t, ω)
where the ϕ ε (•, ω) are bounded in L 1 . The exponent of ε in the first estimate is just a normalization, while the exponent β is the key element of the analysis. One chooses γ in (1.6) to balance the two terms ε|ξ| and ε -β , that is γ = 1/(β + 1), so they are both equal to |ξ| α with α = β/(β + 1). By Gronwall's lemma, the amplification factor for the o.d.e. deduced by Fourier transform of equation Lu = 0, is e C|ξ| α . From here, one deduces the existence of solutions when the Fourier transforms of the data decay faster than e -C|ξ| α , and finally when the data belong to G s when s < 1/α = 1+1/β. For Lipschitz coefficients and uniformly diagonalizable systems, one easily construct symmetrizers such that ∂ t S ε ε -1 S ε , recovering the well posedness for s < 2 obtained in general in [START_REF] Kajitani | The Cauchy problem for uniformly diagonalizable hyperbolic systems in Gevrey classes[END_REF]. All our analysis aims to improve this estimate of the time derivative of the symmetrizers. For this, the new ingredient is the construction of new families of symmetrizers, the coefficients of which are polynomials of ε and of the coefficients of A(t, ω), thus of class C k,µ . Because of this regularity, and using the positivity of S ε , we can use the estimates of [START_REF] Colombini | Well-posedness in the Gevrey classes of the Cauchy problem for a nonstrictly hyperbolic equation with coefficients depending on time[END_REF][START_REF] Tarama | On the Lemma of Colombini, Jannelli and Spagnolo[END_REF], to obtain the bound in (1.7) for ∂ t S ε with a parameter β < 1, at least if k is large enough. The Colombini-Jannelli-Spagnolo-Tarama estimate is recalled in Section 4, while the construction of symmetrizers is performed in Sections 5 to 7, first for 2 × 2 systems, and then in general.

The general case

For general N × N system and maximal multiplicity N , the threshold index is N/(N -1) the index given by Bronštein's theorem.

Proof of Theorem 1.2. The counterexample is in dimension d = 1. Consider the nilpotent N × N matrix (2.1)

A 1 =       0 1 0 0 . . . . . . 1 0      
and the rotations in the plane generated by the first and the last vector of the basis:

(2.2)

Ω(t) =   cos t 0 -sin t 0 Id N -2 0 sin t 0 cos t   .
We consider the system

L = ∂ t + A(t)∂ x with (2.3) A(t) = Ω(t)A 1 Ω -1 (t).
Note that A(t) is an analytic function of t. For v(t) = Ω -1 (t)u(t), the equation Lu = f is transformed to

∂ t v + A 1 ∂ x v + Bv = Ω -1 f,
where

B = Ω -1 ∂ t Ω =   0 0 -1 0 0 0 1 0 0   .
Thus we are reduced to the perturbation of a constant coefficient nilpotent matrix, and we know that the optimal Gevrey index for the well posedness is N/(N -1): the eigenvalues of iξA + B are the roots of

τ N -(-iξ) N -1 + τ N -2 = 0
and their imaginary part grow like c|ξ| (N -1)/N , with c > 0. This implies Theorem 1.2.

3 The general strategy for proving the well posedness

Our analysis follows the original ideas of [START_REF] Colombini | Sur les équations hyperboliques avec des coefficients qui ne dépendent que du temps[END_REF][START_REF] Colombini | Well-posedness in the Gevrey classes of the Cauchy problem for a nonstrictly hyperbolic equation with coefficients depending on time[END_REF][START_REF] Colombini | Two by two strongly hyperbolic systems and Gevrey classes[END_REF][START_REF] Tarama | Une note sur les Systèmes Hyperboliques Uniformément Diagonalisables[END_REF] and relies on an energy method. We consider the Cauchy problem

(3.1) ∂ t u + d j=1 A j (t)∂ x j u + B(t)u = f, u |t=0 = h.
The main part of the analysis consists in solving this equation when f and h have compact support in x. In this case, we perform a Fourier transform in the space variables. Denoting by û(t, ξ) the Fourier transform of u, we are reduced to solve:

(3.2) ∂ t û + iA(t, ξ)û + B(t)û = f , û|t=0 = ĥ.
The main idea from [CDGS] is to use an energy method for this family of ordinary differential system depending on the parameter ξ. More precisely we aim to construct approximate symmetrizers S(t, ξ) which have the following properties:

(S1) For (t, ξ) ∈ [0, T ] × R d , S(t, ξ
) is a self-adjoint positive definite matrix; moreover there is a positive scalar function ∆(t, ξ) and there are constants C and M such that for all (t, ξ)

C -1 ∆(t, ξ)Id ≤ S(t, ξ) ≤ C∆(t, ξ)Id, (3.3) C -1 ξ -M ≤ ∆(t, ξ) ≤ C ξ M . (3.4)
where we use the notation

ξ = (1 + |ξ| 2 ) 1 2 . (S2) For all ξ ∈ R d , S(•, ξ) is absolutely continuous on [0, T ] and there is a function ϕ(•, ξ) ∈ L 1 ([0, T ]), a constant C and α ∈ [0, 1[ such that ϕ(•, ξ) L 1 ([0,T ]) ≤ C (3.5) ∂ t S(t, ξ) ≤ ξ α ϕ(t, ξ)S(t, ξ) a.e. t ∈ [0, T ], (3.6) (S3) There is a constant C such that for all (t, ξ) ∈ [0, T ] × R d (3.7) Im S(t, ξ)A(t, ξ) ≤ C ξ α S(t, ξ).
Theorem 3.1. Suppose that there is a family of approximate symmetrizers which satisfy the properties (S1), (S2) and (S3) above. Suppose that B ∈ L ∞ ([0, T ]). Then, for all index s < 1/α, the Cauchy problem (3.1) is locally and globally well posed in G s .

Proof. a) Estimates for the solutions of

(3.2). Fix ξ ∈ R d . For f ∈ L 1 ([0, T ], C N ) and h ∈ C N consider the solution u ∈ C 0 ([0, T ]; C N ) of the differential system (3.8) ∂ t u + iA(t, ξ)u + B(t)u = f, u |t=0 = h.
The energy

E(t) = S(t, ξ)u(t), u(t) is in W 1,1 ([0, T ]) and (3.9) ∂ t E = 2Re Su, f + ∂ t Su, u + 2 Im SAu, u) -2 Re SBu, u)
For all t and ξ, (3.3) implies that

|(SBu, u)| ≤ C∆ B L ∞ |u| 2 ≤ C (Su, u). Therefore ∂ t E ≤ 2Re Su, f + C(ϕ(t, ξ) + 1) ξ α + C E(t).
This implies that there are constant C 0 , C 1 and C 2 such that (3.10)

S(t, ξ)u(t), u(t) 1 2 ≤ C 0 e C 1 Φ(t) ξ α +tC 2 S(0, ξ)h, h 1 2 + C 0 e C 1 Φ(t) ξ α +tC 2 t 0 S(t , ξ)f (t ), f (t ) 1 2 dt with Φ(t) = t 0 ϕ(t )dt .
By (3.5) the function Φ are uniformly bounded and, using (3.3) and (3.4), we obtain that there is a constant γ such that

(3.11) u(t) ≤ C 1 ξ 2M e γ ξ α h + t 0 f (t ) dt .
b) Existence of solutions for compactly supported data.

If f and h are compactly supported in x are G s functions, there are C and δ > 0 such that (3.12)

∀(t, ξ), f (t, ξ) ≤ Ce -δ ξ 1/s , ĥ(ξ) ≤ Ce -δ|ξ| 1/s ,
for some δ > 0 and finite C. By step a) the solutions û(•, ξ) of the family of o.d.e. (3.2) satisfy

(3.13) û(t, ξ)) ≤ C 1 ξ 2M e γ ξ α e -δ ξ 1/s .
By assumption α < 1/s, and thus for δ < δ there is a constant C 2 such that

(3.14) û(t, ξ) ≤ C 2 e -δ |ξ| 1/s . This implies that û(t) is the Fourier transform of a function u(t) of class G s in x ∈ R d , solution of (3.1) on [0, T ] × R d .
c) Propagation of the support.

Following [CDGS], we use the Paley-Wiener theorem to prove that the solutions found in step b) have compact support in x. Indeed, if f and h are supported in the ball {|x| ≤ R}, their Fourier transform is an entire function of ξ which satisfy for (t, ξ)

∈ [0, T ] × C d : (3.15) f (t, ξ) ≤ Ce -δ ξ 1/s e R|Im ξ| , ĥ(ξ) ≤ Ce -δ|ξ| 1/s e R|Im ξ| .
The solution of the o.d.e. (3.2) is defined for all ξ ∈ C d and clearly holomorphic in ξ. We can estimate it using the symmetrizer S(t, Re ξ). There is a new term in the right hand side of (3.9): Re S(t, Re ξ)A(t, Im ξ)u, u . By (3.3), we know that

S(t, Re ξ)A(t, Im ξ)u, u ≤ C∆(t, Re ξ) |Im ξ| |u| 2 ≤ γ 1 |Im ξ|(S(t, Re ξ)u, u).
Continuing as in step a), instead of (3.13) we obtain that for

ξ ∈ C d , (3.16) û(t, ξ)) ≤ C 1 ξ 2M e γ ξ α e (R+γ 1 t)|Im ξ| e -δ ξ 1/s .
By Paley-Wiener theorem, this implies that u(t, •) is supported in the ball {|x| ≤ R + γ 1 t}.

d)

Local uniqueness and end of the proof of the theorem. By classical duality arguments, solving the backward Cauchy problem for data with compact support, one obtains the local uniqueness of solutions of (3.1), even in spaces of ultra-distributions (G s ) .

When f ∈ C 0 ([0, T ], G s ) and h ∈ G s , we can split them into locally finite sums of compactly supported functions, using a G s partition of unity. We can solve the Cauchy Problem for each piece by step a), and glue the pieces together using the finite speed of propagation of step c) and the local uniqueness result.

Remark 3.2. The constant γ = γ(T ) in (3.13) is estimated by

C sup ξ T 0 ϕ(t, ξ)dt.
Therefore, for the critical exponent s 0 = 1/α, the proof above would provide a local in time solution on [0, T ] × R d in G s 0 , as long as γ = γ(T ) < δ. It is not clear that this can be achieved simply by choosing T small because we do not know wether the ϕ(•, ξ) are uniformly integrable or not.

Remark 3.3. Of course, when α = 0, there is no exponential loss in (3.13), and the Cauchy problem is well posed in C ∞ .

4 Colombini-Jannelli-Spagnolo-Tarama's lemma and extensions

In [START_REF] Tarama | Une note sur les Systèmes Hyperboliques Uniformément Diagonalisables[END_REF], S.Tarama proved the following extension of a result obtained in [CJS] for nonnegative functions.

Lemma 4.1. Given 0 < µ ≤ 1 and k + µ ≥ 1, there is a constant C such that for all a ∈ C k,µ ([0, T ]) and δ > 0:

(4.1)

T 0 |∂ t a(t)| (|a(t)| + δ) 1-1/(k+µ) dt ≤ C a 1/(k+µ) C k,µ
This can be extended to δ = 0 and to functions with values in R n in the following way.

Proposition 4.2. Given 0 < µ ≤ 1 and k + µ ≥ 1, there is a constant C such that for all a ∈ C k,µ ([0, T ], R n ), there is a nonnegative function a in L 1 ([0, T ]) such that for almost all t ∈ [0, T ] (4.2) |∂ t a(t)| ≤ a (t) |a(t)| 1-1/(k+µ) , and
(4.3) a L 1 ([0,T ]) ≤ C a 1/(k+µ) C k,µ
Proof. Consider first the scalar case n = 1. The sequence

|∂ t a(t)| (|a(t)| + 1/j) 1-1/(k+µ)
is nondecreasing in j, hence by Beppo Levi's lemma its limit a (t) ∈ [0, +∞] satisfies

T 0 a (t)dt = lim j→∞ T 0 |∂ t a(t)| (|a(t)| + 1/j) 1-1/(k+µ) dt ≤ C
and hence a ∈ L 1 ([0, T ]). Moreover, for all t and j

|∂ t a(t)| ≤ a (t) (|a(t)| + 1/j) 1-1/(k+µ) .
One can pass to the limit at every point where a is finite, that is almost everywhere, and thus (4.2) follows.

If a = (a 1 , . . . , a n ) takes its values in R n , we can find functions a j in L 1 such that the estimate (4.2) is satisfied for all j. Choosing a of the form C a j , the proposition follows.

Remark 4.3. On the open set {a = 0}, the construction above gives (4.4)

a (t) = |∂ t a(t)| |a(t)| 1-1/(k+µ) .
On the set {a(t) = ∂ t a(t) = 0}, the inequality (4.2) is satisfied for arbitrary finite a (t), and for instance we can choose a (t) = 0. The set {a = 0, ∂ t a = 0} has Lebesgue measure equal to 0, since it is the union of the finite sets {a = 0, |∂ t a| ≥ 1/j}. Therefore, we can define a by (4.4) when a(t) = 0, and by 0 elsewhere. We will use the following corollary of Proposition 4.2

Proposition 4.4. Given 0 < µ ≤ 1 and k + µ ≥ 1, there is a constant C such that for all a ∈ C k,µ ([0, T ], R n ), ∆ ∈ C 0 ([0, T ]; R + ), δ > 0 satisfying (4.5) ∀t ∈ [0, T ], |a(t)| ≤ ∆(t) and δ ≤ ∆(t), the function ϕ(t) = |∂ t a(t)|/∆(t) satisfies (4.6) ϕ L 1 ([0,T ]) ≤ Cδ -1/(k+µ) a 1/(k+µ) C k,µ
.

Proof. By Proposition 4.2, one has

|∂ t a(t)| ≤ a (t)|a(t)| 1-1/(k+µ) ≤ a (t)∆ 1-1/(k+µ) ≤ a (t)δ -1/(k+µ) ∆(t)
and the proposition follows.

5 Proof of Theorem 1.6

In this section we consider 2 × 2 systems and more generally systems with eigenvalues of multiplicity at most two.

Symmetrizers for 2 × 2 systems

Consider general 2 × 2 matrices

(5.1)

A = λId + A , A = a b c -a , λ = 1 2 trA.
This matrix is hyperbolic, that is has real eigenvalues, when λ is real and h := a 2 + bc is real and nonnegative. A set of such matrices is uniformly diagonalizable if there is a positive real number δ, such that for all matrices in this set:

(5.2)

h ≥ δ|A | 2 .
Because A 2 = hId, an exact symmetrizer of A , and thus of A, is

(5.3) S = A * A + hId.
It is self adjoint and non negative since

Su, u = A u 2 + h u 2 .
Moreover, for matrices which satisfy (5.2), one has

(5.4) h u 2 ≤ Su, u = (1 + 1/δ)h u 2 .
The symmetrizer S is not definite positive when h = 0, that is when A has a multiple (double) eigenvalue. Following the ideas of [CDGS, CJS], one adds a (small) corrector to S and consider for ε > 0:

(5.5)

S ε = S + ε ε Id = A * A + (h + ε 2 )Id.
If the condition (5.2) is satisfied, then, with

h ε = h + ε 2 > 0 (5.6) ε|u| 2 ≤ h ε u 2 ≤ Su, u = (1 + 1/δ)h ε u 2 .
Moreover, Im S ε A = ε 2 Im A . Therefore, for matrices which satisfy (5.2), one has

(5.7) Im S ε Au, u ≤ ε 2 (h/δ) 1 2 u 2 ≤ δ -1 2 εh ε |u| 2 ≤ δ -1 2 ε S ε u, u ,
where we have used that εh

1 2 ≤ h + ε 2 = h ε . If the matrices A(t) are C k,µ functions of t ∈ [0, T ], the explicit form (5.3)
shows that S(t) and S ε (t) are C k,µ functions of t. Moreover, Proposition 4.2 implies that there is a constant C 0 and a function ϕ ∈ L 1 ([0, T ]) such that for all C k,µ matrix A (t):

∂ t A (t) ≤ ϕ(t)|A (t)| 1-1/(k+µ)

and

(5.8)

ϕ L 1 ([0,T ]) ≤ C 0 A 1/k+µ C k,µ .
Therefore, if the matrices A(t) satisfy (5.2), one has

∂ t S(t) A (t) ∂ t A (t) ϕ(t)h(t) 1-1/2k ϕ(t)ε -1/k h ε (t)
and thus

(5.9)

∂ t S ε (t)u, u ≤ C 1 ϕ(t)ε -1/k S ε (t)u, u .
where C 1 depends only on k, µ and the constant δ in (5.2).

Similarly, since h is quadratic in the coefficients of A , one has

(5.10)

∂ t h(t) ϕ(t)h(t) 1-1/2(k+µ) ϕ(t)ε -1/(k+µ) h ε (t).
Summing up we have proved the following:

Proposition 5.1. There is a constant C 0 and, given δ > 0, there is another constant C 1 , such that: for all C k+µ family of hyperbolic matrices A(t) which satisfy (5.2), the approximate symmetrizers S ε (t) defined by (5.5) for ε ∈ ]0, 1], satisfy the properties (5.4), (5.7) and there is a function ϕ ∈ L 1 ([0, T ]) satisfying (5.8) such that the inequalities (5.9) and (5.10) hold for almost all t.

The important point in this proposition, is that all the estimates are uniform in A satisfying (5.2) and bounded in C k+µ ([0, T ]).

Symmetrizers for 2 × 2 systems

Proposition 5.2. Consider a 2 × 2 matrix A(t, ξ), homogeneous of degree one in ξ. We assume that it is uniformly diagonalizable and that its coefficients are bounded in C k,µ ([0, T ]) for |ξ| = 1, with k ≥ 1. Then, there is a family of approximate symmetrizers S(t, ξ) which satisfy the properties (S1) (S2) and (S3) listed before Theorem 3.1, with parameter α = 1/(k + µ + 1).

Proof. The statement is trivial when |ξ| ≤ 1, so we only consider the case |ξ| ≥ 1. It is convenient to use polar coordinates ξ = ρω with ρ = |ξ| ≥ 1. We use the symmetrizers S ε (t, ω) associated to the matrices A(t, ω). They satisfy the estimates (5.6), (5.7) and (5.9) with constants independent of ω, with functions ϕ(•, ω) uniformly bounded in L 1 .

Thus k+µ+1) to balance the two terms ε -1/(k+µ) and ε|ξ|, and define

|∂ t S ε | ϕε -1/(k+µ) S ε , Im S ε A(t, ξ) ε|ξ|S ε Now we choose ε = |ξ| -(k+µ)/(
(5.11)

S(t, ξ) = S ε (t, ω), ω = ξ |ξ| , ε = |ξ| -k/(k+1) .
It satisfies the conditions (S2) and ( S3) with α = 1/(k + µ + 1). With this choice h(t, ξ) = h ε (t, ω) satisfies (5.12)

|ξ| -2 ≤ |ξ| -2k/(k+1) ≤ h(t, ξ) ≤ C
and there is a constant C such that

C -1 h(t, ξ)Id ≤ S(t, ξ) ≤ Ch(t, ξ)Id.
which means that the condition (S1) is also satisfied. For further use, we note that (5.13)

∂ t h(t, ξ) ≤ Cϕ(t)ε -1/(k+µ) h(t, ξ) ≤ Cϕ(t)|ξ| 1/(k+µ+1) h(t, ξ).
as a consequence of (5.10).

Symmetrizers for systems with multiplicities at most 2

Consider a N × N system (3.1), uniformly diagonalizable, with coefficients A j ∈ C k+µ ([0, T ]) with k ≥ 1. The eigenvalues of A(t, ξ) can be labelled in increasing order λ 1 (t, ξ) ≤ . . . ≤ λ N (t, ξ). We make the following assumption.

Assumption 5.3. For all j and (t 0 , ξ 0 ) ∈ [0, T ] × R d \{0}, either the multiplicity of λ j (t, ξ) is constant on a neighborhood of (t 0 , ξ 0 ), or it is less than or equal to two.

Proposition 5.2 has the following extension, which, together with Theorem 3.1, finishes the proof of Theorem 1.6.

Theorem 5.4. Under the assumptions above, there is a family of approximate symmetrizers S(t, ξ) which satisfy the properties (S1) (S2) and (S3) with parameter α = 1/(k + µ + 1). Hence, for all B ∈ C 0 ([0, T ]), the Cauchy problem (3.1) is locally and globally well posed in G s for s < 1 + k + µ.

Proof. We consider again the system on the Fourier side, for |ξ| ≥ 1. By finite partition of unity, it is sufficient to prove that for all (t 0 , ξ 0 ) with |ξ 0 | = 1, one can construct a symmetrizer with the desired properties on a conical neighborhood of this point in [0, T ] × R d \{0}.

Given (t 0 , ξ 0 ), there is an invertible matrix P (t, ξ) defined on a conical neighborhood of this point, homogeneous of degree 0 in ξ, C k+µ in t, such that (5.14) Ã(t, ξ) := P (t, ξ)A(t, ξ)P -1 (t, ξ) = diag(A 0 (t, ξ), . . . , A m (t, ξ)) is block diagonal, with i) A 0 has only eigenvalues of constant multiplicity, ii) A 1 , . . . , A m are 2 × 2 matrices. There is an exact C k,µ symmetrizer S 0 (t, ξ) for A 0 (t, ξ), satisfying (5.15)

C -1 Id ≤ S 0 ≤ C, ∂ t S 0 ≤ CS 0 , Im S 0 A 0 = 0.
On the given conical neighborhood, the 2 × 2 block A j have approximate symmetrizers given by Proposition 5.2. They satisfy (5.16)

     ∂ t S j (t, ξ) ≤ ξ 1/(k+µ+1) ϕ j (t, ξ)S j (t, ξ), ϕ j (•, ξ) L 1 ≤ C, Im S j (t, ξ) Ãj (t, ξ) ≤ C|ξ| 1/(k+µ+1) S j (t, ξ), C -1 h j (t, ξ)Id ≤ S j (t, ξ) ≤ Ch j (t, ξ)Id,
where the h j satisfy

(5.17)

|ξ| -2 ≤ h j (t, ξ) ≤ C, ∂ t h j (t, ξ) ≤ Cϕ j (t)|ξ| 1/(k+µ+1) h j (t, ξ).
The first candidate to symmetrize à would be diag(A 0 (t, ξ), . . . , A m (t, ξ)) but it has not necessarily the property (S1) since the h j are different. This can be corrected in the following way. Introduce ∆(t, ξ) = m j=1 h j (t, ξ),

and for j ≥ 1 ∆ j (t, ξ) = l =j h l (t, ξ).
Consider the symmetrizer

(5.18) S(t, ξ) = diag(∆A 0 , ∆ 1 S 1 , . . . , ∆ m S m ).
Each block is a symmetric matrix ≈ ∆Id, and thus there is C such that

(5.19) C -1 ∆(t, ξ)Id ≤ S(t, ξ) ≤ C∆(t, ξ)Id, while ∆ satisfies (5.20) |ξ| -2m ≤ ∆(t, ξ) ≤ C.
By (5.17), one has With (5.16), this implies that (5.21)

∂ t ∆ ≤ Cϕ|ξ| 1/(k+µ+1) ∆, ∂ t ∆ j ≤ Cϕ|ξ| 1/(k+µ+1) ∆ j with ϕ = 1 + ϕ j .
∂ t S(t, ξ) ≤ Cϕ(t, ξ)|ξ| 1/(k+µ+1) S(t, ξ).
The scalar factors ∆ j commute to the block A j and therefore

(5.22) Im S(t, ξ) Ã(t, ξ) ≤ C|ξ| 1/(k+1) S(t, ξ).

Hence S satisfies the properties (S1) (S2) and (S3) for Ã. Let

(5.23) S(t, ξ) = P * (t, ξ) S(t, ξ)P (t, ξ).

The properties (S1) and ( S3) are immediately transported from S to S.

Next, we have

∂ t S = P * ∂ t SP + ∂ t P * SP + P * S∂ t P.
By (5.21), the first term is O(ϕ|ξ| 1/(k+µ+1) S). By (5.19), the second and third terms are O(∆) = O(S). This shows that S also has the property (S2), which finishes the proof of the theorem.

6 Construction of symmetrizers

Preliminary remarks

A possible definition of approximate symmetrizers for a hyperbolic matrix A, that is a matrix with only real eigenvalues, is

(6.1) Σ ε = 2ε ∞ 0 e isA * e -isA e -sε ds = 2ε ∞ 0 e isM * e -isM ds with M = A -i 2 εId. When A is diagonalizable, A = λ j Π j and (6.2) Σ ε → Σ 0 = 2 Π * j Π j .
If K is a compact set of uniformly diagonalizable matrices, there is a constant C such that for A ∈ K and s ∈ R, one has e -isA u ≤ C|u|. Reversing time, this implies that |u| ≤ C e -itA u and therefore there is a constant C such that (6.3)

A ∈ K, ε ≥ 0 ⇒ C -1 |u| 2 ≤ (Σ ε u, u) ≤ C|u| 2 .
If A is a Lipschitz function of t with values in K, differentiating in time the o.d.e ∂ s u + iA(t)u = 0 yields the estimate

|∂ t (e -isA(t) | ≤ Cs|∂ t A(t)| which implies that (6.4) ∂ t Σ ε ≤ Cε -1 .
As mentionned in the introduction the proofs of Theorems 1.4 and 1.6 rely on an improvement of this estimate, and this is clearly what we did in Section 5 for 2 × 2 systems. In general, that is if the multiplicities are not constant, there is no hope that ∂ t Σ ε remains bounded as ε → 0. Our goal is to replace Σ ε by symmetrizers S ε which are smooth functions of the coefficients of the matrix A, up to ε = 0. The basic remark is the following.

Lemma 6.1. There is a polynomial ∆ of ε and the coefficients of A and A * , such that the coefficients of ∆Σ ε are polynomials of ε and the coefficients of A and A * .

Proof. We note that Σ ε satisfies (6.5)

Im (Σ ε A) = 1 2 εΣ ε -εId, Im (Σ ε M ) = -εId.
Denote by M the mapping Σ → ΣM -M * Σ. Because M and M * have no common eigenvalue, this mapping is a bijection on the space of N × N matrices, and Σ ε is the solution of (6.6)

M(Σ ε ) = -2iId.
By uniqueness, the solution of this equation is self adjoint, and we could have defined Σ ε as the solution of (6.6). The determinant ∆ of M is a polynomial of ε and the coefficients of A and A * , and Cramer's formula for the solution of (6.6) imply the Lemma.

The polynomials involved in this result are identified in the next subsection using a different approach which is motivated by the following remark: if one writes ∆M -1 as a polynomial of M and if one notes that M n (Σ) is a linear combination of M * k ΣM j with k + j = n, one obtains that ∆Σ ε has the form

(6.7) ∆Σ ε = σ k,l M * l M l .

The new approach

We start from expressions of the form (6.7), which extends (5.3). Changing the labeling for convenience, they are associated to polynomials

(6.8) S(X, Y ) = m-1 k=0 m-1 l=0 σ k,l X m-1-k Y m-1-l .
Given such a polynomial and a matrix M we define (6.9)

S(M * , M ) := m-1 k=0 m-1 l=0 σ k,l M * m-1-k M m-1-l .
The symmetry condition reads

(6.10) S(Y, X) = S(X, Y )
where, in general, given a polynomial P , we denote by P the polynomial whose coefficients are the complex conjugate of those of P . This condition implies that σ k,l = σ l,k and thus, for all M , that S(M * , M ) is self adjoint.

Lemma 6.2. Let S(X, Y ) be a polynomial (6.8) satisfying (6.10), and let

T (X, Y ) = -1 2i (X -Y )S(X, Y ).
Then for all matrix M , Im S(M * , M )M = T (M * , M ).

Proof. Because the variables M * and M do not commute, one has to be careful and we include a proof. The definition of T is that its coefficients τ k,l are

τ k,l = - 1 2i (σ k-1,l -σ k,l-1 ) with the convention that σ k,l = 0 if k < 0 or l < 0. Because S = S(M * , M ) is self adjoint, one has Im (SM ) = 1 2i (SM -M * S) = 1 2i σ k,l M * k M l+1 -M * k+1 M l = 1 2i (σ k,l-1 -σ k-1,l )M * k M l = T (M * , M ) as claimed.
Thus, to obtain symmetrizers for a hyperbolic matrix A, it is sufficient to consider polynomials S(X, Y ) such that (X -Y )S(X, Y ) can be factorized by the characteristic polynomial P A (Y ) of A on the right, and by the characteristic polynomial, P A (X) = P A (X), of M * on the left. Example 6.3. Suppose that the characteristic polynomial P A of A has real coefficients. For any polynomial Q with real coefficients one can define the polynomial

S(X, Y ) = P A (X)Q(Y ) -Q(X)P A (Y ) X -Y Then S(A * , A
) is self adjoint. Using Lemma 6.2 and the identity P A (A) = P A (A * ) = 0, we see that Im S(A * , A)A = 0.

One looks for approximate symmetrizers for M = A-i 1 2 Id, which satisfy (6.11) Im (SM ) = -ε∆Id were ∆ is real and positive. Denote by P M the characteristic polynomial of M . If Q is a polynomial such that (6.12)

P M (X)Q(X) -Q(X)P M (X) + 2iε∆ = 0, then (6.13) S(X, Y ) = P M (X)Q(Y ) -Q(X)P M (Y ) + 2iε∆ X -Y , (6.13
) is a polynomial and Lemma 6.2 implies that S(M * , M ) is a solution of (6.11).

Remark 6.4. Because P M (M ) = P M (M * ) = 0, it is sufficient to consider polynomials in R(X)/P M . In particular, we can bound the degree m -1 in (6.9) by N -1, where N is the dimension of the matrix.

Positive approximate symmetrizers for hyperbolic matrices

Consider a N × N hyperbolic matrix A with real eigenvalues λ j and denote by P A its characteristic polynomial:

(6.14) P A (X) = X N + N k=1 p k X N -k = N j=1 (X -λ j ).
The characteristic polynomial

P M of M = A -i 2 εId is (6.15) P M (X) = p A (X + iε/2) = N j=1 (X -λ j + iε/2).
The condition (6.12) leads to take ε∆ to be equal (up to a factor) to the resultant of P M and P M , which is (6.16)

Res (P M , P M ) = j,k (λ j -λ k -iε) = (-2iε) N (-1) N (N -1)/2 j<k |λ j -λ k | 2 + ε 2 .
There are too many factors ε, which can be eliminated as follows. Introduce the polynomial R such that (6.17)

P A (X -i 1 2 ε) -P A (X + i 1 2 ε) = -iεR(X, ε).
R has real coefficients, and denoting by R ε (X) = R(X, ε), we have

Res (P M , P M ) = Res (P M , P M -iεR ε ) = Res (P M , -iεR ε ) = (-iε) N Res (P M , R ε ).
Comparing with (6.16), this shows that Res (P M , R ε ) = (-1) N (N -1)/2 ∆ ε where (6.18)

∆ ε = j<k |λ j -λ k | 2 + ε 2 .
We note that ∆ ε is a polynomial in ε and the coefficients p 1 , . . . , p m , and thus a polynomial of ε and the coefficients [a j,k ] of the matrix A.

There are Q and Q 1 , polynomials with coefficients in Z of X, ε and the coefficients of P M and R ε , thus polynomials of X, ε and the coefficients [a j,k ], such that (6.19)

P M (X)Q 1 (X) + R ε Q(X) = 2∆ ε . Therefore 2iε∆ ε = P M (Q + iεQ 1 ) -P M Q.
Because ∆ ε is real:

-2iε∆ ε = P M (Q -iεQ 1 ) -P M Q
and by uniqueness of the decomposition when ∆ = 0, this implies that Q -iεQ 1 = Q and hence (6.20)

P M (X)Q(X) -Q(X)P M (X) + 2iε∆ ε = 0.
Summing up, we have proved the following: Lemma 6.5. There is a polynomial Q(X, ε, [a j,k ]), such that for all hyperbolic matrix A the identity (6.20) is satisfied.

Corollary 6.6. There is a polynomial ∆ of (ε, [a j,k ]) and there is a polynomial S, of (X, Y, ε, [a j,k ]), such that for all hyperbolic matrix A with coefficients [a j,k ], ∆(ε, [a j,k ]) is given by (6.18) and (6.21)

S ε = S(A * + iε/2, A -iε/2, ε, [a j,k ])
is a symmetric matrix such that

(6.22) Im S ε A = 1 2 εS -ε∆ ε Id.
Moreover, for ε > 0, S ε = ∆ ε Σ ε where Σ ε is given by (6.1).

Proof. It only remains to prove the last statement. S = S ε -∆ ε Σ ε satisfies 2iIm (SM ) = SM -M * S = 0. For ε > 0, M and M * have no common eigenvalue and thus S = 0.

Remark 6.7. Because everything is polynomial, we can take ε = 0 in the construction above. In this case, P M = P A , R 0 = P , ∆ 0 = ∆ is the discriminant of P A and S 0 is an exact symmetrizer of A. Moreover, if A is diagonalizable, then (6.23)

S(A * , A) = 2∆(P ) Π * k Π k
where A = λ k Π k is the spectral decomposition of A. This implies in particular that, on the set of strictly hyperbolic matrices, the symmetrizer Π * k Π k is a rational function of the coefficients of A.

Remark 6.8. The construction can be made using the minimal polynomial of A in place of the characteristic polynomial P A . This makes sense, if A is restricted to a set where this minimal polynomial is a smooth function of the coefficients of A.

The next result follows directly from (6.3).

Proposition 6.9. Let K denote a set of uniformly diagonalizable hyperbolic matrices. Then there is a constant C such that for all A ∈ K and all ε ≥ 0 the symmetrizers constructed above satisfy (6.24)

C -1 ∆ ε Id ≤ S ε ≤ C∆ ε Id.

Applications

We consider a system (3.1) with C k,µ coefficients on [0, T ]. We assume that the system is weakly hyperbolic and uniformly diagonalizable.

7.1 Proof of Theorem 1.4

The multiplicity of the eigenvalues is at most N and we first prove Theorem 1.4 with m = N , that is with no assumption on the multiplicities.

Proposition 7.1. There is a family of approximate symmetrizers S(t, ξ) which satisfy the properties (S1), ( S2) and (S3) listed before Theorem 3.1 with parameter α = N (N -1)/(k + µ + N (N -1)).

Proof. We can assume that |ξ| ≥ 1. For ω ∈ S d-1 , denote by S ε (t, ω) the approximate symmetrizers constructed in the previous section associated to the matrix A(t, ω). Then

C -1 ∆ ε (t, ω)Id ≤ S ε (t, ω) ≤ C∆ ε (t, ω)Id, (7.1) Im (S ε (t, ω)A(t, ω) ≤ CεS ε (t, ω). (7.2)
Moreover, by (6.18), (7.3)

C -1 ε N (N -1) ≤ ∆ ε (t, ω) ≤ C
By Corollary 6.6, the coefficients s ε,j,k of S ε (t, ω) are polynomials of the coefficients of A and A * , therefore are bounded in C k,µ ([0, T ]). By Proposition 4.4 they satisfy

|∂ t s ε,j,s (t, ω)| ≤ ϕ j,k (t, ε, ω) s ε,j,s (t, ω) 1-1/(k+µ) with (7.4) ϕ j,k (•, ε, ω) L 1 ([0,T ]) ≤ C
where C is independent of of ω and ε. Using (7.1) and ( 7.3) we obtain that

|∂ t s ε,j,s | ≤ ϕ j,k ∆ 1-1/(k+µ) ≤ ϕ j,k ε -N (N -1)/(k+µ) ∆.
Thus, with ϕ = ϕ j,k , (7.5)

| ∂ t S ε (t, ω)u, u | ≤ Cϕε -N (N -1)/(k+µ) ∆|u| 2 ≤ ϕ(t, ε, ω)ε -N (N -1)/(k+µ) S ε (t, ω)u, u .
To finish the proof, we define

(7.6) S(t, ξ) = S ε (t, ω), ω = ξ |ξ| , ε = |ξ| -k/(k+µ+N (N -1))
where the exponent is chosen to balance the term ε -N (N -1)/(k+µ) in (7.5) and the term ε|ξ| coming from the term Im S ε (t, ω)A(t, ξ). Their final contribution is O(|ξ| (N (N -1)/(k+µ+N (N -1)) ).

With Theorem 3.1 this proposition implies Theorem 1.4 with m = N .

When the maximal multiplicity m is less than N , we argue as in the proof of Theorem 5.4. We can restrict ourselves to |ξ| ≥ 1 and it is sufficient to construct symmetrizers in a conical neighborhood of an arbitrary given point (t 0 , ξ 0 ) with |ξ 0 | = 1. Given such a point, there is such a neighborhood on which one can perform a block reduction of A and write

(7.7) Ã(t, ξ) := P (t, ξ)A(t, ξ)P -1 (t, ξ) = diag(A 1 (t, ξ), . . . , A m (t, ξ))
where the diagonal blocks A j have a size N j ≤ m. Each block A j has approximate symmetrizers given by Proposition 7.1. They satisfy (7.8)

     ∂ t S j (t, ξ) ≤ ξ α j ϕ j (t, ξ)S j (t, ξ), ϕ j (•, ξ) L 1 ≤ C, Im S j (t, ξ) Ãj (t, ξ) ≤ C|ξ| α j S j (t, ξ), C -1 ∆ j (t, ξ)Id ≤ S j (t, ξ) ≤ C∆ j (t, ξ)Id,
where α j = N j (N j -1)/ k + µ + N j (N j -1) , and the ∆ j satisfy (7.9)

|ξ| -N j (N j -1) ≤ ∆ j (t, ξ) ≤ C, ∂ t ∆ j (t, ξ) ≤ Cϕ j (t)|ξ| α j ∆ j (t, ξ). Note that α j ≤ α := m(m -1)/ k + µ + m(m -1) . Introduce ∆(t, ξ) = m j=1 ∆ j (t, ξ), and for j ≥ 1 ∆ j (t, ξ) = l =j ∆ l (t, ξ).
Consider the symmetrizer (7.10) S(t, ξ) = diag(∆ 1 S 1 , . . . , ∆ m S m ).

Each block ∆ j S j is ≈ ∆Id N j , and thus there is C such that (7.11) satisfies the properties (S1), ( S3) and (S2). This finishes the proof of the construction of the symmetrizers and, with Theorem 3.1, the proof the Theorem 1.4 is complete.

C -1 ∆(t, ξ)Id ≤ S(t, ξ) ≤ C∆(t, ξ)Id, while ∆ satisfies (7.12) |ξ| -M ≤ ∆(t, ξ) ≤ C, M = N j (N j -1).

Remarks and additional results

When m = 2, the index 1 + (k + µ)/m(m -1) = 1 + (k + µ)/2 is not optimal as shown in Theorem 1.6. Indeed, for m = 2, we were able to take into account a more precise form of the symmetrizer to improve the estimate (7.5) of ∂ t S. Moreover, Theorem 1.4 is interesting only for k + µ > m(m -1), since the general Bronštein index for uniformly diagonalizable system is 2. This can be seen directly from the properties (6.3), (6.4) and (6.5) which imply that S(t, ξ) = Σ ε (t, ω) with ε = |ξ| -1 2 satisfy the conditions (S1) (S2) and (S3) with parameter α = 1/2. Thus the estimate (7.5) for ∂ t S brings an improvement only for k + µ > m(m -1).

To conclude, we give a class of N × N systems, for which one can get the optimal index 1 + k + µ, thus extending the optimal result of the 2 × 2 case.

Assumption 7.2. We consider a weakly hyperbolic N × N system (3.1) with matrices A j ∈ C k,µ ([0, T ]). For t ∈ [0, T ] and ω ∈ S d-1 , we denote by ∆(t, ω) the discriminant of the characteristic polynomial of A(t, ω) and by A (t, ω) = A(t, ω) -1 N trA(t, ω) Id the traceless part of A(t, ω). We assume that there is a positive constant δ such that for all t and ω (7.16) |∆(t, ω)| ≥ δ|A (t, ω)| N (N -1) .

Note that the discriminant ∆ depends only on the traceless part A . Note also that ∆ is an homogeneous polynomial of degree N (N -1) of the coefficients of A so that the inequality (7.16) is homogeneous in A . Moreover, the set of hyperbolic traceless matrices A such that |A | = 1 and ∆ ≥ δ is a compact set of strictly hyperbolic matrices. Thus, by homogeneity, the set of hyperbolic matrices such that (7.17)

|∆| ≥ δ|A | N (N -1)
is a set of uniformly diagonalizable matrices.

The condition (7.16) implies that A(t) is either strictly hyperbolic when ∆(t) > 0, or has an eigenvalue of multiplicity N at points where ∆ = 0.

Conversely, if A is in a set of uniformly diagonalizable matrices, one has

|A | 2 ≈ λ 2 j
where the λ j are the eigenvalues of A , while

∆ = j<l (λ j -λ l ) 2 .
Thus, in a set of uniformly diagonalizable matrices, the assumption (7.16) is equivalent to (7.18) ∀j, l, (λ j -λ l ) 2 ≥ δ 1 λ 2 j for some positive constant δ 1 .

Remark 7.3. For 2 × 2 systems, the assumption is equivalent to uniform diagonalizabilty.

Theorem 7.4. Under Assumption 7.2 the Cauchy problem 3.1 is well posed in Gevrey spaces G s with s < 1 + k + µ.

Proof. We use the symmetrizers S ε (t, ω) as in the proof of Proposition 7.1. We show how one can improve the estimate of ∂ t S ε , using (7.16). Indeed, S ε is an homogeneous polynomial of degree N (N -1) of ε and the coefficients of A and their complex conjugate, denoted by a = {a α }. Therefore the derivative of its coefficients s j,k are linear combinations of terms of the form ε n ∂ t a α q N (N -1)-n (a), 0 ≤ n < N (N -1),

where q N (N -1)-n is an homogeneous polynomial of degree N (N -1) -n -1 of a. Next we use Proposition 4.4 for each coefficient a α :

|∂ t a α | ≤ ϕ α (t)|a| 1-1/(k+µ)
and obtain the bound |∂ t s j,k | ≤ ε n ϕ(t)|a| N (N -1)-n-1/(k+µ) ≤ ε n ϕ(t)|A | N (N -1)-n-1/(k+µ) ,

where the L 1 norm of ϕ is uniformly controlled by the C k,µ norm of the coefficients. Using (7.16) and the bound (7.3) we conclude that (7.19) |∂ t s j,k | ≤ ϕε -1/(k+µ) ∆ and thus (7.20) |∂ t S ε | ≤ ϕε -1/(k+µ) S ε , which is (5.9). As in Proposition 5.2 we choose ε = |ξ| -(k+µ)/(k+µ+1) and thus obtain a family of symmetrizers which satisfy (S1) (S2) and (S3) listed before Theorem 3.1 with parameter α = 1/k + µ + 1.

We see that the difference with the proof of the general case is that we have improved the exponent of ε in the estimate of ∂ t S. For this, we have used Proposition 4.4 not for the coefficients of S themselves, but, using their polynomial structure, to the coefficients of A . But this procedure is successful only because we have assume a control of these coefficients by ∆, to obtain an estimate of ∂ t S by S.