84 D. Calvo
and hence: | ˆ
uξ |
1 sµ
ex pǫ|ξ |
1 s
P
≤ 1
ǫ .
So we obtain for a suitable constant C 0: | ˆ
uξ | ≤ Cex p−ǫsµ|ξ |
1 s
P
= Cex p−ǫ
′
|ξ |
1 s
P
letting ǫ
′
= ǫsµ. If u ∈ E
′
R
n
satisfies: | ˆ
uξ | ≤ C exp−ǫ|ξ |
1 s
P
= C
exp −
ǫ µ
s |ξ |
1 s
P µ
s
then: | ˆ
uξ |
1 µ
s
exp ǫ
µ s
|ξ |
1 s
P
≤ C
1 µ
s
. Hence, by expanding the exponential into Taylor series we have:
∞
X
N =0
| ˆ uξ |
1 µ
s
ǫ
′ N
|ξ |
1 s
P
N ≤ C
′
with C
′
= C
1 µ
s
, ǫ
′
=
ǫ µ
s
. This implies:
| ˆ uξ |
1 µ
s
ǫ
′ N
|ξ |
N s
P
N ≤ C
′
and hence for a new constant C 0: | ˆ
uξ | ≤ C
′
C N |ξ |
1 s
P
sµN
that means that u ∈ G
sP
R
n
as the conditions of Theorem 4 are satisfied in a neigh- borhood of any x
∈ R
n
.
2. Multi-quasi-hyperbolic operators
For any complete polyhedron P 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 to P by s, P-hyperbolic. Obviously, if PD is multi-quasi-hyperbolic of order s 1 with respect to P, PD
is also multi-quasi-hyperbolic of order r, ∀r, 1 r s with respect to P. We now prove some properties for this class of operators.
Multi-quasi-hyperbolic operators 85
P
ROPOSITION
2. If PD is s, P-hyperbolic for 1 s ∞, then for any λ, ξ ∈ C × R
n
such that Pλ, ξ = 0, there is C 0 such that: 15
|ℑλ| ≤ C|ξ |
1 s
P
. Proof. The coefficient of λ
m−1
in Pλ, ξ is a linear function of ξ . If the zeros of Pλ, ξ are denoted by λ
j
, it follows that P
m j =1
λ
j
is a linear function of ξ . Then also P
m j =1
ℑλ
j
is a linear combination of ξ , and if PD is s, P-hyperbolic, then:
m
X
j =0
ℑλ
j
≥ −mC|ξ |
1 s
P
implies P
m j =0
ℑλ
j
= C for a suitable constant C
. So we obtain for all λ
k
root of Pλ, ξ :
ℑλ
k
= C −
X
j 6=k
ℑλ
j
≤ C + Cm − 1|ξ |
1 s
P
≤ C
′
|ξ |
1 s
P
. That completes the inequality:
|ℑλ
k
| ≤ C|ξ |
1 s
P
for all roots λ
k
of Pλ, ξ . P
ROPOSITION
3. If PD is s, P-hyperbolic for 1 s ∞, then the princi- pal part P
m
D of PD is hyperbolic, i.e. the homogeneous polynomial P
m
λ, ξ satisfies:
16 P
m
λ, ξ = λ, ξ ∈ C × R
n
⇒ ℑλ = 0. Proof. Taking σ 0, λ ∈ C, ξ ∈ R
n
, we get: P
m
λ, ξ = lim
σ →∞
Pσ λ, σ ξ · σ
−m
From Proposition 2 the zeros of Pσ λ, σ ξ must satisfy: |ℑλ
k
| ≤ C |σ ξ |
1 s
P
σ So for σ → ∞, ℑλ = 0 for all the roots λ ∈ C of P
m
λ, ξ , that is P
m
D is hyperbolic. P
ROPOSITION
4. 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.
86 D. Calvo
P
ROPOSITION
5. Let PD be a differential operator of the form: 17
PD = P
m
D +
m−1
X
j =0
a
j
D
x
D
j t
with homogeneous principal part: 18
P
m
D = D
m t
+
m−1
X
j =0
b
j
D
x
D
j t
, with:
or der b
j
D
x
= m − j
or der a
j
D
x
≤ m − j − 1
and assume: 19
P
m
λ, ξ = for λ ∈ C, ξ ∈ R
n
implies ℑλ = 0 |a
j
ξ | ≤ C|ξ |
m−1 P
1 + |ξ |
− j
for j = 1, . . . , m − 1 for a C 0.
Then PD is
m m−1
, P hyperbolic.
Proof. By definition, the terms a
j
D
x
, b
j
D
x
satisfy for a suitable C 0: 20
|b
j
ξ | ≤ C|ξ |
m− j
|a
j
ξ | ≤ C|ξ |
m− j −1
. In the region {ǫ|λ| |ξ |} for ǫ 0 sufficiently small, the following inequality is
satisfied: |Pλ, ξ − λ
m
| ≤ C
m−1
X
j =0
|ξ |
m− j
|λ|
j
λ
m
2 that implies:
|Pλ, ξ | λ
m
2 and consequently Pλ, ξ can’t have roots in this region and the roots must so satisfy
for ǫ 0: 21
|λ| ≤ ǫ
−1
|ξ |. On the other hand, for λ, ξ such that Pλ, ξ = 0:
P
m
λ, ξ = − P − P
m
λ, ξ = −
m−1
X
j =0
a
j
ξ λ
j
.
Multi-quasi-hyperbolic operators 87
In view of the estimates 21 and 19, we obtain: |P
m
λ, ξ | ≤
m−1
X
j =0
|a
j
ξ ||λ|
j
≤ C
m−1
X
j =0
|ξ |
m−1 P
1 + |ξ |
− j
|λ|
j
≤ C
′ m−1
X
j =0
|ξ |
m−1 P
1 + |ξ |
− j
|ξ |
j
≤ C
′′
|ξ |
m−1 P
In view of the hyperbolicity of P
m
we can write: P
m
λ, ξ =
m
Y
j =0
λ − λ
j
, λ
j
∈ R. Hence:
|ℑλ|
m
≤ |P
m
λ, ξ | ≤ C
′′
|ξ |
m−1 P
|ℑλ| ≤ C
′′′
|ξ |
m−1 m
P
i.e. PD is
m m−1
, P hyperbolic.
P
ROPOSITION
6. Any differential operator PD = D
m t
+ P
m−1 j =0
a
j
D
x
D
j t
sat- isfying the condition:
22 |a
j
ξ | ≤ C|ξ |
m− j −1 P
j = 0, 1, . . . , m − 1, for C 0 is
m m−1
, 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:
|Pλ, ξ − λ
m
| ≤ C
m−1
X
j =0
|ξ |
m− j −1 P
|λ|
j
|λ|
m
2 in the region {λ, ξ ∈ C × R
n
: |ξ |
P
ǫ|λ|} for a sufficiently small ǫ 0.
Consequently |Pλ, ξ |
|λ|
m
2
and Pλ, ξ can’t have roots in this region and so they must satisfy:
23 |λ| ≤ ǫ
−1
|ξ |
P
. For λ, ξ such that Pλ, ξ = 0, we write:
λ
m
= −Pλ, ξ − λ
m
= −
m−1
X
j =0
a
j
ξ λ
j
.
88 D. Calvo
In view of the estimate 23 for λ and 22 for a
j
ξ :
|λ|
m
≤ C
′ m−1
X
j =0
|ξ |
m− j −1 P
|λ|
j
≤ C
′′
|ξ |
m−1 P
Hence: |ℑλ|
m
≤ C
′′
|ξ |
m−1 P
|ℑλ| ≤ C
′′′
|ξ |
m−1 m
P
i.e. PD is
m m−1
, P hyperbolic.
R
EMARK
5. A more general version of Proposition 6 is easily obtained by as- suming as in Proposition 5 that PD has hyperbolic homogeneous principal part
P
m
D = P
m−1 j =0
b
j
D
x
D
j t
with: 24
|b
j
ξ | ≤ C|ξ |
m− j P
, j = 0, 1, . . . , m − 1
and keeping condition 22 for the lower order terms. Observe however that 24 implies b
j
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 PD is a differential operator in R
n
with symbol Pξ and Newton polyhe- dron P of formal order µ, then the differential operator in R
n+1
: QD = D
m t
+ PD
x
, with m µ, is multi-quasi-hyperbolic of order
m µ
with respect to P. In fact, the roots of the symbol of QD satisfy:
|ℑλ| ≤ C|ξ |
µ m
P
. 2. A particular case of Proposition 5 is the following:
if P is the polyhedron in R
2
of vertices 0, 0, 0, 2, 1, 0, then µ = 2 and the following operator:
PD
x
, D
t
= P
3
D
x
, D
t
+ C
1
D
2 x
2
+ C
2
D
x
1
+ C
3
D
x
2
+ C
4
D
t
+ C
5
where P
3
D
x
, D
t
is an hyperbolic homogeneous operator of order 3 and C
1
, ..., C
5
∈ C, is multi-quasi-hyperbolic of order
3 2
with respect to P.
Multi-quasi-hyperbolic operators 89
3. Another particular case of Proposition 5 is the following: if P is the polyhedron in R
2
of vertices 0, 0, 0, 3, 1, 2, 2, 0, then the formal order µ = 4 and the operator of order 4:
PD
t
, D
x
= P
4
D
t
, D
x
+ c
1
D
2 x
2
+ c
2
D
x
1
D
x
2
+ c
3
D
x
2
D
t
+ c
4
D
x
1
+ c
5
D
x
2
+ c
6
D
t
+ c
7
where P
4
D
x
, D
t
is an hyperbolic homogeneous operator of order 4 and C
1
, ..., C
7
∈ C, is multi-quasi-hyperbolic of order
4 3
with respect to P. 4. Let PD be a differential operator in R
n
with symbol Pξ , then we consider the differential operator in R
n+1
: QD = D
2 t
+ △
x m
− PD
x
with or der PD 2m. The roots of the symbol of QD satisfy:
λ
2
− |ξ |
2 m
− Pξ = 0 and then, denoting by Pξ
1 m
the generic m − th root of Pξ : ℑλ = |ξ
2
+ Pξ
1 m
|
1 2
senθ where for θ 0:
tg2θ = ℑξ
2
+ Pξ
1 m
ℜξ
2
+ Pξ
1 m
≤ |Pξ |
1 m
|ξ |
2
. We consider the first term of the Taylor expansion to estimate senθ :
senθ ≤ C |Pξ |
1 m
2|ξ |
2
|ℑλ| ≤ C Pξ
1 m
|ξ | Let P
′
be a given complete polyhedron. If for some ρ 1 we have: |Pξ |
1 m
≤ C|ξ |
ρ P
′
|ξ | i.e.
|Pξ | ≤ C|ξ |
ρ m
P
′
|ξ |
m
25 then QD is multi-quasi-hyperbolic of order
1 ρ
with respect to P
′
. If we con- sider in particular the Newton polyhedron associated to PD
x
with formal order µ
2m, then QD 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.
90 D. Calvo
3. Proof of Theorem 2
