Bulletin T.CXXXIII de l’Académie serbe des sciences et des arts − 2006 Classe des Sciences mathématiques et naturelles Sciences mathématiques, N\underline{o} 31

INHOMOGENEOUS GEVREY ULTRADISTRIBUTIONS AND CAUCHY PROBLEM^{1}

DANIELA CALVO, L. RODINO

(Presented at the 2nd Meeting, held on March 24, 2006)

A b s t r a c t. After a short survey on Gevrey functions and ultradistributions, we present the inhomogeneous Gevrey ultradistributions introduced recently by the authors in collaboration with A. Morando, cf. [7]. Their definition depends on a given weight function λ , satisfying suitable hypotheses, according to Liess-Rodino [16]. As an application, we define (s, λ)-hyperbolic partial differential operators with constant coefficients (for s > 1), and prove for them the well-posedness of the Cauchy problem in the frame of the corresponding inhomogeneous ultradistributions. This sets in the dual spaces a similar result of Calvo [4] in the inhomogeneous Gevrey classes, that in turn extends a previous result of Larsson [14] for weakly hyperbolic operators in standard homogeneous Gevrey classes.

AMS Mathematics Subject Classification (2000): 46F05, 35E15, 35S05. Key Words: Gevrey ultradistributions, inhomogeneous Gevrey classes,

Cauchy problem, microlocal analysis

1This paper was presented at the Conference GENERALIZED FUNCTIONS 2004, Topics in PDE, Harmonic Analysis and Mathematical Physics, Novi Sad, September 22– 28, 2004

Introduction

Let us begin this note by a short survey on Gevrey classes and related ultradistributions. Starting from the investigation of the solutions of the heat equation, Gevrey [9] considered subclasses of C^{∞}(Ω), set of the infinitely differentiable functions in an open set Ω ⊂ ℝ^{n}. Namely, with the notation we shall use in the following:

f belongs to G^{s}(Ω), 1 ≤ s < ∞, if for every compact subset K ⊂⊂ Ω it is satisfied:

sup_{x∈K} |D^{α}f(x)| ≤ RC^{|α|}α!^{s}  (1)

for suitable positive constants R and C independent of the multi-index α.

In particular, for s = 1 we recapture the class of the analytic functions of the real variable x ∈ Ω. Somewhat different classes are defined if we assume that for every C > 0 there exists R such that (1) is satisfied (Beurling-type Gevrey functions, we say now with reference to Beurling [1]). Concerning the applications to the heat equation, we have that all the solutions f, also in the weak sense, of

Hf = ∂_{t}f − Δf = 0

are in G^{2}(ℝ^{n+1}). Moreover, assuming space dimension n = 1 and denoting y for the space variable, the Cauchy problem at y = 0:

\{ \begin{matrix}Hf = 0 \\ f(t, 0) = f_{0} \\ ∂_{y}f(t, 0) = f_{1} \end{matrix}  (2)

admits solution f(t, y) , y ∈ ℝ, if and only if f_{0}, f_{1} ∈ G^{s}(ℝ), s < 2 (note that, with respect to the standard parabolic problem with datum at t = 0, one is rather treating here H as a weakly hyperbolic operator, with initial data at y = 0). Starting from the original contribution of Gevrey, researches proceeded then along different lines. Let us distinguish two main streams. From one side, it is natural to generalize (1) as

sup_{x∈K} |D^{α}f(x)| ≤ RC^{|α|}M_{|α|},  (3)

where M = (M_{j}), j = 0, 1, 2, …, is a sequence of positive numbers satisfying suitable properties, cf. f.i. [13, 17]. The corresponding classes are usually denoted by C^{M}(Ω). It is easy to see that when M_{j} = (j!)^{s} we recapture from (3) the Gevrey classes defined by (1). Other interesting ex-

amples are given by the classes for which ∑_{j=0}^{∞} \frac{M_{j−1}}{M_{j}} = ∞, as for the case

M_{j} = j!; they are called quasi-analytic classes, because of their properties, which are similar to those of the analytic class. See for example Mandelbrojt [18], Beurling [1], Rudin [24] and the references there, for details. In the non-quasi-analytic case, functions in C^{M}(Ω) with compact support in Ω exist, and the corresponding set is denoted by C_{0}^{M}(Ω). Giving the natural topology to C_{0}^{M}(Ω) and taking the dual, one obtains the space of the generalized functions D′_{M}(Ω), see for example Roumieu [22], Komatsu [13], Lions-Magenes [17] and Matsumoto [20], containing as a subset the space of the Schwartz distributions D′(Ω) . Researches in this area are still very active, see for example the alternative presentation of Braun-Meise-Taylor [2]. A further generalization is given by Roumieu [23], who considers sequences M_{α} depending on all components of α. In strict connection with such a study, we emphasize a second point of view, where the main objective is given by the analysis of partial differential equations, and the previous spaces play the role of natural functional frames to obtain results of existence, uniqueness and regularity. In this line, the attention is often limited to the Gevrey spaces defined by (1) and to the corresponding ultradistribution spaces D′_{s}(Ω).As an example, let us mention the famous result of Hörmander (see [12]), concerning the regularity of the solutions of the partial differential equations with constant coefficients P (D)u = 0. Namely, all the solutions u ∈ D′(Ω) are actually in G^{s}(Ω) if and only if the symbol P (ξ) satisfies for large |ξ| the estimates |D^{α}P (ξ)|/|P (ξ)| ≤ C|ξ|^{− \frac{|α|}{s}}, i.e. P (D) is s-hypoelliptic. One can easily deduce from this the above-mentioned result of regularity for the solutions of the heat equation. Concerning the Cauchy problem (2), this also can be extended to general operators with constant coefficients. Namely, the Cauchy problem with respect to the time variable is well posed for initial data in the class G^{r}, 1 ≤ r < s, if P (D) is s-hyperbolic (according to the definition of Larsson [14])in the following sense:

P (τ, ξ) = τ^{m} + ∑_{|ν|+j≤m,j≠m} a_{νj}ξ^{ν}τ^{j} = 0 for (τ, ξ) ∈ ℂ × ℝ^{n} ⟹ |ℑτ| ≤ C⟨ξ⟩^{\frac{1}{s}} , (4)

where τ and ξ denote the duals of the time and space variables, respectively, and we write ⟨ξ⟩ = \sqrt{1 + |ξ|^{2}}. For the proof we refer for instance to [12], Theorem 12.7.5, [21], Theorem 2.5.2 and [14]. Generalizing such contributions, several other papers appeared, addressed to linear partial differential equations with analytic-Gevrey coefficients, or non-linear equations with analytic-Gevrey nonlinearity, dealing with the problems of the local existence and regularity of the solutions, the well-posedness of the Cauchy problem and related questions. In particular, in these last thirty years, new techniques of proof and terminology were given by Microlocal Analysis (Gevrey pseudo-differential operators, Gevrey wave front sets, etc). We address to the references in Rodino [21] and Mascarello-Rodino [19] for some recent results in this connection. Finally, to introduce the contents of the present paper, we recall that the Fourier transform f̂ of f ∈ G_{0}^{s}(ℝ^{n}) = G^{s}(ℝ^{n}) ∩ C_{0}^{∞}(ℝ^{n}) satisfies for positive constants C and ε the estimates

|f̂(ξ)| ≤ C exp(−ε⟨ξ⟩^{\frac{1}{s}}).  (5)

Starting from (5), we extend the definition of Gevrey classes by replacing ⟨ξ⟩ by a weight function λ(ξ), satisfying suitable properties. When λ(ξ) = λ̃(|ξ|), one can establish precise connections with generalizations in terms of (3), see for instance Braun-Meise-Taylor [2]. Generic inhomogeneous weight functions λ(ξ) were considered by Liess-Rodino [16] and Rodino [21], Section 1.8; the corresponding classes cannot in general be identified in terms of the estimates (3), however there are natural applications to hypoelliptic partial differential equations, motivating the inhomogeneous approach, cf. [16]. In the next section we shall recall shortly the definition of weight function from Liess-Rodino [16] and present also the corresponding Gevrey classes and ultradistributions, as treated recently by Calvo-Morando-Rodino [7]. We observe that a different approach is presented in Björk [3], under more restrictive hypotheses on the weight function; our setting allows us to include also the multi-anisotropic classes as particular case (cf. [6]). In the last section we shall give a new application, concerning the Cauchy problem for operators with constant coefficients in the frame of inhomogeneous ultradistributions. This extends to the dual spaces the above-mentioned result of Larsson [14] for the standard Gevrey classes, and preceding contributions of Calvo [4, 5] concerning the inhomogeneous and multi-anisotropic Gevrey classes.

1. Inhomogeneous Gevrey ultradistributions

We introduce the weight functions, following Liess-Rodino [16].

Definition 1.1. We say that a function λ : ℝ^{n} → ℝ_{+} is a weight function if there are constants C, C′, δ > 0 such that

\begin{matrix}(i) |λ(ξ) − λ(η)| ≤ C|ξ − η|,  ∀ξ, η ∈ ℝ^{n}, \\ (ii) ⟨ξ⟩^{δ} ≤ C′λ(ξ),  ∀ξ ∈ ℝ^{n}. \end{matrix}

Observe that (i) implies λ(ξ) ≤ C⟨ξ⟩ for a constant C > 0, so that δ must be smaller than 1.

From now on λ is a weight function according to Definition 1.1 and s > 1. For short, in the sequel we refer to classes of functions and distributions in ℝ^{n}; classes in an open subset Ω of ℝ^{n} can be defined similarly.

Definition 1.2. A distribution f ∈ D′(ℝ^{n}) belongs to the inhomogeneous Gevrey class G^{s,λ}(ℝ^{n}) if for any Gevrey function χ ∈ G_{0}^{s}(ℝ^{n}) of the same order s, χf satisfies for suitable C, ε > 0 the condition

|χ̂f(ξ)| ≤ C exp(−ελ(ξ)^{\frac{1}{s}}), ∀ξ ∈ ℝ^{n}.  (6)

Definition 1.3. The inhomogeneous Gevrey classes with compact support are

G_{0}^{s,λ}(ℝ^{n}) = G^{s,λ}(ℝ^{n}) ∩ C_{0}^{∞}(ℝ^{n}).

It follows easily that the space G_{0}^{s,λ}(ℝ^{n}) is defined by the following inhomogeneous version of (5) (cf. [7], Theorem 2.1). A compactly supported distribution f ∈ E′(ℝ^{n}) belongs to G_{0}^{s,λ}(ℝ^{n}) if and only if its Fourier transform satisfies for suitable C, ε > 0 the condition

|f̂(ξ)| ≤ C exp(−ελ(ξ)^{\frac{1}{s}}), ∀ξ ∈ ℝ^{n}.

In order to endow the spaces G_{0}^{s,λ}(ℝ^{n}) and G^{s,λ}(ℝ^{n}) with a topology, we then define the following Banach spaces.

Definition 1.4. Let K be a compact subset of ℝ^{n} and ε a positive number. Then

G_{0}^{s,λ}(K, ε) := \{f ∈ E′(ℝ^{n}) : suppf ⊂ K, sup_{ξ∈ℝ^{n}} exp(ελ(ξ)^{\frac{1}{s}})|f̂(ξ)| < ∞\}.

Therefore, if {K_{j}}_{j∈ℕ} is an exhaustive sequence of compact sets in ℝ^{n} and {ε_{j}}_{j∈ℕ} a decreasing sequence of positive numbers converging to 0, we define the topology in G_{0}^{s,λ}(ℝ^{n}) as

G_{0}^{s,λ}(ℝ^{n}) = indlim_{K_{j}→ℝ^{n},ε_{j}→0}G_{0}^{s,λ}(K_{j}, ε_{j}).

Hence, G_{0}^{s,λ}(ℝ^{n}) is a (DFS)- space, cf. [11]. We now define the topology of the classes G^{s,λ}(ℝ^{n}) with arbitrary support. Firstly let {K_{j}}_{j=1}^{∞} be an exhaustive sequence of compact sets of ℝ^{n} (we set K_{0} := ∅) and, for every j = 1, 2, …, take a function χ_{j} ∈ G_{0}^{s}(K_{j}) such that χ_{j} ≡ 1 on K_{j−1}. For any pair of positive integers l > j we define the continuous map

ρ_{l,j} : G_{0}^{s,λ}(K_{l}) ⟶ G_{0}^{s,λ}(K_{j}), f ⟼ χ_{j}f.

We can then define G^{s,λ}(ℝ^{n}) = projlim_{l>j} (G_{0}^{s,λ}(K_{j}), ρ_{l,j}). The space G^{s,λ}(ℝ^{n}) is a complete Schwartz space, as it is the projective limit of complete Schwartz spaces (cf. [11]). For the properties of the inhomogeneous Gevrey functions and the algebraic operations on them, we refer to Calvo-Morando-Rodino [7]; we just point out that, for δ as in Definition 1.1, the following inclusions involving the inhomogeneous and the standard Gevrey classes hold:

G^{s}(ℝ^{n}) ⊂ G^{s,λ}(ℝ^{n}) ⊂ G^{\frac{s}{δ}} (ℝ^{n}).  (7)

We now pass to define the inhomogeneous Gevrey ultradistributions associated to the inhomogeneous Gevrey classes previously defined.

Definition 1.5. We denote by D′_{s,λ}(ℝ^{n}) := (G_{0}^{s,λ}(ℝ^{n}))′ the topological dual space of G_{0}^{s,λ}(ℝ^{n}). We also set E′_{s,λ}(ℝ^{n}) := (G^{s,λ}(ℝ^{n}))′ for the topological dual of G^{s,λ}(ℝ^{n}).

The space E′_{s,λ}(ℝ^{n}) coincides with the subspace of the ultradistributions in D′_{s,λ}(ℝ^{n}) with compact support. As a consequence of (7), the following inclusions of the standard and inhomogeneous ultradistributions hold:

D′_{\frac{s}{δ}} (ℝ^{n}) ⊂ D′_{s,λ}(ℝ^{n}) ⊂ D′_{s}(ℝ^{n}).

We point out the following property of the Fourier transform on E′_{s,λ}(ℝ^{n}), that we will use to prove the well-posedness of the Cauchy problem in the next section.

Proposition 1.6. If u belongs to E′_{s,λ}(ℝ^{n}), then for every ε > 0 there exists a constant C_{ε} > 0 such that the Fourier transform û of u satisfies

|û(ξ)| ≤ C_{ε} exp (ελ(−ξ)^{\frac{1}{s}} ) , ∀ξ ∈ ℝ^{n}.  (8)

To antitransform a generic function û(ξ) satisfying (8), we need some spaces of inhomogeneous ultradistributions in which the Fourier transform is an automorphism. Namely, we first introduce the space S^{s,λ}(ℝ^{n}), in analogy with [3], [22], [23].

Definition 1.7. We say that a function f ∈ L^{1}(ℝ^{n}) belongs to S^{s,λ}(ℝ^{n}) if f, f̂ ∈ C^{∞}(ℝ^{n}) and there is a constant ε > 0 such that for all α ∈ ℕ^{n} it holds:

p_{α,ε}(f) = sup_{x∈ℝ^{n}} exp(ελ(x)^{\frac{1}{s}})|D^{α}f(x)| < ∞

π_{α,ε}(f) = sup_{ξ∈ℝ^{n}} exp(ελ(ξ)^{\frac{1}{s}})|D^{α}f̂(ξ)| < ∞. (9)

Then we can naturally define a locally convex topology in S^{s,λ}(ℝ^{n}) given by the semi-norms p_{α,ε}, π_{α,ε}. We easily see that G_{0}^{s,λ}(ℝ^{n}) ⊂ S^{s,λ}(ℝ^{n}) ⊂ G^{s,λ}(ℝ^{n}) and S^{s,λ}(ℝ^{n}) is included in the Schwartz space S(ℝ^{n}). In analogy with [3], from (9) we have that the Fourier transform is an automorphism in S^{s,λ}(ℝ^{n}). Therefore, we define the dual spaces of S^{s,λ}(ℝ^{n}).

Definition 1.8. The space of the inhomogeneous ultradistribution S′_{s,λ}(ℝ^{n}) is the topological dual of S^{s,λ}(ℝ^{n}).

We have obviously the inclusions E′_{s,λ}(ℝ^{n}) ⊂ S′_{s,λ}(ℝ^{n}) ⊂ D′_{s,λ}(ℝ^{n}) and S′(ℝ^{n}) ⊂ S′_{s,λ}(ℝ^{n}), where S′(ℝ^{n}) is the space of tempered distributions. We can define the Fourier transform for u ∈ S′_{s,λ}(ℝ^{n}) using Parseval’ formula: û(f) = u(f̂), for all f ∈ S^{s,λ}(ℝ^{n}) (as also f̂ ∈ S^{s,λ}(ℝ^{n})). The Fourier transform is an automorphism in S′_{s,λ}(ℝ^{n}). Note finally that any function û(ξ) satisfying for all ε > 0 the condition (8) can be regarded as an element of S′_{s,λ}(ℝ^{n}), and therefore is the Fourier transform of an inhomogeneous ultradistribution in S′_{s,λ}(ℝ^{n}). It follows that if û satisfies (8), then u belongs to D′_{s,λ}(ℝ^{n}) from the inclusion S′_{s,λ}(ℝ^{n}) ⊂ D′_{s,λ}(ℝ^{n}). For other properties of the inhomogeneous ultradistributions we refer to [7]. An important example of the inhomogeneous Gevrey functions is represented by the multi-anisotropic case, that is widely studied in literature and presents many applications to the theory of partial differential equations, cf. f.i. [5], [10], and the bibliography there; and see [6] for the dual spaces. Here the weight function is defined in terms of a completely regular polyhedron, according to

Definition 1.9. A convex polyhedron P ⊂ ℝ^{n} is complete if it satisfies the following conditions:

i) all vertices (we denote their set by V(P)) have rational non-negative coordinates; ii) the origin (0, …, 0) belongs to P; iii) P has one vertex in each coordinate axis (different from the origin); iv) the outer normals (we denote by N_{1}(P) their set) to the non-coordinate faces of P have strictly positive components.

Then we define μ = max_{j=1,...,n} max_{ν∈N_{1}(P)} ν_{j}^{−1} the formal order of P.

Therefore, the function |ξ|_{P} = ∑_{v∈V(P)} |ξ^{v}|^{\frac{1}{μ}} is a weight function according to Definition 1.1. The multi-anisotropic Gevrey classes obtained by fixing λ(ξ) = |ξ|_{P} in (6), can be defined also in terms of estimates on the derivatives, cf. the previous references (with respect to (3), bounds are given in terms of a suitable multi-sequence M_{α}, depending on each component of α).

2. Cauchy problem in inhomogeneous ultradistributions

As an application of the inhomogeneous Gevrey ultradistributions, we study in this setting the well-posedness of the Cauchy problem, by considering a class of weakly hyperbolic operators, called (s, λ)-hyperbolic: these operators were introduced in [4], modeled to have well-posedness in the inhomogeneous Gevrey classes. They are an extension of the s-hyperbolic operators of Larsson [14] (see formula (4) and of the multi-quasi-hyperbolic operators in [5]). For simplicity, we suppose that λ(ξ) = λ(−ξ), for all ξ ∈ ℝ^{n}, as satisfied in the standard homogeneous and multi-anisotropic cases.

Definition 2.1. We say that a differential operator with constant coefficients in ℝ_{t} × ℝ_{x}^{n}:

P (D) = P (D_{t}, D_{x}) = D_{t}^{m} + ∑_{|ν|+j≤m,j≠m} a_{νj}D_{x}^{ν}D_{t}^{j}  (10)

is (s, λ)-hyperbolic if there exists a constant C > 0 such that for any (τ, ξ) ∈ ℂ × ℝ^{n} the symbol of P (D) satisfies the condition:

P (τ, ξ) = τ^{m}+ ∑_{|ν|+j≤m,j≠m} a_{νj}ξ^{ν}τ^{j} = 0 for (τ, ξ) ∈ ℂ×ℝ^{n} ⟹ |ℑτ| ≤ Cλ(ξ)^{\frac{1}{s}} .

Obviously, if P (D) is (s, λ)-hyperbolic, then P (D) is also (r, λ)-hyperbolic for 1 < r < s and s-hyperbolic (cf. [14]). The (s, λ)-hyperbolic operators are obviously weakly hyperbolic, i.e. all the roots τ of the characteristic equation P_{m}(τ, ξ) = τ^{m} + ∑_{|ν|+j=m,j≠m} ξ^{ν}τ^{j} = 0 are real. In the opposite direction, if P_{m}(τ, ξ) satisfies the weakly hyperbolic assumption, in order to have the (s, λ)-hyperbolicity, we need to ask some Levi-type conditions on the lower order terms, modeled on the weight function λ, as shown for instance by the following result. We omit the proof, detailed in [4] together with many other examples.

Proposition 2.2. Let P (D) in (10) be weakly hyperbolic. Assume that the multiplicity of the roots of its principal symbol P_{m}(τ, ξ) is equal to M ≤ m and the lower order terms satisfy for some k < M and a constant C > 0:

|a_{νj}ξ^{ν}| ≤ Cλ(ξ)^{k}⟨ξ⟩^{m−M−j} for |ν| + j ≤ m − 1.

Then P (D) is (\frac{M}{k} , λ)-hyperbolic.

Finally, we can state the announced new result, of which we give a short proof.

Theorem 2.3. Let P (D) be an (s, λ)-hyperbolic differential operator in ℝ_{t} × ℝ_{x}^{n}. Let 1 < r < s and u_{k} ∈ D′_{r,λ}(ℝ_{x}^{n}) (k = 0, 1, …, m − 1). Then the Cauchy problem:

\{ \begin{matrix}P (D)u = D_{t}^{m}u + ∑_{|ν|+j≤m, j≠m} a_{νj}D_{x}^{ν}D_{t}^{j}u = 0 \\ D_{t}^{k}u(0, x) = u_{k}(x),  k = 0, 1, …, m − 1 \end{matrix}  (11)

admits a unique solution

u ∈ C^{∞}([−T, T], D′_{r,λ}(ℝ_{x}^{n}))

for any T > 0. If r = s, the solution is only local in time.

P r o o f. As P (D) is a weakly hyperbolic operator with constant coefficients, then the phenomena related to P (D) propagate with a finite speed, and therefore it is not restrictive to prove the theorem for data in the set of compactly supported inhomogeneous ultradistributions E′_{r,λ}(ℝ^{n}). Analogously to [4], [12] and [14], after performing the partial Fourier transform with respect to the space variable in the Cauchy problem (11), the unique solution u(t, x) is obtained by anti-transforming

û(t, ξ) = ∑_{j=0}^{m−1} û_{j}(ξ)F_{j}(t, ξ),  (12)

where û_{j} (j = 0, …, m − 1) are the Fourier transforms of the data of (11) and F_{j} are the unique solutions of the Cauchy problems

\{ \begin{matrix}P (D_{t}, ξ)F_{j} = 0 \\ D_{t}^{k}F_{j}(0, ξ) = δ_{jk}, k = 0, …, m − 1, \end{matrix}

for δ_{jk} = 1 if j = k and 0 otherwise. As u_{j} belong to E′_{r,λ}(ℝ^{n}), then from (8) it follows that for all ε > 0 there is a constant C_{ε} > 0 such that |û_{j}(ξ)| ≤ C_{ε} exp(ελ(ξ)^{\frac{1}{r}}). To estimate F_{j} we use the following result (cf. [12], Lemma 12.7.7):

Lemma 2.4. Let L(D) = D^{m} + ∑_{j=0}^{m−1} a_{j}D^{j} be an ordinary differential operator with constant coefficients a_{j} ∈ ℂ. Write Λ = {τ ∈ ℂ : L(τ) = 0} and assume: max_{τ∈Λ} |τ| ≤ A and max_{τ∈Λ} |ℑτ| ≤ B. Then the solutions v_{j}(t), j = 0, 1, …, m − 1, of the Cauchy problems:

\{ \begin{matrix}L(D)v_{j} = 0 \\ (D^{k}v_{j})(0) = δ_{jk},  k = 0, …, m − 1 \end{matrix}

satisfy the estimates:

|D^{N}v_{j}(t)| ≤ 2^{m}(A + 1)^{N+m+1}e^{(B+1)|t|},  N = 0, 1, …, t ∈ ℝ. (13)

We now determine the constants A, B in order to apply the estimate (13), for N = 0, to the functions F_{j}(t, ξ), j = 0, 1, …, m−1 (with ξ as parameter). As P (D) is (s, λ)-hyperbolic, then we can take B = C′λ(ξ)^{\frac{1}{s}}, while it is easy to see that A = C_{1}⟨ξ⟩. This leads to have for new constants:

|F_{j}(t, ξ)| ≤ (C_{1}⟨ξ⟩)^{m+1}c_{1} exp(C′|t|λ(ξ)^{\frac{1}{s}}) ≤ c_{2} exp(C_{2}(1 + |t|)λ(ξ)^{\frac{1}{s}}).

Then we can estimate û(t, ξ) given by (12) as follows: for all ε > 0 there is a constant C_{ε} > 0 such that

|û(t, ξ)| ≤ ∑_{j=0}^{m−1} c_{2}C_{ε} exp(ελ(ξ)^{\frac{1}{r}}) exp(C_{2}(1 + |t|)λ(ξ)^{\frac{1}{s}}).

As r < s, for t ∈ [−T, T] (for T > 0 fixed), for all ε′ > 0, taking ε sufficiently small (depending on ε′, r, s, T), then there is a constant C_{ε′} > 0 such that holds |û(t, ξ)| ≤ C_{ε′} exp(ε′λ(ξ)^{\frac{1}{r}}), ∀ξ ∈ ℝ^{n}, implying that u belongs to C^{∞}([−T, T], D′_{r,λ}(ℝ^{n})). We observe that for r = s the solution is only local in time. □

To conclude our treatment, let us show the application of Theorem 2.3 to some operators. Let P (D_{x}) = ∑ c_{α}D_{x}^{α} be a hypoelliptic operator with constant coefficients in ℝ^{n} and let P be its Newton polyhedron, i.e. the convex hull of the points {0} ∪ {α : c_{α} ≠ 0}, it is a complete polyhedron, according to the classical result of Friberg [8]. If μ is the formal order of P and m > μ, then the operator

D_{t}^{m} + P (D_{x})

is (\frac{m}{μ}, | · |_{P})-hyperbolic, and therefore the Cauchy problem (11) admits a unique solution u ∈ C^{∞}([−T, T], D′_{r,P}(ℝ^{n})) (for all T > 0) for any data u_{k} ∈ D′_{r,P}(ℝ^{n}) (k = 0, …, m − 1), if r < s = \frac{m}{μ}.

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