Weakly hyperbolic equations with non-analytic coefficients and lower order terms

In this paper we consider weakly hyperbolic equations of higher orders in arbitrary dimensions with time-dependent coefficients and lower order terms. We prove the Gevrey well-posedness of the Cauchy problem under $C^k$-regularity of coefficients of the principal part and natural Levi conditions on lower order terms which may be only continuous. In the case of analytic coefficients in the principal part we establish the $C^\infty$ well-posedness. The proofs are based on using the quasi-symmetriser for the corresponding system. The main novelty compared to the existing literature is the possibility to include lower order terms to the equation as well as considering any space dimensions. We also give results on the ultradistributional and distributional well-posedness of the problem, and we look at new effects for equations with discontinuous lower order terms.


Introduction
In this paper we study the well-posedness of the weakly hyperbolic Cauchy problem where each A m−j (t, D x ) is a differential operator of order m − j with continuous coefficients depending only on t. Later we will also relax the continuity assumption replacing it by the boundedness. As usual, D t = 1 i ∂ t and D x = 1 i ∂ x . Let A (m−j) denote the principal part of the operator A m−j and let λ l (t, ξ), l = 1, ..., m, be the real-valued roots of the characteristic polynomial which we write as This means that τ m + m−1 j=0 |γ|=m−j a m−j,γ (t)ξ γ τ j = m l=1 (τ − λ l (t, ξ)).
The well-posedness of the weakly hyperbolic equations has been a challenging problem for a long time. For example, even for the second order Cauchy problem in one space dimension, (1.2) ∂ 2 t u − a(t, x)∂ 2 x u = 0, u(0, x) = g 1 (x), ∂ t u(0, x) = g 2 (x), up until now there is no characterisation of smooth functions a(t, x) ≥ 0 for which (1.2) would be C ∞ well-posed. On one hand, there are sufficient conditions. For example, Oleinik has shown in [16] that (1.2) is C ∞ well-posed provided there is a constant C > 0 such that Ca(t, x) + ∂ t a(t, x) ≥ 0. In the case of a(t, x) = a(t) depending only on t, when the problem becomes (1.3) ∂ 2 t u − a(t)∂ 2 x u = 0, u(0, x) = g 1 (x), ∂ t u(0, x) = g 2 (x), the Oleinik's condition is satisfied for a(t) ≥ 0 with a ′ (t) ≥ 0. On the other hand, in the celebrated paper [8], Colombini and Spagnolo constucted a C ∞ function a(t) ≥ 0 such that (1.3) is not C ∞ well-posed. The situation becomes even more complicated if one adds mixed terms to (1.3), even depending only on t and analytic. For example, the Cauchy problem for the equation x u = 0 is Gevrey G s well-posed for s < 2 while it is ill-posed for any s > 2. For other positive and negative results for second order equations with time-dependent coefficients we refer to seminal papers of Colombini, De Giorgi and Spagnolo [3] and Colombini, Jannelli and Spagnolo [5], and to Nishitani [15] for the necessary and sufficient conditions for the C ∞ well-posedness of (1.2) with analytic a(t, x) ≥ 0 in one dimension.
A reasonable substitute for the C ∞ well-posedness in the weakly hyperbolic setting is the well-posedness in the space G ∞ = s>1 G s . Thus, Colombini, Jannelli and Spagnolo proved in [4] that for every C ∞ function a(t) ≥ 0, the Cauchy problem (1.3) is G ∞ well-posed. More precisely, they showed that if a(t) is in C k , it is well posed in G s with s ≤ 1 + k/2, and if a(t) is analytic, it is C ∞ well-posed.
From another direction, there are also general results for (1.1). For example, it was shown by Bronshtein in [2] that, in paticular, the Cauchy problem (1.1) with C ∞ coefficients is G s well-posed provided that 1 ≤ s < 1+ 1 m−1 . In some cases, this can be improved. For example, for constant multiplicities, see paper [6] by Colombini and Kinoshita in one-dimension (see also D'Ancona and Kinoshita [9]), and the authors' paper [11] for further improvements of Gevrey indices and all dimensions, with a survey of literature therein.
In this paper our interest in analysing the Cauchy problem (1.1) is motivated by (A) allowing any space dimension n ≥ 1; (B) considering the effect of lower order terms or, rather, the properties of the lower order terms which do not influence the results on the Gevrey wellposedness (we will look at new effects for both continuous and discontinuous lower order terms); the inclusion of lower order terms in this setting has been untreatable by previous methods; (C) providing well-posedness results in spaces of distributions and ultradistributions.
Our main reference here is the paper [13] of Kinoshita and Spagnolo who have studied the Cauchy problem (1.1) for operators with homogeneous symbols in one dimension, x ∈ R. Under the condition (1.4) ∃M > 0 : on the roots λ j (t, ξ) they have obtained the following well-posedness result: Theorem 1.1 ( [13]). Assume that n = 1 and that the differential operator is homogeneous, i.e. .
The proof is based on the construction of a quasi-symmetriser Q (m) ε which thanks to the condition (1.4) is nearly diagonal. Previously, equations of second and third order with analytic coefficients, still with n = 1 and without low order terms, have been analysed by Colombini and Orrú [7]. They have shown the C ∞ well-posedness of (1.1) under assumption (1.4). Moreover, if all the coefficients a m−j (t) vanish at t = 0, they showed that the condition (1.4) is also necessary. So, for us it will be natural to adopt (1.4) for our analysis.
Let us briefly discuss the difficulties of aims (A)-(C) above. For the dimensional extension (A), even under condition (1.4) for the characteristic roots, for space dependent coefficients such an extension is impossible, see e.g. Bernardi and Bove [1], for examples of second order operators with polynomial coefficients for which the C ∞ well-posedness fails for any n ≥ 2. It is interesting to note that for these examples the usual Ivrii-Petkov conditions on lower order terms are also satisfied. As we will show, the C ∞ (and other) well-posedness holds in our case in any dimension n ≥ 1 since the coefficients depend only on time. In part (B), the proof of the well-posedness for equations with lower order terms highlights several interesting and somewhat surprising phenomena. For example, if the coefficients of the principal part are analytic and the lower order terms are only bounded (in particular, they may be discontinuous, or may exhibit more irregular oscillating behaviour), but the Cauchy data is Gevrey, we still obtain the solution in Gevrey spaces. Indeed, the Levi conditions in this paper control the zeros of the lower order terms but not their regularity. Finally, aim (C) is motivated by an interesting and challenging problem for weakly hyperbolic equations: analysing the propagation of singularities. For this, in order to be able to use also non-Gevrey techniques, we need to have first well-posedness in some bigger space. This will be achieved for the Cauchy problem (1.1) in the spaces of Beurling Gevrey ultradistributions. A subtle point of this construction is that we will have to use the Beurling Gevrey ultradistributions and not the usual Roumieu Gevrey ultradistributional class. In the case of the analytic principal part we will obtain well-posedness in the usual space of distributions.
In particular, in this paper we extend Theorem 1.1 to weakly hyperbolic equations with non-homogeneous symbols and in any space dimension n ≥ 1, and find suitable assumptions on the lower order terms for the Gevrey well-posedness. Already from the beginning we deviate from [13] by using pseudo-differential techniques to reduce the equation to the system. This will allow us to treat all the dimensions n ≥ 1. However, the main challenge in the present paper is the analysis of the lower order terms. In fact, in most (if not all) the literature on the application of the quasi-symmetriser to weakly hyperbolic equations the considered equations are always assumed to have homogeneous symbols. It is our intension to show that the quasi-symmetriser can be effectively used to control parts of the energy corresponding to the lower order terms. It is interesting to see the appearing Levi conditions expressing the dependence of the lower order terms on the principal part of the operator. Such control becomes possible by exploiting the Sylvester form of the system corresponding to equation (1.1), and the structure of the quasi-symmetriser.
An interesting effect that we observe is that the results remain true assuming just the continuity of the lower order terms in time. For example, we will have the C ∞ well-posedness for equations with analytic coefficients in the principal part and only continuous lower order terms. Moreover, we give a variant of our results with only assuming the boundedness of lower order terms in time (instead of continuity).
In this paper we formulate the conditions on the lower order terms in terms of the symbols A m−j+1 . Note that in (1.1), the operator A m−j+1 (t, D x ) is the coefficient in front of the derivative D j−1 t . We assume that there is some constant C > 0 such that we have for all t ∈ [0, T ], j = 1, . . . , m and for ξ away from 0 (i.e., for |ξ| ≥ R for some R > 0). For j = m, this is the condition on the low order terms coming from the coefficient in front of D m−1 t , in which case A 1 − A (1) is independent of ξ, and assumption (1.5) should read as which will be automatically satisfied due to the boundedness of A 1 in t. In Section 2 we will give examples of the condition (1.5). We will also show in treating the case m = 3 that from the point of view of the desired energy inequality for (1.1) the assumption (1.5) is rather natural.
We are now ready to formulate the well-posedness results. Part (i) of Theorem 1.2 is the extension of Theorem 1 in [13] to any space-dimension and to equations with low order terms. In the sequel D ′ (s) (R n ) (E ′ (s) (R n )) denotes the space of Gevrey Beurling (compactly supported) ultradistributions. For the relevant details on these spaces of ultradistributions and their characterisations, with their appearance in the analysis of weakly hyperbolic equations, we refer to our paper [11], where these have been applied to the low (Hölder) regularity constant multiplicities case. Theorem 1.2. Let n ≥ 1. If the coefficients satisfy A j (·, ξ) ∈ C([0, T ]) and A (j) (·, ξ) ∈ C ∞ ([0, T ]) for all ξ, the characteristic roots are real and satisfy (1.4), and the low order terms satisfy (1.5), then the Cauchy problem (1.1) is well-posed in any Gevrey space. More precisely, for A j (·, ξ) ∈ C([0, T ]), we have: for some k ≥ 2 and g j ∈ G s (R n ) for j = 1, ..., m, then there exists a unique solution u ∈ C m ([0, T ]; G s (R n )) provided that .
In the case of analytic coefficients, we have C ∞ and distributional well-posedness. By W ∞,m we denote the Sobolev space of functions having m derivatives in L ∞ . In the case of discontinuous but bounded lower order terms we have the following: (ii) Assume the conditions of Theorem 1.3 with A j (·, ξ) ∈ C([0, T ]) replaced by A j (·, ξ) ∈ L ∞ ([0, T ]), j = 1, . . . , m. Then the C ∞ well-posedness remains true provided that we replace the conclusion u ∈ C m ([0, T ]; C ∞ (R n )) by We refer to Remark 4.1 for a brief discussion of the strictly hyperbolic case. In this case, even in the situation of the low regularity of coefficients (C 1 ), one can analyse the global behaviour of solutions with respect to time (see [14]). The cases of constant coefficients and systems with controlled oscillations have been treated in [17] and [18], respectively.
Finally, we describe the contents of the sections in more details. Section 2 collects some motivating examples of applications of our results. In Section 3 we recall the required facts about the quasi-symmetriser and in Section 4 we use it to derive the energy estimate for the solutions of the hyperbolic system in Sylvester form corresponding to the Cauchy problem (1.1). The estimate on the part of the energy corresponding to lower order terms is given in Section 5. In Section 6 we prove Theorems 1.2, 1.3 and 1.4 and we end the paper with a final remark on the Levi conditions (1.5).

Examples
Let us first give an example of the Levi conditions (1.5) for the equations of third order, m = 3. In this case (1.5) become for some C > 0. It is convenient in certain applications, whenever possible, to write conditions (1.4) and (1.5) in terms of the coefficients of the equation. Such analysis for (1.1) has been recently carried out by Jannelli and Taglialatela [12]. In Example 3 below we will give an example of the meaning of conditions (2.1).
Since the hyperbolic equations above have homogeneous symbols, the coefficients are real. We refer to Colombini-Orrú [7] and Kinoshita-Spagnolo [13] for more examples of equations without lower order terms in one dimension n = 1.
We now give more examples, which correspond to the new possibility, ensured by Theorems 1.2 and 1.3, to consider equations with lower order terms and equations in higher dimensions n ≥ 1.
Example 1. As a first example we consider the second order equation Assume a 2,2 (t) is real and a 2,2 (t) ≤ 0. The condition (1.4) is trivially satisfied by the roots The well-posedness results of Section 1 are obtained under the conditions (1.5) on the lower order terms. In this case (1.5) means that the coefficient a 1,0 (t) is bounded on [0, T ] and that there exists a constant c > 0 such that |a 2,1 (t)ξ +a 2,0 (t)| 2 ≤ −ca 2,2 (t)ξ 2 for all t ∈ [0, T ] and for ξ away from 0. Note that this last condition holds if Take now the general second order equation ,0 (t)u = 0. As observed above condition (1.4) coincides with the bound from below . Here a 1,1 , a 2,2 are assumed real. The conditions (1.5) on the lower order terms are of the type |a 1,0 (t)| ≤ c 1 and |a 2, and for ξ away from 0.
The equation is an n-dimensional version of the previous example, with real a 1,j and a 2 . The condition (1.4) is trivially satisfied when a 2 (t) ≤ 0. The conditions (1.5) on the lower order terms are as follows: for t ∈ [0, T ] and ξ away from 0.
Example 3. We finally give an example of a higher order equation. Let where a(t), b(t) and c(t) are real-valued functions with b and c bounded above and from below by a (for instance, The Levi conditions (1.5) on the lower order terms are of the following type: for t ∈ [0, T ] and ξ away from 0.
where the matrices Q We finally recall that a family {Q α } of nonnegative Hermitian matrices is called nearly diagonal if there exists a positive constant c 0 such that 11 , ..., q α,mm }. The following linear algebra result is proven in [13, Lemma 1].
Lemma 3.2 is employed to prove that the family Q (m) ε (λ) of quasi-symmetrisers defined above is nearly diagonal when λ belongs to a suitable set. The following statement is proven in [13,Proposition 3].
We conclude this section with a result on nearly diagonal matrices depending on 3 parameters (i.e. ε, t, ξ) which will be crucial in the next section. Note that this is a straightforward extension of Lemma 2 in [13] valid for 2 parameter (i.e. ε, t) dependent matrices.
be a nearly diagonal family of coercive Hermitian matrices of class C k in t, k ≥ 1. Then, there exists a constant C T > 0 such that for any function V :

Reduction to a first order system and energy estimate
We now go back to the Cauchy problem (1.1) and perform a reduction of the morder equation to a first order system as in [19]. Let D x be the pseudo-differential operator with symbol ξ = (1 + |ξ| 2 ) 1 2 . The transformation with l = 1, ..., m, makes the Cauchy problem (1.1) equivalent to the following system The matrix in (4.1) can be written as By Fourier transforming both sides of (4.1) in x we obtain the system From now on we will concentrate on the system (4.3) and on the matrix for which we will construct a quasi-symmetriser. Note that the eigenvalues of the matrix A 1 are exactly the roots λ j (t, ξ), j = 1, ..., m. It is clear that the condition (1.4) holds for the eigenvalues ξ −1 λ j (t, ξ) of the 0-order matrix A(t, ξ) as well. Let us define the energy We recall that from Proposition 3.1 Q (m) ε (t, ξ) is a family of smooth nonnegative Hermitian matrices such that In addition there exists a constant C m > 0 such that for all t ∈ [0, T ], ξ ∈ R n and ε ∈ (0, 1] the following estimates hold uniformly in V : In the sequel we assume that the coefficients a j in the equation (1.1) are of class C k , or in other words that the matrix A(t, ξ) has entries of class C k in t ∈ [0, T ]. It follows by construction that the quasi-symmetriser has the same regularity property. We now estimate the three terms of the right hand side of (4.5).

4.2.
Second term. From the property (4.8) we have that , which is the main task in this paper. By Proposition 3.1(iv) and the definition of the matrix B(t, ξ) we have that where we notice that (Q . Note that from Proposition 3.1(i) we have that (Q 0 V, V ) ≤ E ε . In the next section we will show that the conditions on B corresponding to (1.5) imply that for some constant C 3 > 0 independent of t ∈ [0, T ], ξ ∈ R n and V ∈ C m .
holds for all V ∈ C m . From the hypothesis of strict hyperbolicity it easily follows that This bound from below implies and |ξ| ≥ 1 and hence the estimate holds trivially in the strictly hyperbolic case for any lower order term B (for our purposes it will not be restrictive to assume |ξ| ≥ 1). Concluding, when the roots λ i are distinct the Gevrey and ultradistributional well-posedness results in Theorem 1.2 and Theorem 1.3 can be stated without additional conditions on the lower order terms. Strictly hyperbolic equations under low regularity (Hölder C α , 0 < α < 1) of the coefficients have been analysed by the authors in [11], to which we refer for general statements on the Gevrey and ultradistributional well-posedness in this setting.

Estimates for the lower order terms
We begin by rewriting ((Q It follows that for some constant C > 0 independent of t, ξ and V , then the condition (4.9) will hold. It is our task to show that the condition (1.5) on the matrix B of the lower order terms implies the estimate (5.1). Before dealing with the general case of B m×m-matrix, let us consider the instructive case m = 3, which will illustrate the general argument in a simplified setting. In the sequel, for f and g real-valued functions (in the variable y) we write f (y) ≺ g(y) if there exists a constant C > 0 such that f (y) ≤ Cg(y) for all y. More precisely, we will set y = (t, ξ) or y = (t, ξ, V ). 5.1. The case m = 3. By definition of the row vectors W where λ i , i = 1, 2, 3, are the 0-order normalised roots. Hence Thus, instead of working on proving (5.1) we can work on the equivalent inequality In terms of the coefficients of the matrix B the Levi conditions (2.1) on the lower order terms can be written as Under these conditions we now want to prove that (5.2) holds for all vectors V . We note here that actually for the right hand side of (5.2) by the triangle inequality we have the upper bound in which the right hand side of (5.3) appears naturally.
Our strategy is to proceed by 3 steps making use of the following partition of R 3 : Estimate on Σ δ 1 1 .
Making use of the conditions (5.3) we have that where also in the last line we make use of the condition (1.4) on the roots λ i . Hence, by applying this to different combinations of terms, we get From the bound from below (5.5) and the estimate (5.4) one has that the inequality (5.2) holds true in the region Σ δ 1 1 for all δ 1 > 0.

One immediately has
More precisely, We estimate the right-hand side of (5.2) as By using condition (1.4) we get the estimate and then Combining this with (5.6) we conclude that for any δ 2 and for δ 1 big enough the right-hand side of (5.2) can be estimated from below by |V 2 | 2 and, therefore, (5.2) holds true on Σ δ 1 Since on Σ δ 2 2 c we have it follows that We conclude that for δ 1 and δ 2 big enough, and, therefore, (5.2) holds in the area Σ δ 1 The next table describes and summarises the proof above: for j = 1, ..., m. Then we have |WBV | ≺ |WV | uniformly over all V ∈ C m . More precisely, define for k = 1, ..., m − 1. Then, there exist suitable δ j > 0, j = 1, ..., m − 1, such that Note that (5.7) is a reformulation of the condition (2.2) on the lower order terms. The proof of Theorem 5.1 makes use of the following two lemmas.
This leads immediately to the formula (5.8).
Lemma 5.3. For all k = 1, ..., m, we have Proof. We give a proof by induction on the order m. Setting m = 2 the estimates above makes sense for k = 1. Hence we have to prove that This is clear since by the condition (1.4) we have that Assume now that (5.9) holds for m − 1. Estimating the left-hand side of (5.9) with the differences between two arbitrary summands we can write By applying Lemma 5.2 and the condition (1.4) we obtain the following bound from below: Noting that m−1−j (π l 2 (π l 1 λ)) for j = k, ..., m − 2, we write the estimate above as By now applying the inductive hypothesis to the last two summands in (5.10) we obtain which completes the proof.
Proof of Theorem 5.1. By definition of the matrices W and B we have that |WBV | 2 ≺ |WV | 2 is equivalent to Making use of the conditions (5.7) we have that the following estimate is valid on the area Σ δ 1 1 : Setting k = 1 in Lemma 5.3 we obtain the bound from below This proves the inequality (5.11) on Σ δ 1 1 for any δ 1 > 0. Let us now assume that By iteration one can easily prove the following bound We now pass to estimate the right-hand side of (5.11) making use of Lemma 5.3 and of the bound (5.12). We obtain Therefore, the estimate (5.11) holds in the region Σ δ 1 for any δ k > 0 choosing δ 1 , δ 2 , ..., δ k−1 big enough. We conclude the proof by assuming for 1 ≤ h ≤ m − 1, arguing as above and taking δ h ≥ 1 we obtain the estimate This means that the inequality (5.11) holds on Σ δ 1 sufficiently large values of δ 1 , δ 2 , ..., δ m−1 .

Well-posedness results
We are now ready to prove the well-posedness results given in Theorem 1.2. For the advantage of the reader and the sake of simplicity we reformulate Theorem 1.2 as the following Theorem 6.1 where we make use of the language and notations introduced in Theorem 5.1.
Proof. As usual, the well-posedness in the case of s = 1 follows from the result of Bony and Shapira, so we may assume s > 1. By finite propagation speed for hyperbolic equations it is not restrictive to take compactly supported initial data and therefore to have that the solution u is compactly supported in x. Combining the energy estimate (4.5) with the estimates of the first, second and third term in Section 4 we obtain the estimate valid for some positive constants C 1 , C 2 , C 3 , for t ∈ [0, T ] and |ξ| ≥ R. Here, the estimate for the third term is provided by Theorem 5.1 for |ξ| ≥ R. A straightforward application of Gronwall's lemma leads to ]. Finally, making use of the inequality (4.7) we arrive at for some new constant C > 0, for t ∈ [0, T ] and |ξ| ≥ R.
where U is the u j 's column vector. If the initial data g l belong to G s 0 (R n ) from the Fourier transform characterisation of Gevrey functions ([11, Proposition 2.2]) we have that |V (0, ξ)| ≤ c e −δ ξ 1 s for some constants c > 0 and δ > 0. Hence, It follows that for some c ′ , δ > 0 and for |ξ| large enough. This is sufficient to prove that U(t, x) belongs to the Gevrey class G s (R n ) for all t ∈ [0, T ] and that the Cauchy problem (1.1) has a unique solution u ∈ C m ([0, T ]; G s (R n )) for s < σ under the assumptions of case (i).
(ii) If the initial data g l are Gevrey Beurling ultradistributions in E ′ (s) (R n ), from the Fourier transform characterisation of ultradistributions ([11, Proposition 2.13]) we have that there exist δ > 0 and c > 0 such that |V (0, ξ)| ≤ c e δ ξ 1 s for all ξ ∈ R n . Hence, taking s ≤ σ, we obtain the estimate for some c ′ , δ ′ > 0. This proves that the Cauchy problem (1.1) has a unique solution u ∈ C m ([0, T ]; D ′ (s) (R n )) for s ≤ σ under the assumptions of case (ii). We pass to consider the case of analytic coefficients. We prove C ∞ and distributional well-posedness of the Cauchy problem (1.1) providing an extension of Theorem 1 in [13] to any space dimension. Our proof makes use of the following lemma on analytic functions, a parameter-dependent version of the statement (61)-(62) in [13]. Lemma 6.2. Let f (t, ξ) be an analytic function in t ∈ [0, T ], continuous and homogeneous of order 0 in ξ ∈ R n . Then, (i) for all ξ there exists a partition (τ h(ξ) ) of the interval [0, T ] such that for all t ∈ (τ h(ξ) , τ h+1(ξ) ), ξ ∈ R n with ξ = 0 and 0 ≤ h(ξ) ≤ N(ξ). Proof.
Since the function f is homogeneous of order 0 in ξ we can assume |ξ| = 1.
In the case of analytic coefficients, Theorem 1.3 follows from the following Theorem 6.3. Proof. By the finite propagation speed for hyperbolic equations it is not restrictive to assume that the initial data g l are compactly supported. If the coefficients a j are analytic in t on [0, T ] then by construction the entries of the quasi-symmetriser Q (m) ε are analytic as well. In particular, by Proposition 3.1 q ε,ij (t, ξ) = q 0,ij (t, ξ) + ε 2 q 1,ij (t, ξ) + · · · + ε 2(m−1) q m−1,ij (t, ξ).
We use the partition of the interval [0, T ] in Lemma 6.2 (applied to any q ε,ij (t, ξ) or more precisely to any q ε,ij (t, ξ) = q ε,ij (t, ξ) ξ /|ξ|, homogeneous function of order 0 in ξ having the same zeros in t of q ε,ij (t, ξ)). Considering the first interval [0, as in [13, p.567]. Hence It follows by the Gronwall inequaity that there exists a constant α > 0 such that On the interval [ε, τ 1 − ε] we proceed as in the proof for the Gevrey well-posedness under the conditions (5.7) on the lower order terms for |ξ| ≥ R. We have Since the family Q ε (λ) is nearly diagonal when the roots λ j satisfy the condition (1.4) we have that Q ε ≥ c 0 diag Q ε , i.e., This fact combined with the inequality q ε,hh |V h | 2 and Lemma 6.2 yields |∂ t q ε,ij (t, ξ)| |q ε,ij (t, ξ)| dt for some constant C 1 independent of t and ξ = 0. Going back to estimate (6.4) we obtain for t ∈ [ε, τ 1 − ε] and |ξ| ≥ R. This implies, by Gronwall's lemma, that Remembering that from Lemma 6.2 the function N(ξ) is bounded in ξ we conclude that there exist some κ ∈ N and C > 0 such that (6.6) |V (t, ξ)| ≤ C ξ κ |V (0, ξ)| on [0, T ] for all |ξ| ≥ R. It is clear that the estimate (6.6) implies C ∞ and distributional well-posedness of the Cauchy problem (1.1).
Finally, given the energy estimates established above the proof of Theorem 1.4 is simple: Proof of Theorem 1.4. We observe that the estimates (6.2) and (6.6) imply that V (t, ξ) is bounded in ξ if the lower order terms A(·, ξ) are bounded on [0, T ]. Coming back to the solution u of (1.1) and the definition of V we get that the solution u(t, x) is in the class C m−1 ([0, T ]) with respect to t. Finally, from the equality D m t u = − m−1 j=0 A m−j (t, D x )D j t u we see that the right hand side is bounded in t, implying that u(t, x) is in W ∞,m ([0, T ]) with respect to t.
We conclude the paper with the following remark on how the results change if we assume less than the Levi conditions (1.5). We thank T. Kinoshita for drawing our attention to this question.
In other words, for any fixed k ∈ N by involving sufficiently enough matrices B −h in the Levi condition (6.8) (how many depend on the equation order m and the regularity k of the coefficients) one can still obtain G s well-posedness for but not the well-posedness in any Gevrey space even if k increases to infinity. This is due to the fact that condition (6.9) implies 2(m−1)(k−1) k+2(m−1) ≤ m − 1 and, therefore, gives the restriction k ≤ 2m.
(ii) Assume now that the equation coefficients are smooth. This implies that for any a > 0 we can take k large enough such that ε −2(m−1)/k ≤ ξ a . Hence, ε −2(m−1)/k ≤ ξ −h−1 ε −2(m−1) with h = 0. Setting then ε ξ = ξ −1 ε −2(m−1) we get that under the Levi condition (6.8) with h = 0 the Cauchy problem (1.1) is well-posed in G s with In terms of Gevrey order this result is worse than the one stated in Theorem 1.2 but it is obtained with Levi conditions only on the coefficients appearing in the matrix B 0 . We note that it is better than the Bronstein's result due to the extra assumption (1.4) and the Levi condition (6.8) with h = 0.