On the Cauchy problem for a class of hyperbolic operators with triple characteristics

The Cauchy problem for a class of hyperbolic operators with triple characteristics is analyzed. Some a priori estimates in Sobolev spaces with negative indexes are proved. Subsequently, an existence result for the Cauchy problem is obtained.


Introduction
In the past, many authors studied widely hyperbolic operators with double characteristics, both in the case when there is no transition between different types on the set where the principal symbol vanishes of order 2 (see for instance [5,8] for a general survey) and when there is transition (see [1][2][3][4]). The operators are called effectively hyperbolic if the propagation cone C is transversal to the manifold of multiple points (see [8]). Moreover, if this occurs and lower order terms satisfy a generic Ivrii-Petkov vanishing condition, we have well posedness in C ∞ (see [7]).
The aim of the paper is to analyze the following class of operators with triple characteristics where D x j = 1 i ∂ x j , j = 0, 1, 2, under hyperbolicity assumptions, namely |b| ≤ 2 3 . Such a class of operators has been considered in [6], for example operators whose propagation cone is not transversal to the triple characteristic manifold. The authors B Annamaria Barbagallo annamaria.barbagallo@unina.it 1 Department of Mathematics and Applications "R. Caccioppoli", University of Naples Federico II, Via Cinthia -Monte S. Angelo, 80126 Naples, Italy prove a well posedness result in the Gevrey category for a simple hyperbolic operator with triple characteristics and whose propagation cone is not transversal to the triple manifold. Furthermore they estimate the precise Gevrey threshold, by exhibiting a special class of solutions, through which we can violate weak necessary solvability conditions. More precisely, let x = (x 0 , x ) where x = (x 1 , x 2 ), let ξ = (ξ 0 , ξ ), where ξ = (ξ 1 , ξ 2 ). In [6], the authors study the well posedness of the following Cauchy problem is the Gevrey s class. They obtained that the Cauchy problem for P is well posed in the Gevrey 2 class assuming such that the Cauchy problem for P is not locally solvable at the origin in the Gevrey s class.
In this paper, instead, we investigate on the well posedness of the Cauchy problem with f ∈ H r (Ω), in the Sobolev spaces, obtaining an existence result for solutions. Let us set

It results
As a consequence, problem (1) becomes where we set g = i f , in Ω, with g real function. The main result of the paper is the following.
Theorem 1 Let f ∈ H r loc (Ω), with r ≥ 5. For every h, T > 0, the Cauchy problem The rest of the paper is organized as follows. Section 2 deals with some preliminary notations and definitions. In Sect. 3 some a priori estimates are established. Section 4 is devoted to obtain a priori estimates in Sobolev spaces with negative indexes. Finally, the existence result for solutions to the Cauchy problem are proved in Sect. 5.

Notations and preliminaries
Let α = (α 0 , α 1 , α 2 ) ∈ N 3 0 . Let ∂ α be the derivative of order |α|, let ∂ h x j be the derivative of order h with respect to x j and let ∂ h x j ,x p be the derivative of order h with respect to x j and x p .
We indicate the L 2 -scalar product, the L 2 -norm and the H r -norm (r ∈ N 0 ) by (·, ·), · and · H r respectively. Let Ω be an open subset of R 3 . Let C ∞ 0 (Ω) be the space of the restrictions to Ω of functions belonging to C ∞ 0 (R 3 ). For each K ⊆ Ω compact set, let C ∞ 0 (K ) be the set of functions ϕ ∈ C ∞ 0 (Ω) having support contained in K . Let S(R 3 ) be the space of rapidly decreasing functions. In particular, let S(Ω) be the space of the restrictions to Ω of functions belonging to S(R 3 ). Let − ∞, +∞[ and let s ∈ R, let us denote by · H 0,0,s (Ω) the norm given by where the Fourier transform is performed only with respect to the variable x 2 . Moreover, let us denote by A s the pseudodifferential operator given by Let us recall that A s : C ∞ 0 (Ω) → C ∞ (Ω). For every ϕ(x 2 ) ∈ C ∞ 0 (R), the operator ϕ A s u extends to a linear continuous operator from H 0,0,r comp.
Moreover, denoted by U x 2 the projection of supp u on the axis x 2 , if supp ϕ ⊆ R\U x 2 , then ϕ A s u is regularizing with respect to the variable x 2 , namely it results: The norms u H 0,0,s (Ω) and A s u L 2 (Ω) are equivalent for any s ∈ R.
Let s ∈ R and p ≥ 0. Let H p,s (R 3 ) be the space of distributions U into R 3 such that

Lemma 1 Let u ∈ S(Ω) and let p,
Now, we establish a useful estimate.
Proof In order to obtain the claim, we follow analogous techniques used in the proof of Lemma 3.2 in [3]. For the reader's convenience, we present the demonstration. We have where m ∈ N and By using (5), we get and also where Easily, we deduce and, then, Making use of (6) and (7), we obtain From the previous inequality and the Peetre inequality (see [9], pag. 17), it follows where c p,r ,s is independent of n and T .
Taking into account Lemma 2, we deduce For every ε > 0, for every r ≤ 0 and s ∈ R there exists n > 1 such that In the following, we establish a priori estimates in

Theorem 2 For every h, T > 0, there exists a positive constant c such that
Proof By means of a translation with respect to x 2 in T , we consider the function We extend the function v in even We consider the following Fourier development of the function v: where ω 0 = 2π 4T = π 2T and We remark that the Fourier coefficients c n are real. We apply the operator Q to v n obtaining where we set

It results
Qv We estimate the Fourier coefficients c n (x 0 , x 1 ) by means of L n c n (x 0 , x 1 ) in L 2 . To this aim, let us consider the inner products From which we have Let us evaluate the inner products Proceeding as in (10), we obtain Hence, we deduce As a consequence, we have Making use of (11) and (12), we get Hence, it results Applying the Parseval inequality, we have Taking into account (13) and (14), we obtain We remark that For the Parseval inequality, it results Moreover, we remark that Applying, again, the Parseval inequality, we have Making use of (15), (16) and (17), we obtain On the other hand, it results

From which it follows
Hence, we have From (18) and (19), we deduce By using Lemma 1, it results Let us remark As a consequence, we have Furthermore, we obtain By using (21), (22), (23) and (24), we deduce Now, we want to estimate directly the norms. Let us start from It results Let us compute the other norm remembering that We have From which, it follows Moreover, making use of (25) and (20), we obtain from which the claim follows.
Let us remark that the positive constant c in (26) does not depend on T but only on x 1 . As a consequence, the following result holds:

A priori estimate in Sobolev spaces
In the following, we establish a priori estimate in the Sobolev spaces.

Theorem 3 For every s > 0, it results
where where A s is the pseudodifferential operator defined as: with s > 0. Applying (27) to v s , we have where R 1 , R 2 and R 3 are regularizing operators with respect to the variable x 2 of type with ψ ∈ C ∞ 0 (R) such that ψ = 0 in [−1, 1], as in Lemma 3, and having used Making use of Lemmas 1, 3 and (29), we deduce where R 4 and R 4 are regularizing operators with respect to the variable x 2 of type (30). Now, written the operator Q as: where L is the wave operator, namely L = ∂ 2 Integrating by parts, we have easily: Making use of Lemma 1, it follows Taking into account (31), (32) and Lemma 3, we deduce

Conclusions
The paper deals with a class of hyperbolic operators with triple characteristics. A priori estimate in Sobolev spaces with negative indexes are obtained. Thanks to this estimate, the existence of solutions to the associated Cauchy problem can be established.