438 G. De Donno - L. Rodino
having the same J as the non-hypoelliptic operators 9, 10. If r
∗
is even, then i, ii and 8 imply ℑ J
r
≡ 0 for odd r r
∗
; as corresponding example of hypoelliptic operator consider 15
D
m x
1
− D
m−1 x
2
+ i x
2h 1
D
m−1 x
2
+ i D
2 x
1
D
m−3 x
2
r
∗
= 2, having the same J
as 11, 12. In 13, 14, 15, the order m has to be chosen sufficiently large, to satisfy the assumption
m−1 2
r
∗
. Returning to general operators, we may regard Theorem 1 as extension of a result of Liess-Rodino[7],Theorem 6.3, which prove the same
order of Gevrey hypoellipticity, requiring i and ℑ J
r
= 0 for all r r
∗
, which is stronger than ii; see also Tulovsky [17] for hypoellipticity in the C
∞
-sense. Observe however that Liess- Rodino [7] allow 0 r
∗
≤ m − 2, whereas we do not know whether our result is valid for
m−1 2
≤ r
∗
≤ m − 2. The proof of Theorem 1 will be reduced, after conjugation by classical Fourier integral op-
erators, to a simple S
m ρ,δ
argument let us refer in particular to the result of Kajitani-Wakabayashi [5] in the Gevrey frame.
2. Gevrey hypoellipticity for a class of differential polynomials.
In this section we begin to study a pseudo-differential model in suitable simplectic co-ordinates. The conclusion of the proof of Theorem 1 will be given in the subsequent Section 3. As before
we denote by x = x
1
, ..., x
n
the real variables in , open subset of R
n
; ξ = ξ
1
, ξ
2
, ..., ξ
n
, ξ
2
0, the dual variables of x. We consider the conic neighborhood 3 = {0 ξ
2 1
+ |ζ |
2
Cξ
2 2
} of the axis ξ
2
0, where ζ = ξ
3
, ..., ξ
n
∈ R
n−2
, for a suitable constant C. Moreover we take q, m, r, s ∈ N such that m ≥ 2, 1 ≤ q m, and r, s belong to the set
I = { r, s ∈ N
2
: 0 qr + ms qm }. Let the function in × 3 = Ŵ
16 px, ξ = ξ
m 1
− h
0,q
x, ξ ξ
q 2
+ X
r,s∈I
h
r,s
x, ξ ξ
r 1
ξ
s 2
, be a differential polynomial, symbol of a micro pseudo-differential operator Px, D, where
h
·,·
: Ŵ → C , h
·,·
= ℜ h
·,·
+ i ℑ h
·,·
, ℜ h
·,·
, ℑ h
·,·
: Ŵ → R, ℜ h
·,·
, ℑ h
·,·
∈ G
1
Ŵ , see below.
We define the sets, for k ∈ N, 0 k qm: I
k
= { r, s ∈ N
2
: qr + ms = k } and fix k = k
∗
such that qm −
1 2
k
∗
qm. We use the notation k
−
for all k k
∗
and k
+
for all k k
∗
. We may split I = I
−
S I
k
∗
S I
+
, with I
−
= S
I
k
−
, I
+
= S
I
k
+
. L
EMMA
1. Let px, ξ be the function 16,where h
·,·
is assumed to be homogeneous of order zero with respect to ξ and analytic, which implies for some constant L 0
17 |D
α x
D
β ξ
h
·,·
| ≤ L
|α| + |β| + 1
α β 1 + |ξ |
−|β|
. Assume moreover I
k
∗
consists of one couple r
∗
, s
∗
, k
∗
= qr
∗
+ ms
∗
, such that: i ℑ h
r
∗
, s
∗
x, ξ 6= 0, for all x, ξ ∈ Ŵ,
Gevrey hypoellipticity 439
ii ℑ h
r
∗
, s
∗
x, ξ ℑ h
r,s
x, ξ ξ
r
∗
+r 1
ξ
s+s
∗
2
≥ 0, for all x, ξ ∈ Ŵ, k
∗
k
+
= qr +ms qm ,
iii ℑ h
0,q
x, ξ ℑ h
r
∗
, s
∗
x, ξ ξ
r
∗
1
ξ
q+s
∗
2
≤ 0, for all x, ξ ∈ Ŵ, iv ℜ h
0,q
x, ξ 6= 0, for all x, ξ ∈ Ŵ. Then for all α, β ∈ Z
n
+
, for all K ⊂⊂ , we have for new positive constants L and B independent of α , β:
18 |D
α x
D
β ξ
px, ξ | |ξ |
ρ|β| − δ|α|
| px, ξ | ≤ L
|α| + |β| + 1
α β , |ξ | B ,
where ρ =
k
∗
− q m−1 m
, δ =
qm − k
∗
m
. Observe that we have δ ρ, since we have assumed k
∗
qm −
1 2
R
EMARK
1. Hypothesis ii implies that ℑ h
r
∗
, s
∗
x, ξ and ℑ h
r,s
x, ξ are both positive or both negative ℑ h
r,s
x, ξ may vanish, too, and that r is according both even or both odd to r
∗
for all r such that k
∗
k
+
. Otherwise r is not according to r
∗
, ℑ h
r,s
x, ξ has to vanish in Ŵ.
Hypothesis iii induces ℑ h
0,q
x, ξ ≡ 0 if r
∗
is odd. R
EMARK
2. By formula 18 and by Kajitani-Wakabayashi[5], Theorem 1.9, we have that the operator Px, D, associated to the symbol px, ξ in 16, is G
d
-microlocally hypoelliptic in Ŵ for d ≥ max
n
1 ρ
,
1 1−δ
o =
1 ρ
. R
EMARK
3. When ρ 1, and δ 0, one can prove by means of interpolation theory as in Wakabayashi[18], Theorem 2.6 that 18 is valid for any α, β ∈ Z
n
+
, if 18 holds for |α + β| = 1. Hence it is sufficient to verify 18 for |α + β| = 1 because ρ =
k
∗
−qm−1 m
q m
1, and δ =
qm−k
∗
m
0. R
EMARK
4. For the proof of Theorem 1 it will be sufficient to apply Lemma 1 for q = m − 1. The general case 1 ≤ q m leads to a more involved geometric invariant statement,
which we shall detail in a future paper. Proof of Lemma 1. We first estimate the numerator of 18, then we give some lemmas to esti-
mate the denominator of 18. If |α| = 1, |β| = 0, we get
|D
x
j
px, ξ | |ξ |
−δ
= P
r,s ∈ I
D
x
j
h
r,s
x, ξ ξ
r 1
ξ
s 2
− D
x
j
h
0,q
x, ξ ξ
q 2
|ξ|
−δ
≤ L
1
P
r,s ∈ I
|ξ
1
|
r
ξ
s 2
+ ξ
q 2
|ξ |
−δ
, j = 1, ..., n;
for a suitable constant L
1
in view of the assuption 17. If |α| = 0, |β| = 1, then
|D
ξ
j
px, ξ | |ξ |
ρ
≤ L
2
X
r,s ∈ I
|ξ
1
|
r
ξ
s 2
+ ξ
q 2
|ξ|
ρ
1 + |ξ |
−1
, 19
j = 3, ..., n;
440 G. De Donno - L. Rodino
for a suitable constant L
2
, in view of 17.
Moreover: 20
|D
ξ
1
px, ξ | |ξ |
ρ
≤ m|ξ
1
|
m−1
+ L
3
P
r,s ∈ I
|ξ
1
|
r−1
ξ
s 2
|ξ |
ρ
+L
4
P
r,s ∈ I
|ξ
1
|
r
ξ
s 2
+ ξ
q 2
|ξ |
ρ
1 + |ξ |
−1
and 21
|D
ξ
2
px, ξ | |ξ |
ρ
≤ q L
0,q
ξ
2 q−1
+ L
5
P
r,s ∈ I
|ξ
1
|
r
ξ
s−1 2
|ξ |
ρ
+L
6
P
r,s ∈ I
|ξ
1
|
r
ξ
s 2
+ ξ
q 2
|ξ |
ρ
1 + |ξ |
−1
, for suitable constants L
3
, L
4
, L
5
, L
6
, L
0,q
in view of 17. On the other hand, we have:
22 |ξ |
ρ
1 + |ξ |
−1
≤ |ξ |
−δ
, f or all ξ ∈ 3,
in fact, by multiplying by |ξ |
δ
1 + |ξ | on both sides of 22, we obtain |ξ | − |ξ |
p
+ 1 ≥ 0 , f or all ξ ∈ 3, where p = ρ + δ 1.
Then in the right-hand side of 19, 20, 21 we may further estimate |ξ |
ρ
1 + |ξ |
−1
by |ξ |
−δ
. Therefore, to prove 18, it will be sufficient to show the boundedness in Ŵ, for |ξ | B, of the
functions Q
1
ξ = P
r,s ∈ I
|ξ
1
|
r
ξ
s 2
+ ξ
q 2
|ξ |
−δ
| px, ξ | ,
Q
2
ξ = m|ξ
1
|
m−1
+ L
3
P
r,s ∈ I
|ξ
1
|
r−1
ξ
s 2
|ξ |
ρ
| px, ξ | ,
Q
3
ξ = q L
0,q
|ξ
2
|
q−1
+ L
5
P
r,s ∈ I
|ξ
1
|
r
ξ
s−1 2
|ξ |
ρ
| px, ξ | we observe that terms of the type Q
2
, Q
3
were already considered in De Donno [2]. First introduce in the cone 3, three regions:
23 R
1
: c ξ
q 2
≤ |ξ
1
|
m
≤ C ξ
q 2
, R
2
: |ξ
1
|
m
≥ C ξ
q 2
, R
3
: |ξ
1
|
m
≤ c ξ
q 2
; where the constants c, C satisfy c min
n
1 2
mi n
x ,ξ ∈Ŵ
|ℜ h
0,q
x, ξ |, 1 o
, and C max
2 max
x ,ξ ∈Ŵ
|ℜ h
0,q
x, ξ |, 1 .
The following inequalities then hold: 24
|ξ |
−δ
≤
C
δ q
|ξ
1
|
−δ
m q
, ξ ∈ 3
T R
1
I |ξ
1
|
−δ
, ξ ∈ 3
T R
2
I I ξ
−δ 2
, ξ ∈ 3
T R
3
; I I I
note that II and III hold for all ξ ∈ 3, but for our aim we may limit ourselves to consider them respectively in 3
T R
2
and in 3 T
R
3
. By abuse of notation, in the following we shall also denote by R
1
, R
2
, R
3
the sets × R
1
, × R
2
, × R
3
; recall that Ŵ = × 3.
Gevrey hypoellipticity 441
We will show in Lemma 2, Lemma 3 and Lemma 4, that there are positive constants K
1
1, K
2
1, K
3
1, B, such that: 25
| px, ξ | ≥ K
1
ℑh
r
∗
, s
∗
x, ξ |ξ
1
|
r
∗
ξ
s
∗
2
, i n Ŵ
\ R
1
, |ξ | B,
26 | px, ξ | ≥ K
2
|ξ
1
|
m
, i n Ŵ
\ R
2
, |ξ | B,
27 | px, ξ | ≥ K
3
ξ
2 q
, i n Ŵ
\ R
3
, |ξ | B.
In 25 we may further estimate ℑh
r
∗
, s
∗
x, ξ λ for λ 0, in view of i in Lemma 1.
We first consider Q
1
ξ separately in the regions R
1
, R
2
, R
3
, to prove boundedness. In R
1
by 24,25, we get easily, writing as before k = qr + ms: Q
1
ξ ≤ const
X
k
1 |ξ
1
|
m−
k q
+ 1
, |ξ| B , where m −
k q
0 by definition of I and I
k
− set s. In the regions R
2
, R
3
by using respectively 24,26 and 24,27, we have for a constant ǫ 0 which we may take as small as we want by fixing B sufficiently large:
Q
1
ξ ≤ const
X
k
1 |ξ
1
|
m+q−
k q
−
k∗ m
+ 1
|ξ
1
|
δ
ǫ , |ξ| B ,
and Q
1
ξ ≤ const
X
k
1 |ξ
2
|
2q−
k m
−
k∗ m
+ 1
|ξ
2
|
δ
ǫ , |ξ| B .
We have therefore proved that Q
1
ξ is bounded. Let us estimate Q
2
ξ , Q
3
ξ . As above in
the regions R
1
, R
2
, R
3
, we obtain
Q
2
ξ ≤ const
1 +
X
k
1 |ξ
1
|
m−
k q
,
Q
3
ξ ≤ const
1 |ξ
1
|
m q
−1
+ X
k
1 |ξ
1
|
m+
m q
−1−
k q
ǫ ,
in R
1
for |ξ | B, Q
2
ξ ≤ const
1 |ξ
1
|
m−
k∗ q
+ X
k
1 |ξ
1
|
2m−
k q
−
k∗ q
ǫ ,
Q
3
ξ ≤ const
1 |ξ
1
|
m+
m q
−1−
k∗ q
+ X
k
1 |ξ
1
|
2m+
m q
−1−
k q
+
k∗ q
ǫ ,
442 G. De Donno - L. Rodino
in R
2
for |ξ | B, Q
2
ξ ≤ const
1 |ξ
2
|
q−
k∗ m
+ X
k
1 |ξ
2
|
2q−
k m
−
k∗ m
ǫ ,
Q
3
ξ ≤ const
1 |ξ
2
|
1+q−
q m
−
k∗ m
+ X
k
1 |ξ
2
|
2q+1−
q m
−
k m
−
k∗ m
ǫ ,
in R
3
for |ξ | B. Now Lemma 2, Lemma 3 and Lemma 4 complete the proof.
L
EMMA
2. Let px, ξ be the function 16, such that 17 and i, ii, iii in Lemma 1 hold. Then there are positive constants K
1
1, B, such that: | px, ξ | ≥ K
1
ℑh
r
∗
, s
∗
x, ξ |ξ
1
|
r
∗
ξ
s
∗
2
, x, ξ ∈ Ŵ
\ R
1
, |ξ | B.
Proof. We have that
28 | px, ξ |
2
= ξ
m 1
− ℜ h
0,q
x, ξ ξ
q 2
+ P
r,s∈I
ℜ h
r,s
x, ξ ξ
r 1
ξ
s 2
2
+ +
ℑ h
r
∗
, s
∗
x, ξ ξ
r
∗
1
ξ
s
∗
2
+ P
r,s∈I
−
ℑ h
r,s
x, ξ ξ
r 1
ξ
s 2
+ +
P
r,s∈I
+
ℑ h
r,s
x, ξ ξ
r 1
ξ
s 2
− ℑ h
0,q
x, ξ ξ
q 2
2
; by removing the terms rising from the real part of px, ξ , we can write
| px, ξ |
2
≥ ℑ h
r
∗
, s
∗
x, ξ
2
ξ
2r
∗
1
ξ
2s
∗
2
+
4
X
j =1
J
j
x, ξ where
J
1
x, ξ =
X
r,s∈I
−
ℑ h
r,s
x, ξ ξ
r 1
ξ
s 2
+ 29
X
r,s∈I
+
ℑh
r,s
x, ξ ξ
r 1
ξ
s 2
− ℑh
0,q
x, ξ ξ
q 2
2
,
30 J
2
x, ξ = 2ℑ h
r
∗
, s
∗
x, ξ X
r,s∈I
−
ℑ h
r,s
x, ξ ξ
r
∗
+r 1
ξ
s
∗
+s 2
,
31 J
3
x, ξ = 2ℑ h
r
∗
, s
∗
x, ξ X
r,s∈I
+
ℑ h
r,s
x, ξ ξ
r
∗
+r 1
ξ
s
∗
+s 2
,
32 J
4
x, ξ = −2ℑ h
r
∗
, s
∗
x, ξ ℑ h
0,q
x, ξ ξ
r
∗
1
ξ
s
∗
+q 2
.
Gevrey hypoellipticity 443
29 is non-negative for all x, ξ ∈ Ŵ, 31 and 32 are also non negative by hypotheses ii , iii for all x, ξ ∈ Ŵ.
Let us fix attention on J
2
x, ξ defined by 30. We have for all ǫ 0 ℑh
r
∗
, s
∗
x, ξ
2
ξ
1 2r
∗
ξ
2s
∗
2
+ J
2
x, ξ ≥ 1 − ǫ ℑh
r
∗
, s
∗
x, ξ
2
ξ
1 2r
∗
ξ
2s
∗
2
, in Ŵ
T R
1
, |ξ | B. In fact, assuming for simplicity ξ
1
≥ 0,by 17, 23 in Ŵ T
R
1
and hypothesis i, for all ǫ 0 we get for B sufficiently large
|J
2
x, ξ | ℑh
r
∗
, s
∗
x, ξ
2
ξ
1 2r
∗
ξ
2s
∗
2
≤ const X
r,s∈I
−
ξ
r
∗
+r 1
ξ
s
∗
+s 2
ξ
1 2r
∗
ξ
2s
∗
2
≤
≤ const X
r,s∈I
−
ξ
r
∗
+r+s
∗
+s
m q
1
ξ
2r
∗
+2s
∗ m q
1
ǫ , |ξ | B;
we remark that k
∗
= qr
∗
+ ms
∗
k
−
= qr + ms. Then,
| px, ξ | ≥ K
1
ℑh
r
∗
, s
∗
x, ξ |ξ
1
|
r
∗
ξ
s
∗
2
, x, ξ ∈ Ŵ
\ R
1
, |ξ | B,
for a suitable constant K
1
. L
EMMA
3. Let px, ξ be the function 16,such that 17 holds. Then there are positive constants K
2
1, B, such that: | px, ξ | ≥ K
2
|ξ
1
|
m
, x, ξ ∈ Ŵ
\ R
2
, |ξ | B.
Proof. We write | px, ξ |
2
as in 28; by removing the terms arising from the imaginary part of px, ξ , we get
33 | px, ξ |
2
≥ ξ
m 1
− ℜ h
0,q
x, ξ ξ
q 2
2
+ W
1
x, ξ + W
2
x, ξ where
34 W
1
x, ξ =
X
r,s∈I
ℜ h
r,s
x, ξ ξ
r 1
ξ
s 2
2
,
35 W
2
x, ξ = 2 X
r,s∈I
ℜ h
r,s
x, ξ ξ
r+m 1
ξ
s 2
− 2ℜ h
0,q
x, ξ X
r,s∈I
ℜ h
r,s
x, ξ ξ
r 1
ξ
s+q 2
. Observe first that for λ 0 sufficiently small
ξ
m 1
− ℜ h
0,q
x, ξ ξ
q 2
2
λ ξ
2m 1
; in fact
ξ
m 1
− ℜ h
0,q
x, ξ ξ
q 2
2
≥ ξ
2m 1
− 2ℜ h
0,q
x, ξ ξ
m 1
ξ
q 2
,
444 G. De Donno - L. Rodino
and using 23 in Ŵ T
R
2
, we have for ℜ h
0,q
ξ
1
≥ 0 ξ
2m 1
− 2ℜ h
0,q
x, ξ ξ
m 1
ξ
q 2
≥ 1 −
2 C
ℜ h
0,q
x, ξ ξ
2m 1
λ ξ
2m 1
, since C 2 max
x ,ξ ∈Ŵ
|ℜ h
0,q
x, ξ |. 34 is non negative for all x, ξ ∈ Ŵ. We denote 35 by ϒ
1
x, ξ − ϒ
2
x, ξ , then | px, ξ |
2
≥ λξ
2m 1
+ ϒ
1
x, ξ − ϒ
2
x, ξ . Arguing on ϒ
1
, ϒ
2
in the same way as we have done in Lemma 2, it is possible to show that for all ǫ 0
λξ
2m 1
+ ϒ
1
x, ξ − ϒ
2
x, ξ ≥ λ − ǫξ
2m 1
, x, ξ ∈ Ŵ
\ R
2
, |ξ | B ,
then | px, ξ | ≥ K
2
|ξ
1
|
m
, x, ξ ∈ Ŵ
\ R
2
, |ξ | B ,
where K
2
= λ − ǫ
1 2
. L
EMMA
4. Let px, ξ be the function 16, such that 17 and iv in Lemma 1 hold. Then there are positive constants K
3
1, B, such that: | px, ξ | ≥ K
3
ξ
2 q
, x, ξ ∈ Ŵ
\ R
3
, |ξ | B.
Proof. We apply again 33, 34, 35 to | px, ξ |
2
. Observe that in Ŵ T
R
3
, arguing as above, since c
1 2
min
x ,ξ ∈Ŵ
|ℜ h
0,q
x, ξ |, we obtain for a suitable constant µ 0 ξ
m 1
− ℜ h
0,q
x, ξ ξ
q 2
2
µ ξ
2q 2
. About the terms in 34 and 35, the remarks we have done in Lemma 3 hold by replacing λ ξ
2m 1
with µ ξ
2q 2
, then we have | px, ξ | ≥ K
3
ξ
2 q
, x, ξ ∈ Ŵ
\ R
3
, |ξ | B ,
where K
3
= µ − ǫ
1 2
.
3. Fourier integral operators and proof of Theorem 1