GENERALIZED GEVREY CLASSES AND MULTI-QUASI-HYPERBOLIC OPERATORS

D. Calvo

GENERALIZED GEVREY CLASSES AND MULTI-QUASI-HYPERBOLIC OPERATORS

Abstract. In this paper we consider generalized Gevrey classes defined in terms of Newton polyhedra. In such functional frame we prove a theorem of solvability of the Cauchy Problem for a class of partial differential op- erators, called multi-quasi-hyperbolic. We then give a result of regularity of the solution with respect to the space variables and finally analyze the regularity with respect to the time variable.

Introduction

It is well known from the Cauchy-Kovalevsky theorem that the Cauchy Problem for partial differential equations with constant coefficients or analytic coefficients, and an- alytic data admits a unique, analytic solution.

But there are problems that are not C^∞well-posed, i.e. starting with C^∞data, there is not a C^∞solution. In these cases it is natural to consider the behaviour of the operator in the Gevrey classes G^s,1 <s < ∞(for definition and properties see for example Rodino [11]). Solvability of the Cauchy Problem in Gevrey spaces has been obtained for a class of partial differential operators with constant coefficients, the so called s- hyperbolic operators.

More precisely, we recall the following definition and the corresponding result, for the proof see Cattabriga [3], H¨ormander [9], Rodino [11].

DEFINITION1. Let’s consider partial differential operators inRⁿ⁺¹ =R_t×Rⁿ_x, non-characteristic with respect of the t-hyperplanes, i.e. operators that can be written in the form:

(1) P(Dt,Dx)=D_t^m+

m−1X

j=1

aj(Dx)D_t^j

with or der(aj(Dx))≤m−j .

We say that P(D)is s-hyperbolic (with respect to the t variable), 1 <s <∞, if its symbol satisfies for a suitable C>0 the condition:

if λ^m+P_m−1

j=0aj(ξ )λ^j =0 for(λ, ξ )∈C_t×Rⁿ_x, then =λ≥ −C(1+ |ξ|^1s).

In the case=λ≥ −C we say that P(D)is hyperbolic.

73

THEOREM1. Let P(D)be a differential operator inR_t ×Rⁿ_x of the form (1) and let P be s-hyperbolic with respect to t, with 1<s <∞. Let 1 <r < s and assume fk(x) ∈ G^r₀(Rⁿ_x)for k = 0,1, . . . ,m−1. Then there exists a Gevrey function u ∈ G^r(Rⁿ⁺¹)satisfying the Cauchy Problem:

(2)

(P(D)u =D_t^mu+Pm−1

j=0 a_j(D_x)D_t^ju=0

D^k_tu(0,x)= f_k(x) ∀x∈Rⁿ,∀k=0,1, . . . ,m−1.

In the case P(D)is hyperbolic , we have the corresponding result of existence in the C^∞class.

The previous Theorem 1 can be extended to operators with variable coefficients, for example we refer to the important contribution of Bronstein [2].

Here, remaining in the frame of constant coefficients, we want to extend the previous theorem in order to assure the solvability of the Cauchy Problem for a larger class of data.

To this end, we define generalized Gevrey classes G^sP, 1 < s < ∞, based on a complete polyhedronP, following Zanghirati [13], Corli [6], and give equivalent defi- nitions of these classes (for details see Section 1).

Let us observe that G^s ⊂ G^sP. The classes G^sP allow to express a precise result of regularity for the so called multi-quasi-elliptic equations, defined in terms of the norm

|ξ|Passociated toP, see Cattabriga [4], Hakobyan-Margaryan [8], Boggiatto-Buzano- Rodino [1] and the subsequent Section 1.

We then introduce a class of differential operators with constant coefficients, modelled on a complete polyhedronP, that is natural to name as multi-quasi-hyperbolic opera- tors.

DEFINITION2. Let 1<s<∞and letP be a complete polyhedron. We say that a differential operator with constant coefficients inRⁿ_x ×R_t of the form (1) is multi- quasi-hyperbolic of order s with respect toPif there exists a constant C >0 such that for(λ, ξ )∈C×Rⁿthe condition:

P(λ, ξ )=λ^m+

m−1X

j=0

a_j(ξ )λ^j =0 implies:

=λ≥ −C|ξ|

1 s

P

where|ξ|Pis the weight associated toPas in Section 1.

The algebraic properties of the symbol of multi-quasi-hyperbolic operators will be studied in Section 2, where we shall also give some examples.

Since|ξ|P ≤const(1+ |ξ|), the previous definition implies s-hyperbolicity; therefore we may apply to P(D)the previous Theorem 1 and conclude the well-posedness of (2) in G^r with r <s. However, for multi-quasi-hyperbolic operators of order s we have

the well-posedness of the Cauchy Problem in the larger classes G^rP, r < s. More precisely, in Section 3 we will prove the following theorem:

THEOREM2. Let P(D)be a differential operator inR_t×Rⁿ_xas in (1) and let P be multi-quasi-hyperbolic of order s with respect to a complete polyhedronPinRⁿ, with 1<s<∞. Let 1<r <s and assume f_k ∈G^r₀^P(Rⁿ_x)for k=0,1, . . . ,m−1. Then there exists u(t,·)∈G^rP(Rⁿ_x)for t ∈Rsatisfying the Cauchy Problem (2).

This gives a regularity of the solution u with respect to the space variables. To test the regularity with respect to the time variable, we need to define a new polyhedron P⁰ that extends the polyhedronP toRⁿ⁺¹. We shall then be able to conclude u ∈ G^rP0(Rⁿ⁺¹), see to Section 4 for details.

1. Complete polyhedra and generalized Gevrey classes

A convex polyhedronP inRⁿis the convex hull of a finite set of points inRⁿ. There is univocally determined byP a finite setV(P)of linearly independent points, called the set of vertices ofP, as the smallest set whose convex hull isP.

Moreover, ifPhas non-empty interior, there exists a finite set:

N(P)=N₀(P)SN₁(P) such that :

|ν| =1,∀ν∈N₀(P) and

P = {z∈Rⁿ|ν·z≥0,∀ν∈N₀(P)∧ν·z≤1,∀ν∈N₁(P)}.

The boundary ofP is made of facesF_νof equation:

ν·z=0 ifν∈N₀(P) ν·z=1 ifν∈N₁(P).

We now introduce a class of polyhedra that will be very useful in the following.

DEFINITION3. A complete polyhedron is a convex polyhedronP ⊂Rⁿ₊such that:

1. V(P)⊂Nⁿ (i.e. all vertices have integer coordinates);

2. the origin(0,0, . . . ,0)belongs toP; 3. di m(P)=n;

4. N₀(P) = {e₁,e₂, . . . ,e_n}, with e_j = (0,0, . . . ,0,1_{j−t h},0, . . . ,0) ∈ Rⁿ for j=1, . . . ,n;

5. N₁(P)⊂Rⁿ₊.

We note that 5. means that the set: Q(x)= {y∈ Rⁿ|0≤ y≤ x} ⊂P if x ∈ P and if s belongs to a face ofPand r >s then r 6∈P.

We can consider also polyhedra with rational vertices instead of integer vertices, as in Zanghirati (see [13]); the properties below remain valid.

PROPOSITION1. LetPbe a complete polyhedron inRⁿwith natural (or rational) vertices s^l =(s^l₁, . . . ,s_n^l), l =1, . . . ,n(P), where n(P)is the number of the vertices ofP, then:

1. for every j =1,2, . . . ,n, there is a vertex s^lj ofPsuch that:

(0, . . . ,0,s^l_j^j,0, . . . ,0)=s^l_j^je_j, s^l_j^j =max

s∈Ps_j =: m_j(P).

2. there is a finite non-empty setN₁(P)⊂Qⁿ₊\{0}such that:

P = \

ν∈N₁(P)

{s∈Rⁿ₊:ν·s≤1};

3. for every j =1, . . . ,n there is at least oneν∈N₁(P)such that:

mj =mj(P)=ν⁻¹_j ; 4. if s∈P, then:

|ξ^s| ≤

n(P)X

l=1

|ξ^sl|



ξ^s = Yn j=1

ξ^s_j^j



.

The proof is trivial and we need only to point out that 4. is a consequence of the following lemma, for whose proof we refer to Boggiatto-Buzano-Rodino [1], Lemma 1.1.

LEMMA 1. Given a subset A ⊂ (R⁺₀)ⁿ and a linear convex combinationβ = P

α∈Ac_αα, then for any x∈(R⁺₀)ⁿthe following inequality is satisfied:

(3) x^β ≤X

α∈A

c_αx^α

We now give some notations related to a complete polyhedronP.

Let’s denote by L(P)the cardinality of the smallest set N₁(P)that satisfies 2. of Proposition 1.

We denote:

F_ν(P)= {s∈P :ν·s=1}, ∀ν∈N₁(P) a face ofP; F=S

ν∈N₁(P)F_ν(P)the boundary ofP; V(P)the set of vertices ofP;

δP = {s∈Rⁿ₊:δ⁻¹s∈P}, δ >0;

k(s,P)=inf{t >0 : t⁻¹s∈P} =max_ν∈N₁(P)ν·s, s∈Rⁿ₊. Now letPbe a complete polyhedron, we say:

µ_j(P)=max_ν∈N₁(P)ν⁻¹_j ;

µ=µ(P)=max_j=1,...,nµj the formal order ofP; µ⁽⁰⁾(P)=min_γ∈V(P)\{0}|γ| the minimum order ofP;

µ⁽¹⁾(P)=max_γ∈V(P)|γ| the maximum order ofP; q(P)=

µ(P)

µ₁(P), . . . ,_µ^µ(P)

n(P)

;

|ξ|P =(P

s∈V(P)ξ^2s)^2µ1 ,∀ξ ∈Rⁿ the weight ofξassociated to the polyhedronP. Considering a polynomial with complex coefficients, we can regard it as the symbol of a differential operator, and associate a polyhedron to it, as in the following.

DEFINITION 4. Let P(D) = P

|α|≤mcαD^α, cα ∈ Cbe a differential operator with complex coefficients inRⁿand P(ξ ) =P

|α|≤mc_αξ^α, ξ ∈ Rⁿits characteristic polynomial. The Newton polyhedron or characteristic polyhedron associated to P(D) is the convex hull of the set:

{0}[

{α∈Zⁿ₊: c_α 6=0}.

There follow some examples of Newton polyhedra related to differential operators:

1. If P(ξ )is an elliptic operator of order m, then its Newton polyhedron is complete and is the polyhedron of vertices{0,mej, j =1, . . . ,n}and so:P = {ξ ∈Rⁿ: ξ ≥0, Pn

i=1ξi ≤m}.

The setN₁(P)is reduced to a point:

ν=m⁻¹Pm

j=1ej =(m⁻¹, . . . ,m⁻¹).

m_j(P)=µ_j(P)=µ⁽⁰⁾(P)=µ⁽¹⁾(P)=µ(P)=m, j=1,2, . . . ,n;

q(P)=(1,1, . . . ,1);

k(s,P)=m⁻¹|s| =m⁻¹Pn

j=1s_j, s∈Rⁿ₊.

2. If P(ξ )is a quasi-elliptic polynomial of order m (see for example H¨ormander [9], Rodino [11], Zanghirati [12]), its characteristic polyhedronP is complete and has vertices{0,m_je_j, j =1, . . . ,n}where m_j =m_j(P)are fixed integers.

The setN₁(P)is again reduced to a point:

ν=Pn

j=1m⁻¹_j e_j. P = {ξ ∈Rⁿ:ξ ≥0,Pn

j=1m⁻¹_j ξ_j ≤1};

µ_j(P)=m_j, j =1, . . . ,n;

µ⁽⁰⁾(P)=minj=1,...,nmj;

µ(P)=µ⁽¹⁾(P)=max_j=1,...,nmj =m;

q(P)=(_m^m

1, . . . ,_m^m

n);

k(s,P)=µ(P)⁻¹q·s, s∈Rⁿ₊.

In this case the unique face ofPis defined by the equation:

1

m₁x1+. . .+ 1

m_nxn=1.

We note in general that s belongs to the boundary of k(s,P)Pand k(s,P)is univocally determined for complete polyhedra.

k(s,P)satisfies the following inequality that will be very useful in the following:

(4) |s|

µ⁽¹⁾ ≤k(s,P)≤ | |s|

µ⁽⁰⁾ ≤ |s|, ∀s∈Rⁿ₊.

We remember (see [1]) that the polyhedron of a hypoelliptic polynomial is complete, but the converse is not true in general.

We now introduce a class of generalized Gevrey functions associated to a complete polyhedron, as in Corli[6], Zanghirati [13].

They can be regarded as a particular case of inhomogeneous Gevrey classes with weight λ(ξ )= |ξ|P, in the sense of the definition of Liess-Rodino [11], and can be expressed also by means of the derivatives of u.

Following Corli [6] we give the following definition:

DEFINITION5. LetPbe a complete polyhedron inRⁿ. Letbe an open set inRⁿ and s∈R, s>1. We denote by G^sP()the set of all u∈C^∞()such that:

(5) ∀K ⊂⊂, ∃C >0 :

|D^αu(x)| ≤C^|α|+1(µk(α,P))^sµk(α,P), ∀α∈Zⁿ₊, ∀x∈K. We also define:

G^s₀^P()=G^sP()∩C₀^∞().

The space G^sP()can be endowed with a natural topology. Namely, we denote by C^∞(P,s,K,C)the space of funcions u∈C^∞()such that:

(6) suppu⊂K kukK,C =sup_α∈Zⁿ

+sup_x∈KC^−|α|(µk(α,P))^{−sµk(α,P)}|D^αu(x)|<∞ With such a norm, C^∞(P,s,K,C)is a Banach space. Then:

G^sP()= \

K⊂⊂

[

C>0

C^∞(P,s,K,C)

endowed with the topology of projective limit of inductive limit.

REMARK1. IfP is the Newton polyhedron of an elliptic operator, then G^sP() coincides with G^s(), the set of the standard s-Gevrey functions in.

REMARK2. IfPis the Newton polyhedron of a quasi-elliptic operator, then:

G^sP()=G^sq(), where q= m

m₁, . . . , m m_n

the set of the anisotropic Gevrey functions, for definition see H¨ormander [9], Rodino [11], Zanghirati [12].

REMARK3. We have the following inclusion:

G^s

µ

µ(1) ⊂G^sP ⊂G^s

µ

µ(0), ∀s>1, ∀P as follows immediately from Definition 5 and the inequality (4).

We give now equivalent definitions of generalized Gevrey classes.

The arguments are similar to those in Corli [6], Zanghirati [13], but simpler, since for our purposes we need to consider only classes for s >1; to be definite, we prefer to give here self-contained proofs.

LetP be a complete polyhedron inRⁿand let K be a compact set inRⁿ. DEFINITION6. Ifν∈N₁(P), let:

C(ν)= {α∈Zⁿ₊: k(α,P)=α·ν}.

C(ν)is a cone ofZⁿ₊and C(ν)TF=F_ν. This means that k(α,P)⁻¹α∈F_ν. LEMMA2. Let s>1, there is a functionχ ∈C₀^∞(Rⁿ)such that:

χ (x)=1, x∈ K,

|D^αχ| ≤C(C N^sµ)^α·ν, ifα·ν≤N, ∀N =1,2, . . . , ∀ν∈N₁(P).

(7)

Proof. Every u∈ G^s₀(Rⁿ)satisfies the conditions 7. In fact, every u ∈ G^s₀(Rⁿ)satis- fies:

|D^αu(x)| ≤CC^|α||α|^s|α|≤CC^|α|N^s|α| if|α| ≤N.

In fact, as 0< νj ≤1,∀j=1, . . . ,n,∀ν∈N₁(P)andα·ν≤ |α|, we get:

|α| ≤α·νmax(νj)⁻¹=α·νµ

|α| ≤N ⇒α·ν≤N.

So, taking R=C^µµ^sµ, we obtain:

|D^αu(x)| ≤C(R N^sµ)^α·ν, ∀ν ∈N₁(P), ∀α:α·ν≤N.

Then we can proceed as in the C^∞case to constructχ ∈ G^s₀(Rⁿ)such thatχ ≡1 in K .

LEMMA 3. With the previous notations, if u ∈ G^sP(Rⁿ), then taking χ as in Lemma 2, we obtain the estimate:

(8) |χu(ξ )| ≤c C

C N^s

|ξ|P+N^s µN

N =1,2, . . . Proof. By Leibniz formula we can write:

|D^α(χu)| ≤X

β≤α

α β

|D^α−βχ||D^βu|

Let’s choose anyβ ≤α, thenβ ∈C(ν)for someν∈N₁(P)(not necessarily unique) and for thatνwe get:

sup

x∈suppχ

|D^βχ (x)| ≤C(C N^sµ)^β·ν

and by Lemma 2:

sup

x∈suppχ

|D^α−βχ (x)| ≤C(C N^sµ)^{(α−β)·ν} ifα·ν≤ N, N=1,2, . . .

So we get:

sup

x∈suppχ

|D^α−βχ (x)||D^βu(x)| ≤C(C N^sµ)^{(α−β)·ν}C₁^|α−β|+1(µk(β,P))^sµk(β,P)

≤C(C N^sµ)^{(α−β)·ν}C₁(C₂µk(β,P))^sµk(β,P) as|β| ≤µk(β,P), taking C₂=C

1 s

1.

But we have supposed that k(β,P)=β·νandβ ≤α, moreoverα·ν ≤ N implies β·ν≤ N . We now proceed to estimate:

sup

x∈suppχ

|D^α−βχ (x)||D^βu(x)| ≤C⁰(C N^sµ)^{(α−β)·ν}(C1N^sµ)^β·ν

≤C⁰(C⁰⁰N^sµ)^α·ν

∀α, ifα·ν ≤ N, ∀ν ∈N₁(P), N =1,2, . . .. Taking C⁰⁰ =max{C,C₁}, using the linearity of scalar product and observing that:

(α−β)·ν+β·ν=α·ν, k(α,P)=max{α·ν, ν∈N₁(P)}

we get the inequality:

|D^α(χu)| ≤C⁰(C⁰⁰N^s)^µk(α,P). On the other hand we have:

|D\^α(χu)| =|

Z

e^{−i x·ξ}D^α(χu)|

≤ Z

suppχ

|D^α(χu)| ≤C sup suppχ

|D^α(χu)|

asχhas compact support. Using the properties of the Fourier transform we conclude:

|D\^α(χu)| = |ξ^αχu| ≤c C sup suppχ

|D^α(χu)| ≤C(C N^s)^µk(α,P).

Let nowα=vN , for anyv∈V(P), the set of vertices ofP, summing up the previous inequalities forα=0, α=vN,∀v ∈V(P), we obtain:

|χu(ξ )|Nc ^sµN+ X

v∈V(P)

|χuc(ξ )ξ^vN| ≤C(C N^s)^µN.

Using the following inequality:

X

v∈V(P)

|ξ^vN|n(P)^Nµ−1≤ |ξ|^Nµ_P ≤2^{n(P)(µN−1)} X

v∈V(P)

|ξ^vN| where n(P)denotes the number of vertices ofPdifferent from the origin.

So we can conclude that:

|χu(ξ )| ≤c C(C N^s)^µN N^sµN+P

s∈V(P)|ξ^{s N}|

≤ C(C N^s)^µN N^sµN+ ^|ξ|

Nµ P

2^{n(P)(µN−1)}

≤C⁰

C⁰N^s

|ξ|P+N^s µN

, N=1,2, . . . .

THEOREM 3. Let be an open set inRⁿ, x0 ∈ , u ∈ D⁰(), then u is of class G^sP in a neighborhood of x0if and only if there is a neighborhood U of x0and v∈E⁰()orv∈S⁰(Rⁿ)such that:

1. v=u in U 2. vˆsatisfies:

(9) | ˆv(ξ )| ≤C C N^s

|ξ|P

µN

=C



C⁰N

|ξ|

1 s

P





sµN

, N =1,2, . . . . REMARK4. The previous Theorem 3 admits the more general formulation:

Let K ⊂⊂,u∈D⁰(), then u is of class G^sPin a neighborhood U of K if and only if there isv∈E⁰()orv∈S⁰(Rⁿ), v=u in U such thatvˆsatisfies the estimate (9).

The proof is analogous to that of Theorem 3.

Proof. Proof of necessity: Let u∈G^sP in the set{x :|x−x⁰| ≤3r}, 0<r≤1, χ as in Lemma 3, with K = {x :|x−x₀| ≤r}and suppχ ⊂ {x :|x−x₀| ≤2r}. Then the functionv=χu satisfies conditions 1.,2. of the theorem.

Proof of sufficiency:

Letv ∈ E⁰()satisfy the conditions 1.,2.. Then there are two constants M₀,C > 0 such that:

| ˆv(ξ )| ≤C(1+ |ξ|)^M0. So:

| ˆv(ξ )| ≤C|ξ|_P^M, M =µM₀ Let’s fixα∈Zⁿ₊, the integralR

|ξ^αv(ξ )|dˆ ξ converges by condition 2..

By 1., if x∈U , then:

D^αu(x)=(2π )⁻ⁿ Z

e^{i x·ξ}ξ^αv(ξ )dξ .ˆ

Now we use the property:

(10) |ξ^α| ≤ |ξ|^µk(α,_P ^P).

In fact, givenα ∈Zⁿ₊, then _k(α,P)^α ∈F and so, by the definition of convex hull, told s^l1, . . . ,s^lr the vertices of the face where_k(α,P)^α lies, we have:

α=k(α,P) Xr i=1

λ_is^li, Xr i=1

λ_i =1, λ_i ≥0, and hence by Lemma 1:

|ξ^α| = Yn j=1

|ξ^α_j^j| ≤ Xr i=1

λi



 Yn j=1

|ξj|^{sl jj}





k(α,P)

≤



 X

s^l∈V(P)

Yn j=1

|ξj|^{2sl jj}





1 2k(α,P)

≤ |ξ|^µk(α,_P ^P). (11)

Now, splitting the integral into the two regions:

|ξ|P <N^s, |ξ|P >N^s we get:

|D^αu(x)| ≤(2π )⁻ⁿ(1+N^s)^M+sµk(α,P) Z

|ξ|P<N^s

dξ +C(C N^s)^µN

Z

|ξ|P>N^s

|ξ|^{µk(α,P)−µN}_P dξ .

The first integral is limited for all N and the second converges for large N , namely we set N =k(α,P)+R for R depending only onPand M. Then:

|D^αu(x)| ≤C⁰(C⁰µk(α,P)+R)^{sµ(k(α,P)+R)} implies:

|D^αu(x)| ≤C^|α|+1(µk(α,P))^sµk(α,P).

We now give a characterization of generalized Gevrey functions by means of expo- nential estimates for the Fourier transform, that is possible if s>1 and will be of main interest in the proof of Theorem 2.

THEOREM4. 1. Let u∈G^sP₀ (Rⁿ), then there exist two constants C>0, >

0 such that:

(12) | ˆu(ξ )| ≤C exp(−|ξ|

1 s

P);

2. if the Fourier transform of u ∈ E⁰(Rⁿ), or u ∈ S⁰(Rⁿ)satisfies (12), then u ∈ G^sP(Rⁿ).

In the proof of Theorem 4 we shall use the following lemma:

LEMMA4. The estimates:

(13) | ˆu(ξ )| ≤C



C N

|ξ|

1

Ps





N

, N =1,2, . . .

are equivalent to the following:

(14) | ˆu(ξ )| ≤C^N+1N !|ξ|⁻

N s

P , N =1,2, . . . for suitable different constants C>0 independent of N .

The proof of the lemma is trivial and based on the inequalities: N ! ≤N^N, ∀N = 1,2, . . . and N^N ≤e^NN !. Now we prove Theorem 4.

Proof. Let’s suppose that u ∈ G^sP₀ (Rⁿ), then taking suppu ⊂ K in Remark 4, we obtain:

| ˆu(ξ )|^sµ1 ≤C



C N

|ξ|

1 s

P





N

, N=1,2, . . .

Then by the previous lemma, u satisfies for a suitable constant C⁰>0:

| ˆu(ξ )|^sµ1 ≤(C⁰)^N+1N !|ξ|⁻

N s

P

and fixing= _2C¹0 we get:

| ˆu(ξ )|^{sµ1 N}|ξ|

N s

P

N ! ≤ 1 2

1 2^N . Summing up for N=1,2, . . .:

| ˆu(ξ )|^sµ1 X∞ N=0

^N|ξ|

N s

P

N ! ≤ 1 2

X∞ N=0

1 2^N

and hence:

| ˆu(ξ )|^sµ1 ex p(|ξ|

1

Ps )≤ 1 . So we obtain for a suitable constant C>0:

| ˆu(ξ )| ≤Cex p(−sµ|ξ|

1 s

P)=Cex p(−⁰|ξ|

1 s

P) letting⁰=sµ.

If u∈E⁰(Rⁿ)satisfies:

| ˆu(ξ )| ≤C exp(−|ξ|

1

Ps )=C

exp

− µs|ξ|

1

Ps

µs

then:

| ˆu(ξ )|^µs1 exp

µs|ξ|

1 s

P

≤C^µs1 . Hence, by expanding the exponential into Taylor series we have:

X∞ N=0

| ˆu(ξ )|^µs1 (⁰)^N|ξ|

1 s

P

N ! ≤C⁰ with C⁰=C_µs¹, ⁰= _µs .

This implies:

| ˆu(ξ )|^µs1 (⁰)^N|ξ|

N s

P

N ! ≤C⁰ and hence for a new constant C>0:

| ˆu(ξ )| ≤C⁰



C N

|ξ|

1 s

P





sµN

that means that u∈ G^sP(Rⁿ)as the conditions of Theorem 4 are satisfied in a neigh- borhood of any x₀∈Rⁿ.

2. Multi-quasi-hyperbolic operators

For any complete polyhedronP we define the corresponding class of multi-quasi- hyperbolic operators, according to Definition 2. For short, we denote multi-quasi- hyperbolic operators of order s with respect toP by(s,P)-hyperbolic.

Obviously, if P(D)is multi-quasi-hyperbolic of order s >1 with respect toP, P(D) is also multi-quasi-hyperbolic of order r, ∀r, 1<r<s with respect toP.

We now prove some properties for this class of operators.

PROPOSITION 2. If P(D) is (s,P)-hyperbolic for 1 < s < ∞, then for any (λ, ξ )∈C×Rⁿsuch that P(λ, ξ )=0, there is C>0 such that:

(15) |=λ| ≤C|ξ|

1 s

P.

Proof. The coefficient ofλ^m−1 in P(λ, ξ ) is a linear function ofξ. If the zeros of P(λ, ξ )are denoted byλ_j, it follows thatPm

j=1λ_j is a linear function ofξ. Then also Pm

j=1=λj is a linear combination ofξ, and if P(D)is(s,P)-hyperbolic, then:

Xm j=0

=λj ≥ −mC|ξ|

1 s

P

impliesPm

j=0=λ_j = C₀ for a suitable constant C₀. So we obtain for allλ_k root of P(λ, ξ ):

=λk =C₀−X

j6=k

=λj ≤C₀+C(m−1)|ξ|

1 s

P≤C⁰|ξ|

1 s

P .

That completes the inequality:

|=λ_k| ≤C|ξ|

1 s

P

for all rootsλk of P(λ, ξ ).

PROPOSITION3. If P(D)is(s,P)-hyperbolic for 1 <s < ∞, then the princi- pal part P_m(D)of P(D) is hyperbolic, i.e. the homogeneous polynomial P_m(λ, ξ ) satisfies:

(16) Pm(λ, ξ )=0 (λ, ξ )∈C×Rⁿ ⇒ =λ=0.

Proof. Takingσ >0, λ∈C, ξ ∈Rⁿ, we get:

P_m(λ, ξ )= lim

σ→∞P(σ λ, σ ξ )·σ^−m From Proposition 2 the zeros of P(σ λ, σ ξ )must satisfy:

|=λk| ≤C|σ ξ|

1 s

P

σ

So forσ → ∞,=λ=0 for all the rootsλ∈Cof P_m(λ, ξ ), that is P_m(D)is hyperbolic.

PROPOSITION4. For a differential operator P_m(D)associated to an homogeneous polynomial P_m(λ, ξ ), the notion of hyperbolicity and(s,P)-hyperbolicity coincide.

The proof follows easily from Proposition 3.

PROPOSITION5. Let P(D)be a differential operator of the form:

(17) P(D)=Pm(D)+

m−1X

j=0

aj(Dx)D_t^j

with homogeneous principal part:

(18) Pm(D)=D_t^m+

m−1X

j=0

bj(Dx)D_t^j,

with:

or der(b_j(D_x))=m− j or der(a_j(D_x))≤m−j−1 and assume:

(19) Pm(λ, ξ )=0 forλ∈C, ξ∈Rⁿimplies=λ=0

|aj(ξ )| ≤C|ξ|^m−1_P (1+ |ξ|)^−j for j=1, . . . ,m−1 (for a C>0).

Then P(D)is(_m−1^m ,P)hyperbolic.

Proof. By definition, the terms aj(Dx), bj(Dx)satisfy for a suitable C >0:

(20) |b_j(ξ )| ≤C|ξ|^m−j

|aj(ξ )| ≤C|ξ|^m−j−1.

In the region{|λ| > |ξ|}(for > 0 sufficiently small), the following inequality is satisfied:

|P(λ, ξ )−λ^m| ≤C

m−1X

j=0

|ξ|^m−j|λ|^j <λ^m 2 that implies:

|P(λ, ξ )|>λ^m 2

and consequently P(λ, ξ )can’t have roots in this region and the roots must so satisfy for >0:

(21) |λ| ≤⁻¹|ξ|.

On the other hand, for(λ, ξ )such that P(λ, ξ )=0:

Pm(λ, ξ )= −(P−Pm)(λ, ξ )= −

m−1X

j=0

aj(ξ )λ^j.

In view of the estimates (21) and (19), we obtain:

|Pm(λ, ξ )| ≤

m−1X

j=0

|aj(ξ )||λ|^j ≤C

m−1X

j=0

|ξ|^m−1_P (1+ |ξ|)^−j|λ|^j

≤C⁰

m−1X

j=0

|ξ|^m−1_P (1+ |ξ|)^−j|ξ|^j ≤C⁰⁰|ξ|^m−1_P

In view of the hyperbolicity of P_m we can write:

Pm(λ, ξ )= Ym j=0

(λ−λj), λj ∈R.

Hence:

|=λ|^m ≤ |P_m(λ, ξ )| ≤C⁰⁰|ξ|^m−1_P

|=λ| ≤C⁰⁰⁰|ξ|

m−1 m

P

i.e. P(D)is(_m−1^m ,P)hyperbolic.

PROPOSITION6. Any differential operator P(D)=D^m_t +Pm−1

j=0 a_j(D_x)D_t^j sat- isfying the condition:

(22) |a_j(ξ )| ≤C|ξ|^m−_P ^j−1 j =0,1, . . . ,m−1, for C >0 is(_m−1^m ,P)hyperbolic.

We note that the principal part is only D^m_t and is obviously hyperbolic; Proposition 6 states that in this particular case we may replace (19) with the weaker assumption (22).

Proof. By the estimates (22) we have:

|P(λ, ξ )−λ^m| ≤C

m−1X

j=0

|ξ|^m−j_P ⁻¹|λ|^j <|λ|^m 2

in the region{(λ, ξ )∈C×Rⁿ:|ξ|P < |λ|}for a sufficiently small >0.

Consequently|P(λ, ξ )|> ^|λ|₂^m and P(λ, ξ )can’t have roots in this region and so they must satisfy:

(23) |λ| ≤⁻¹|ξ|P.

For(λ, ξ )such that P(λ, ξ )=0, we write:

λ^m= −(P(λ, ξ )−λ^m)= −

m−1X

j=0

aj(ξ )λ^j .

In view of the estimate (23) forλand (22) for aj(ξ ):

|λ|^m ≤C⁰

m−1X

j=0

|ξ|^m−j_P ⁻¹|λ|^j ≤C⁰⁰|ξ|^m−1_P

Hence:

|=λ|^m≤C⁰⁰|ξ|^m−1_P

|=λ| ≤C⁰⁰⁰|ξ|

m−1 m

P

i.e. P(D)is(_m−1^m ,P)hyperbolic.

REMARK 5. A more general version of Proposition 6 is easily obtained by as- suming as in Proposition 5 that P(D) has hyperbolic homogeneous principal part P_m(D)=Pm−1

j=0 b_j(D_x)D_t^j with:

(24) |bj(ξ )| ≤C|ξ|^m−_P ^j, j =0,1, . . . ,m−1 and keeping condition (22) for the lower order terms.

Observe however that (24) implies bj(ξ )≡0, but in the quasi-homogeneous case.

There follow some examples of multi-quasi-hyperbolic operators, that follow from the previous propositions.

1. If P(D)is a differential operator inRⁿwith symbol P(ξ )and Newton polyhe- dronP of formal orderµ, then the differential operator inRⁿ⁺¹:

Q(D)=D^m_t +P(D_x),

with m> µ, is multi-quasi-hyperbolic of order^m_µ with respect toP. In fact, the roots of the symbol of Q(D)satisfy:

|=λ| ≤C|ξ|

µ m

P.

2. A particular case of Proposition 5 is the following:

ifPis the polyhedron inR²of vertices(0,0), (0,2), (1,0), thenµ=2 and the following operator:

P(D_x,D_t)=P₃(D_x,D_t)+C₁D²_x

2+C₂D_x1 +C₃D_x2+C₄D_t+C₅ where P3(Dx,Dt) is an hyperbolic homogeneous operator of order 3 and C1, ...,C5∈C, is multi-quasi-hyperbolic of order³₂ with respect toP.

3. Another particular case of Proposition 5 is the following:

ifP is the polyhedron inR² of vertices(0,0), (0,3), (1,2), (2,0), then the formal orderµ=4 and the operator of order 4:

P(Dt,Dx)=P4(Dt,Dx)+c1D_x²

2+c2Dx₁Dx₂ +c3Dx₂Dt

+c₄D_x1 +c₅D_x2 +c₆D_t+c₇

where P4(Dx,Dt) is an hyperbolic homogeneous operator of order 4 and C1, ...,C7∈C, is multi-quasi-hyperbolic of order⁴₃ with respect toP.

4. Let P(D)be a differential operator inRⁿwith symbol P(ξ ), then we consider the differential operator inRⁿ⁺¹:

Q(D)=(D_t²+ 4_x)^m−P(D_x) with or der P(D) <2m.

The roots of the symbol of Q(D)satisfy:

(λ²− |ξ|²)^m−P(ξ )=0

and then, denoting by P(ξ )^m1 the generic m−th root of P(ξ ):

=λ= |ξ²+P(ξ )^m1|¹²senθ where forθ >0:

tg2θ = =(ξ²+P(ξ )^m1)

<(ξ²+P(ξ )^m1)

≤ |P(ξ )|^m1

|ξ|² .

We consider the first term of the Taylor expansion to estimate senθ:

senθ≤C|P(ξ )|^m1 2|ξ|²

|=λ| ≤CP(ξ )^m1

|ξ|

LetP⁰be a given complete polyhedron. If for someρ <1 we have:

|P(ξ )|^m1 ≤C|ξ|^ρ_P0|ξ| i.e.

|P(ξ )| ≤C|ξ|^ρm_P0|ξ|^m (25)

then Q(D)is multi-quasi-hyperbolic of order _ρ¹ with respect toP⁰. If we con- sider in particular the Newton polyhedron associated to P(Dx)with formal order µ <2m, then Q(D)is multi-quasi-hyperbolic of order^m_µ, but we can consider also a larger class of polyhedra satisfying condition (25), and in any case stronger with respect to what we may deduce from Proposition 5.

3. Proof of Theorem 2

Now we prove Theorem 2.

Proof. We try to satisfy the Cauchy Problem:

(

P(D)u =D^m_t u+P_m−1

k=0 ak(Dx)D^k_tu=0

D^k_tu(0,x)= f_k(x) ∀x ∈Rⁿ,∀k=0,1, . . . ,m−1 by a function u(t,x)such that u(t,x)∈S(Rⁿ_x)for any fixed t∈R.

We apply partial Fourier transform with respect to x , considering t as a parameter, so the Cauchy Problem admits the following equivalent formulation:

(26)

(P(D_t, ξ )uˆ=D_t^muˆ+Pm−1

k=0 a_k(ξ )D_t^juˆ=0

D^k_tu(0, ξ )ˆ = ˆfk(ξ ) ∀ξ ∈Rⁿ, k=0,1, . . . ,m−1.

This makes sense as f_khave compact support,∀k=0,1, . . . ,m−1 and u ∈S(Rⁿ),∀t fixed.

Now we consider the Cauchy Problem (26) as an ordinary differential problem in t, depending on the parameterξ. A solution to problem (26) is given by:

(27) u(t, ξ )ˆ =

m−1X

j=0

fˆj(ξ )Fj(t, ξ ),

where F_j(t, ξ ), j = 0,1. . . . ,m−1, satisfy the ordinary Cauchy Problem on t de- pending on the parameterξ ∈Rⁿ:

(28)

(

P(Dt, ξ )Fj =0

D^k_tF_j(0, ξ )=δ_{j k} k=0,1, . . . ,m−1 whereδ_{j k}denote the Kronecker delta.

The solution of (28) exists and is unique by the Cauchy theorem for ordinary dif- ferential equations, and the functionu defined in (27) gives indeed a solution to theˆ Cauchy Problem (26), as is easy to check. Now we want to estimate|D^α_xu(t,x)|or, equivalently,| ˆu(t, ξ )|to obtain generalized Gevrey estimates with respect to the space variables.

By assumption fˆ_j(ξ ) ∈ G^r₀^P(Rⁿ), so, in view of Theorem 4,(1), there are constants j,Cj >0(j =0,1, . . . ,m−1)such that for everyξ ∈Rⁿ:

| ˆf_j(ξ )| ≤C_jexp(−_j|ξ|

1

Pr )≤C exp(−|ξ|

1

Pr ),

taking:

C =max{C_j, j=0, . . . ,m−1}, =max{j, j=0, . . . ,m−1}.

To estimate Fj we use the following lemma (for the proof see for example H¨or- mander[9], Lemma 12.7.7).

LEMMA 5. Let L(D)= D^m +P_m−1

j=0 ajD^j be an ordinary differential operator with constant coefficients aj ∈C. Write3= {λ∈C: L(λ)=0}and assume:

maxλ∈3|λ| ≤A,

maxλ∈3|=λ| ≤B forλ∈3.

(29)

Then the solutionsv_j(t), j=0,1, . . . ,m−1 of the Cauchy Problems:

(30)

(L(D)v_j =0

(D^kvj)(0)=δj k, k=0, . . . ,m−1 satisfy the following estimates:

|D^Nvj(t)| ≤2^m(A+1)^N+m+1e^(B+1)|t|, N=0,1, . . . , t ∈R.

(31)

We now apply the estimates of Lemma 5 for N = 0 to the functions F_j(t, ξ )in (28), j =0,1, . . . ,m−1, takingξ as a parameter. If P(D)is(s,P)-hyperbolic, then

∃C⁰>0 such that the roots of P(λ)satisfy:

|=λ| ≤C⁰|ξ|

1 s

P,

consequently we may take B=C⁰|ξ|

1 s

P.

Now we determine A. Let’s consider the characteristic polynomial of P:

P(λ, ξ )=λ^m+

m−1X

j=0

aj(ξ )λ^j

where a_j(ξ )is a polynomial of degree at most equal to m− j . So there are constants C_j such that:

|a_j(ξ )| ≤C_j(1+ |ξ|)^m−j.

It follows easily that for >0 sufficiently small the zeros of P(λ, ξ )cannot belong to the region{(1+ |ξ|) < |λ|}and must necessarily satisfy:

(32) |λ| ≤⁻¹(1+ |ξ|) .

So we can take:

(33) A=⁻¹(1+ |ξ|)

and estimate for a suitable C>0:

(34) |Fj(t, ξ )| ≤(⁻¹(1+ |ξ|)+1))^m+1C exp(C(|t| +1)|ξ|

1 s

P)

By summing up the estimates for fˆj,Fj we get the following estimates foru:ˆ

| ˆu(t, ξ )| ≤

m−1X

j=0

| ˆfj(ξ )||Fj(t, ξ )|

≤C

m−1X

j=0

exp(−|ξ|

1 r

P)exp(C(1+ |t|)|ξ|

1 s

P) . (35)

By assumption, r <s, and so¹_r > ¹_s implies that:

|ξ|→+∞lim

|ξ|

1

Ps

|ξ|

1 r

P

=0

Then there exist positive constants C₁⁰ =C₁⁰(|t|),C⁰₂=C₂⁰(|t|)such that:

C(1+ |t|)|ξ|

1 s

P−|ξ|

1 r

P ≤ −C₁⁰|ξ|

1 r

P+C₂⁰ . Hence we get the following estimate foru:ˆ

| ˆu(t, ξ )| ≤C⁰⁰exp(−C₁⁰|ξ|

1 r

P).

So we have obtained that u∈G^rPfor any t∈Rin view of Theorem 4,2). We observe that the constants C⁰₁,C⁰⁰may depend on t, but are locally bounded, for|t| ≤T,∀T >

0.

REMARK6. We have supposed that r >s to get the result of regularity. In the case r =s, the regularity is only local in time, as evident from the previous computations.

4. Regularity with respect to the time variable

We know that the solution of the Cauchy Problem is in C^∞([−T,T ],G^rP(Rⁿ)),∀T >

0 ; now we will discuss its regularity with respect to the time variable in generalized Gevrey classes. To do so, it is necessary to extend the polyhedron to(n+1)variables, that is possible by means of the following proposition.

PROPOSITION7. Given a complete polyhedronPinRⁿ, we defineP⁰as the convex hull inRⁿ⁺¹of the vertices ofP plus the vector(µ₀,0, . . . ,0)withµ₀ ∈ Q₊, 0 <

µ0 ≤ µ, cf. figure. ThenP⁰ is a complete polyhedron inRⁿ⁺¹with the same formal orderµofP.