Rend. Sem. Mat. Univ. Pol. Torino
Vol. 59, 4 (2001)
N.M. Tri
ON LOCAL PROPERTIES OF SOME CLASSES OF
INFINITELY DEGENERATE ELLIPTIC DIFFERENTIAL
OPERATORS
Abstract. We give necessary and sufficient conditions for local solvability
and hypoellipticity of some classes of infinitely degenerate elliptic differential operators.
1. Introduction
We deal with local properties of differential operators
1 ∂
G ca,b = X 2 X 1 + i c(x)|x|−4 + (a(x) − b(x))|x|−3 e− |x| ,
∂y
√
where (x, y) ∈ R2 , i = −1, the functions a(x), b(x), c(x) satisfy
a+ ∈ C if x > 0
a(x) =
=: a,
a− ∈ C if x < 0
b+ ∈ C if x > 0
b(x) =
=: b,
b− ∈ C if x < 0
c+ ∈ C if x > 0
c(x) =
=: c
c− ∈ C if x < 0
(1)
and
X1
=
X2
=
1 ∂
∂
− i b(x)sign(x)|x|−2e− |x| ,
∂x
∂y
1 ∂
∂
− i a(x)sign(x)|x|−2e− |x| .
∂x
∂y
We will assume that Re a+ · Re a− · Re b+ · Re b− 6= 0. The form (1) is motivated by [1],
[2] (see also [3])where the authors studied the hypoellipticity and the local solvability
of finitely degenerate differential operators. Hypoellipticity and local solvability of
G ca,b were investigated in [4] when a+ = a− = −1, b+ = b− = 1, c+ = c− , or
a+ = a− = −1, b+ = b− = 1, c+ + c− = 2, and in [5] where a+ = a− =
−eiϕ , b+ = b− = eiϕ , c+ = (1 − λ+ )eiϕ , c− = (1 − λ− )eiϕ , where ϕ ∈ [0, π/2),
λ+ , λ− ∈ C. For those cases in [1], [2], [4] and [5], we have given another proof
277
278
N.M. Tri
of non-hypoellipticity, recently in [6], [7], [8], by constructing explicit non-smooth
solutions of homogeneous equations. In [9], by the same method we give the values of
a+ , a− , b+ , b− , c+ , c− where G ca,b is not hypoelliptic. Here we shall use the method
of constructing parametrixes. Since G ca,b is elliptic away from the line x = 0, we
will consider only the case when |x| ≤ 1. Throughout the paper we denote by C a
general positive constant which may vary from place to place. The paper is organized
as follows. In section 2 we consider the case Re a+ < 0, Re a− < 0, Re b+ > 0,
Re b− > 0, which directly generalizes the results of Hoshiro-Yagdjian. In section 3 we
investigate the case Re a+ > 0, Re a− > 0, Re b+ > 0, Re b− > 0. In section 4 we
deal with the case Re a+ > 0, Re a− < 0, Re b+ > 0, Re b− < 0. Finally, in section
5 we consider the case Re a+ < 0, Re a− > 0, Re b+ > 0, Re b− > 0. The results in
sections 3-5 have completely new characteristics comparing with those considered in
[4] and [5].
2. The case Re a+ < 0, Re a− < 0, Re b+ > 0, Re b− > 0
T HEOREM 1. Assume that Re a+ < 0, Re a− < 0, Re b+ > 0, Re b− > 0. Then
G ca,b is not hypoelliptic (nor local solvable) if and only if
or
c+
= k,
a+ − b+
c+
= −k − 1,
a+ − b+
where k and l are non-negative integers.
c−
=l
a− − b−
c−
= −l − 1
a− − b−
Proof. I) In this part we prove the hypoellipticity and the local solvability by constructing right and left parametrices. Let us consider the Fouruer transform of u with respect
to y
Z
û(x, η) =
+∞
e−iyη u(x, y)d y.
−∞
By this transform G ca,b become
1
1
d
d
d
Ĝ ca,b x,
,η
=
+ ax −2 sign(x)e− |x| η
+ bx −2 sign(x)e− |x| η
dx
dx
dx
1
=
(c|x|−4 + (a − b)|x|−3)e− |x| η =
1
d2
d
+ (a + b)x −2sign(x)e− |x| η
2
dx
dx
+
abx −4e− |x| η2 + (b − c)|x|−4e− |x| η − (a + b)|x|−3 e− |x| η.
−
2
We would like to study solutions of the equation
d
c
(2)
Ĝ a,b x,
, η û(x, η) = 0.
dx
1
1
279
On local properties
1
− |x|
Put û(x, η) = xe−be η f (z), where z = (b − a)e
following confluent hypergeometric equation for f (z)
1
− |x|
η. Then we will have the
d 2 f (z)
d f (z)
c
+ (1 − z)
−
f (z) = 0.
2
dz
dz
b−a
The equation has two linearly independent solutions
c
c
, 1, z , f 2 (z) = e z 9 1 −
, 1, −z
f 1 (z) = 9
b−a
b−a
z
where 9(α, γ , z) is the Tricomi function (see [10], p. 255). Therefore we have the
following pair of solutions of the equation (2)
− 1
1
c+
− |x|
−b+ ηe |x|
,
if x ≥ 0 :
û +
(x,
η)
:=
xe
9
,
1,
(b
−
a
)ηe
+
+
1
b+ − a+
− 1
1
c+
− |x|
+
−a+ ηe |x|
u 2 (x, η) := xe
,
9 1−
, 1, (a+ − b+ )ηe
b+ − a+
(3)
− 1
1
c−
− |x|
−b− ηe |x|
,
1,
(b
−
a
)ηe
9
,
if x ≤ 0 :
û −
(x,
η)
:=
xe
−
−
1
b− − a−
− 1
1
c−
− |x|
−a− ηe |x|
,
1,
(a
−
b
)ηe
û −
(x,
η)
:=
xe
.
9
1
−
−
−
2
b− − a−
±
As x → 0 we have the following asymptotics for û ±
1 (x, η), û 2 (x, η)
û +
1 (x, η) ≈
1 − x (log[(b+ − a+ )η] + 8(c+ /(b+ − a+ )) − 28(1))
,
0 (c+ /(b+ − a+ ))
û +
2 (x, η) ≈
1 − x (log[(a+ − b+ )η] + 8(1 − c+ /(b+ − a+ )) − 28(1))
,
0 (1 − c+ /(b+ − a+ ))
û −
1 (x, η) ≈
−1 − x (log[(b− − a− )η] + 8(c− /(b− − a− )) − 28(1))
,
0 (c− /(b− − a− ))
û −
2 (x, η) ≈
−1 − x (log[(a− − b− )η] + 8(1 − c− /(b− − a− )) − 28(1))
,
0 (1 − c− /(b− − a− ))
(4)
where 8(z) is the Gauss polygramma function (8(z) = 0(z)/ 0 0 (z)). Define D + (x, η)
+
−
(respectively D − (x, η)) as the Wronskian of û +
1 (x, η), û 2 (x, η) (resp. û 1 (x, η),
−
û 2 (x, η)). Similarly (as in [4], Proposition 2) it is not difficult to see that
  − ctg πc+ − i sin πc+
if arg(b+ − a+ )η ∈ (0, π]
b+ −a+
b+ −a+
D + (0, η) =
πc
πc
+
+
 − ctg
if arg(b+ − a+ )η ∈ (−π, 0]
b+ −a+ + i sin b+ −a+
  − ctg πc− − i sin πc−
if arg(b− − a− )η ∈ (0, π]
b− −a−
b− −a−
D − (0, η) =
πc
πc
−
−
 − ctg
if arg(b− − a− )η ∈ (−π, 0].
b− −a− + i sin b− −a−
280
N.M. Tri
+
Note that D + (0, η) · D − (0, η) 6= 0 when η 6= 0. Therefor û +
1 (x, η), û 2 (x, η) are
−
linearly independent solutions of (2) when x ≥ 0, η 6= 0, and û 1 (x, η), û −
2 (x, η) are
linearly independent solutions of (2) when x ≤ 0, η 6= 0. Next for η > 0 we define
the following pair of solutions of (2)
(
û +
1 (x, η) if x ≥ 0
(5)
u + (x, η) =
u + (η)û − (x, η) + c u + (η)û − (x, η) if x ≤ 0
c1,−
1
2,−
2
+
v (x, η) =
(6)
+
(
+
+
v (η)û + (x, η) + c v (η)û + (x, η) if x ≥ 0
c1,+
1
2,+
2
û −
1 (x, η) if x ≤ 0
+
+
+
u
u
v , cv
+
+
where coefficients c1,−
, c2,−
, c1,+
2,+ are chosen such that u (x, η), v (x, η) are
continously differentiable at x = 0
c−
−
log
(a
−
b
)η
−
log
(b
−
a
)η
−
8
1
−
−
−
+
+
b− −a−
u+
c1,−
(η) =
c+
c−
−
0 b+ −a+ 0 1 − b− −a− D (0, η)
+
+ 48(1)
−8 b+c−a
+
,
+
c−
+
− (0, η)
0 b+c−a
0
1
−
D
b− −a−
+
+
u
(η) =
c2,−
+
+
v
c1,+
(η)
=
+
+
v
c2,+
(η)
=
+
+
log (b+ − a+ )η + log (b− − a− )η + 8 b+c−a
+
c−
c+
−
0 b− −a− 0 b+ −a+ D (0, η)
−
− 48(1)
8 b−c−a
−
,
c+
−
− (0, η)
0 b−c−a
0
D
b+ −a+
−
log (a+ − b+ )η + log (b− − a− )η + 8 1 −
−
+
0 b−c−a
0 1 − b+c−a
D + (0, η)
−
+
−
8 b−c−a
− 48(1)
−
,
c+
−
+ (0, η)
0
1
−
D
0 b−c−a
b+ −a+
−
c+
b+ −a+
+
− log (b+ − a+ )η − log (b− − a− )η − 8 b+c−a
+
c−
c+
0 b− −a− 0 b+ −a+ D + (0, η)
−
−8 b−c−a
+ 48(1)
−
.
−
+
0 b−c−a
0 b+c−a
D + (0, η)
−
+
281
On local properties
For η < 0 we define the following pair of solutions of (2)
(
û +
−
2 (x, η) if x ≥ 0 −
(7)
u (x, η) =
−
u−
u
c1,−
(η)û −
1 (x, η) + c2,− (η)û 2 (x, η) if x ≤ 0
−
v (x, η) =
(8)
(
−
−
−
v (η)û + (x, η) + c v (η)û + (x, η) if x ≥ 0
c1,+
2,+
2
1
(x,
η)
if
x
≤
0
û −
2
−
−
−
u
u
v , cv
−
−
where the coefficients c1,−
, c2,−
, c1,+
2,+ are chosen such that u (x, η), v (x, η)
are continously differentiable at x = 0
c−
−
log
(a
−
b
)η
−
log
(a
−
b
)η
−
8
1
−
−
−
+
+
b− −a−
u−
c1,−
(η) =
c−
c+
−
0 1 − b+ −a+ 0 1 − b− −a− D (0, η)
+
−8 1 − b+c−a
+ 48(1)
+
,
+
c−
+
− (0, η)
0
1
−
D
0 1 − b+c−a
b− −a−
+
u−
c2,− (η)
=
+
v−
c1,+ (η)
=
+
−
v
c2,+
(η)
=
+
log (a+ − b+ )η + log (b− − a− )η + 8 1 −
c+
−
−
0
1
−
0 b−c−a
b+ −a+ D (0, η)
−
−
8 b−c−a
− 48(1)
−
,
c+
−
− (0, η)
0 b−c−a
0
1
−
D
b
−a
−
+
+
c+
b+ −a+
+
log (a+ − b+ )η + log (a− − b− )η + 8 1 − b+c−a
+
c−
c+
+
0 1 − b− −a− 0 1 − b+ −a+ D (0, η)
−
8 1 − b−c−a
− 48(1)
−
,
c+
−
+ (0, η)
0 1 − b−c−a
0
1
−
D
b
−a
−
+
+
+
− log (b+ − a+ )η − log (a− − b− )η − 8 b+c−a
+
c+
c−
0 1 − b− −a− 0 b+ −a+ D + (0, η)
−
−8 1 − b−c−a
+ 48(1)
−
.
−
+
0 1 − b−c−a
0 b+c−a
D + (0, η)
−
+
282
N.M. Tri
Put W ± (0, η) = u ± (0, η)vx± (0, η) − v ± (0, η)u ±
x (0, η). From (5), (6), (7), (8) we
deduce that
+
v
W + (0, η) = c2,+
(η)D + (0, η),
−
u
W − (0, η) = c1,−
(η)D − (0, η).
+
−
We see that W ± (0, η) = 0 for |η| > C if and only if a+c−b
= k, a−c−b
= l or
+
−
c+
c−
=
−k
−
1,
=
−l
−
1
where
k
and
l
are
non-negative
integers.
Hence
a+ −b+
a− −b−
±
±
u (x, η), v (x, η) are two linearly independent solutions of (2) for |η| > C if and
+
−
+
−
only if a+c−b
6= k, a−c−b
6= l or a+c−b
6= −k − 1, a−c−b
6= −l − 1 where k and l
+
−
+
−
are non-negative integers. Now if we denote the Wronskian of u ± (x, η), v ± (x, η) by
W ± (x, η) then by the Liouville theorem we have
Z x
±
±
p(s, η)ds ,
W (x, η) = W (0, η) exp −
0
where
p(s, η) =
Hence
(a+ + b+ )x −2 sign(x)e−1/|x|η if x ≥ 0,
(a− + b− )x −2 sign(x)e−1/|x|η if x ≤ 0.

−1/|x|
v + (η)D + (0, η)e −η(a+ +b+ )e−1/|x|

W + (0, η)e−η(a++b+ )e
= c2,+



if x ≥ 0,
W + (x, η) =
+ (0, η)e −η(a− +b− )e−1/|x| = c v + (η)D + (0, η)e −η(a− +b− )e−1/|x|

W

2,+


if x ≤ 0,

−1/|x|
u − (η)D − (0, η)e −η(a+ +b+ )e−1/|x|

W − (0, η)e−η(a++b+ )e
= c1,−



if x ≥ 0,
W − (x, η) =
− (0, η)e −η(a− +b− )e−1/|x| = c u − (η)D − (0, η)e −η(a− +b− )e−1/|x|

W

1,−


if x ≤ 0.
+
−
Since Re a+ < 0, Re a− < 0, Re b+ > 0, Re b− > 0, if a+c−b
6= k, a−c−b
6= l
+
−
c−
c+
or a+ −b+ 6= −k − 1, a− −b− 6= −l − 1 where k and l are non-negative integers, then
u + (−1, η), v + (1, η) exponentially increase when η → +∞, and u − (−1, η), v − (1, η)
exponentially increase when η → −∞. Hence we construct the Green function as
follows
+
G (x, x 0 , η) if η ≥ C,
G(x, x 0 , η) =
G − (x, x 0 , η) if η ≤ −C,
where
+
0
for η ≥ C : G (x, x , η) =
−
0
(
for η ≤ −C : G (x, x , η) =
(
v + (x,η)u + (x 0 ,η)
W + (x 0 ,η)
v + (x 0 ,η)u + (x,η)
W + (x 0 ,η)
v − (x,η)u − (x 0 ,η)
W − (x 0 ,η)
v − (x 0 ,η)u − (x,η)
W − (x 0 ,η)
if x ≤ x 0 ,
if x 0 ≤ x,
if x ≤ x 0 ,
if x 0 ≤ x.
283
On local properties
+
v (η)/W + (0, η)| < C and using the asymptotic behaviors of
By noting that |c2,+
9(α, γ , z) at zero and at infinity we can show that (in the similar way as in [4])
Z
1
Z
1
−1
−1
|G + (x, x 0 , η)|d x ≤ C ;
−
0
|G (x, x , η)|d x ≤ C ;
Z
1
Z
1
−1
−1
|G + (x, x 0 , η)|d x 0 ≤ C ,
|G − (x, x 0 , η)|d x 0 ≤ C ,
Z 1
∂ G + (x, x 0 , η)
∂ G + (x, x 0 , η)
|
|
|d x ≤ C ;
|d x 0 ≤ C ,
∂x
∂x
−1
−1
Z 1
Z 1
∂ G − (x, x 0 , η)
∂ G − (x, x 0 , η)
|
|
|d x ≤ C ;
|d x 0 ≤ C.
∂x
∂x
−1
−1
Z
1
More general, for an arbitrary natural number n we will have
Z
Z
|Dηn G + (x, x 0 , η)|d x ≤ Cη−n ;
Z
1
|Dηn G − (x, x 0 , η)|d x ≤ C|η|−n ;
Z
1
1
−1
1
−1
−1
−1
|Dηn G + (x, x 0 , η)|d x 0 ≤ Cη−n ,
|Dηn G − (x, x 0 , η)|d x 0 ≤ C|η|−n .
Finally if we define the operator Q
Qu(x, y) =
Z
1
Z
1
Z
∞
0
eiη(y−y ) φ(η)G(x, x 0 , η)u(x 0 , y 0 )d y 0 d x 0 dη,
−1 −1 −∞
where φ(η) is a cut-off function φ(η) ∈ C ∞ (R), φ(η) = 0 if |η| ≤ C, φ(η) = 1
if |η| ≥ 2C, then Q will serve a right parametrix for G ca,b , and Q ∗ will serve a left
c̄
parametrix for G c∗
a,b = −G b̄,ā . Hence the hypoellipticity and local solvability follow.
II) In this part we will prove the theorem in the non-hypoellipticity and non-local solv+
−
able cases. We argue only the cases when a+c−b
= k, a−c−b
= l, where k, l are
+
−
non-negative numbers. The other case can be treated similarly. By the theorem of
Hörmander if G ca,b is local solvable at the origin for some its neighborhood ω, there
exist constants C, m such that
Z
X
X
(9)
|
f vd xd y| ≤ C sup
|Dxα D βy f |
|Dxα D βy G c∗
a,b v|
α+β≤m
α+β≤m
for all f, v ∈ C0∞ (ω). Functions which violate this inequality will be constructed. For
large λ let fλ = F(λ2 x, λ2 y)λ5 , where function F(x, y) ∈ C0∞ (R) and
(10)
Z
∞
Z
∞
−∞ −∞
F(x, y)d xd y = 1.
284
N.M. Tri
Next it is easy to see that the function
(
−1/|x| 0
xe−b̄+ηe
L k (b̄+ − ā+ )ηe−1/|x| if x ≥ 0,
U (x, η) =
−1/|x| 0
xe−b̄−ηe
L l (b̄− − ā− )ηe−1/|x| if x ≤ 0,
where L 0k (z) =
Put
1 z n −z n
n! e Dz (e z )
are the Laguerre polynomials, solve the equation (2).
Z ∞
2
g(λρ)eiλ yρ U (x, λ2 ρ)dρ,
vλ = χ(x)χ(y)
0
R
where g ∈ C0∞ (0, ∞), χ ∈ C0∞ (−∞, ∞), g(ρ)dρ = 1, χ(x) = 1, when |x| ≤ ,
with a fixed small enough positive number . Now we will show that the inequality (9)
does not hold for f = ∂x f λ and v = vλ with the parameter λ large positive enough.
Indeed, for λ large enough, f λ ∈ C0∞ (ω) and
Z Z
Z Z
∂vλ
∂ fλ
vλ (x, y)d xd y =
f λ (x, y)d xd y = A + B,
(11)
−
∂x
∂x
where
A=
Z Z Z
f λ (x, y)χ(y)χx (x)g(λρ)U (x, λ2 ρ)eiλ
B=
Z Z Z
fλ (x, y)χ(y)χ(x)g(λρ)Ux (x, λ2 ρ)eiλ
2 yρ
2 yρ
dρd xd y,
dρd xd y.
It is not difficult to show that limλ→∞ B = 1 and for every positive number N, there
exists a number C N such that |A| ≤ C N (1 + λ)−N . Next it is easy to check that the
function
Z ∞
wλ (x, y) =
g(λρ)eiλ
2 yρ
U (x, λ2 ρ)dρ
0
solves the equation G c̄b̄,ā wλ (x, y) = 0. Hence
G c∗
a,b χ(x)χ(y)wλ (x, y) = Q(x, y, D x , D y )wλ (x, y)
where Q(x, y, Dx , D y ) is a first order differential operator with coefficients vanishing
in S = [−, ] × [−, ]. Next by similar argument in [2], [5] it is shown that
|Dxα D βy wλ (x, y)| ≤ C N,m (1 + λ)−N
for any (x, y) ∈
/ S . Therefore we deduce that
X
−N
(12)
sup
|Dxα D βy G c∗
.
a,b vλ | ≤ Cn (1 + λ)
α+β≤m
Finally we see that (10), (11), (12) contradict (9).
R EMARK 1. The following cases Re a+ > 0, Re a− < 0, Re b+ < 0, Re b− > 0;
Re a+ > 0, Re a− > 0, Re b+ < 0, Re b− < 0; Re a+ < 0, Re a− > 0, Re b+ > 0,
Re b− < 0 can be considered analogously.
285
On local properties
3. The case Re a+ > 0, Re a− > 0, Re b+ > 0, Re b− > 0
T HEOREM 2. Assume that Re a+ > 0, Re a− > 0, Re b+ > 0, Re b− > 0. Then
G ca,b is not hypoelliptic nor local solvable at the origin.
Proof. We separate the proof into some cases
I) The non-resonance case a+ 6= b+ , a− 6= b− . We retain all notations used previously.
For η > 0 we define the following solution of (2) V (x, η) = u + (x, η) is defined as
−
in (5). Note that when x ≤ 0 the solutions û −
1 (x, η), û 2 (x, η) exponentially decrease
when η → +∞.
+
A) If b+c−a
∈
/ Z− ∪ 0 then we set f = fλ , v = vλ as in section 2, with U (x, η)
+
+
∈ Z− ∪ 0 then we set f = ∂x fλ , v = vλ as in section
replaced by V (x, η). B) If b+c−a
+
2, with U (x, η) replaced by V (x, η).
Then in a similar way as in section 2 we can contradict (9) by using f, v with large
enough λ.
II) The resonance case a+ = b+ , a− = b− . A) When c+ 6= 0, c− 6= 0 by taking
the limit when a+ → b+ , a− → b− in (3) it is easy to see that the following pair is
solutions of (2) (see [10], p. 266)
1
− |x|
1 1
− |x|
)2
1 1 9 21 , 1, 4 c+ ηe− |x| 2 ,
− 1 1
− 1
1 1
−a+ ηe |x| +2(c+ ηe |x| ) 2 9 1 , 1, −4 c ηe − |x| 2 ,
ũ +
(x,
η)
:=
xe
+
2
2
1
1 1
− |x|
− |x|
1 1
−a− ηe
−2(c− ηe
) 2 9 1 , 1, 4 c ηe − |x| 2 ,
when x ≤ 0 : ũ −
−
1 (x, η) := xe
2
− 1
− 1 1
1 1 −a− ηe |x| +2(c− ηe |x| ) 2 9 1 , 1, −4 c ηe − |x| 2 .
ũ −
(x,
η)
:=
xe
−
2
2
when x ≥ 0 :
−a+ ηe
ũ +
1 (x, η) := xe
−2(c+ ηe
Next for η > 0 let us define the following solution of (2)
( +
ũ 1 (x, η) if x ≥ 0,
Ṽ (x, η) =
−
Ṽ
Ṽ
c1,−
(η)ũ −
1 (x, η) + c2,− (η)ũ 2 (x, η) if x ≤ 0,
Ṽ
Ṽ
where c1,−
(η), c2,−
(η) are chosen as in section 2. Now we can repeat the proof in
section 2.
+
B) When c+ 6= 0, c− = 0 then we have solutions ũ +
1 (x, η), ũ 2 (x, η) when x ≥ 0, and
e−a− ηe
1
|x|
, xe−a− ηe
1
|x|
when x ≤ 0.
C) When c+ = 0, c− 6= 0 then we have solutions e−a+ ηe
−
and ũ −
1 (x, η), ũ 2 (x, η) when x ≤ 0.
D) When c+ = 0, c− = 0 then we have solutions e−a+ ηe
1
ηe |x|
1
ηe |x|
1
|x|
1
|x|
, xe−a+ ηe
, xe−a+ ηe
1
|x|
1
|x|
when x ≥ 0,
when x ≥ 0,
and e−a−
, xe−a−
when x ≤ 0.
III) The half-resonance case a+ 6= b+ , a− = b− . This case can be treated by using the
solutions in part I) when x ≥ 0, and the solutions in part II) when x ≤ 0.
286
N.M. Tri
IV) The half-resonance case a+ = b+ , a− 6= b− . This case can be treated by using the
solutions in part II) when x ≥ 0, and the solutions in part I) when x ≤ 0.
R EMARK 2. The following case Re a+ < 0, Re a− < 0, Re b+ < 0, Re b− < 0
can be considered analogously.
4. The case Re a+ > 0, Re a− < 0, Re b+ > 0, Re b− < 0
T HEOREM 3. Assume that Re a+ > 0, Re a− < 0, Re b+ > 0, Re b− < 0. Then
G ca,b is always hypoelliptic and local solvable at the origin.
Proof. We consider only the non-resonance case a+ 6= b+ , a− 6= b− . The other cases
(resonance and half-resonance) can be treated analogously. Next for η > 0 we define
the following pair of solutions of (2)
(13)
(14)
+
(
+
(
U (x, η) =
V (x, η) =
û +
1 (x, η) if x ≥ 0,
+
+
−
U
U
c1,−
(η)û −
1 (x, η) + c2,− (η)û 2 (x, η) if x ≤ 0,
û +
2 (x, η) if x ≥ 0,
+
+
−
V
V
c1,−
(η)û −
1 (x, η) + c2,− (η)û 2 (x, η) if x ≤ 0,
+
+
+
+
U
U
V
V
where the coefficients c1,−
, c2,−
, c1,−
, c2,−
are chosen as in section 2 such that
+
+
U (x, η), V (x, η) are continuously differentiable at x = 0.
For η < 0 we define the following pair of solutions of (2)
(15)
(16)
−
(
−
(
U (x, η) =
V (x, η) =
−
−
−
+
U
U
c1,+
(η)û +
1 (x, η) + c2,+ (η)û 2 (x, η) if x ≥ 0,
û −
1 (x, η) if x ≤ 0,
−
−
+
V
V
c1,+
(η)û +
1 (x, η) + c2,+ (η)û 2 (x, η) if x ≥ 0,
û −
2 (x, η) if x ≤ 0,
−
−
−
U , cU , c V , c V are chosen such that U − (x, η),
where the coefficients c1,+
2,+ 1,+ 2,+
−
V (x, η) are continuously differentiable at x = 0. Since Re a+ > 0, Re a− <
0, Re b+ > 0, Re b− < 0, then U + (−1, η), V + (−1, η) exponentially increase when
η → +∞, and U − (1, η), V − (1, η) exponentially increase when η → −∞. Therefore
we construct the Green function as follows
( +
G (x, x 0 , η) if η ≥ C,
0
G(x, x , η) =
G − (x, x 0 , η) if η ≤ −C,
On local properties
287
where, for η ≥ C:

+
+ 0
+ 0
+

 V (x, η)U (x , η) − V (x , η)U (x, η) if x ≤ x 0 ,
+
0
+
0
D (x , η)
G (x, x , η) =

 0 if x 0 ≤ x,
and for η ≤ −C:

−
− 0
− 0
−

 V (x, η)U (x , η) − V (x , η)U (x, η) if x ≤ x 0 ,
−
0
−
0
D (x , η)
G (x, x , η) =

 0 if x 0 ≤ x.
Now the rest of the proof goes through as in section 2. We leave the details to the
readers.
R EMARK 3. The following case Re a+ < 0, Re a− > 0, Re b+ < 0, Re b− > 0
can be treated analogously.
5. The case Re a+ < 0, Re a− > 0, Re b+ > 0, Re b− > 0
T HEOREM 4. Assume that Re a+ < 0, Re a− > 0, Re b+ > 0, Re b− > 0. Then
G ca,b is not hypoelliptic nor local solvable at the origin.
Proof. We argue only for the non-resonance case a+ 6= b+ , a− 6= b− . The halfresonance case can be treated analogously. Put
( +
ũ 1 (x, η) if x ≥ 0,
Ũ (x, η) =
−
Ũ
Ũ
c1,−
(η)ũ −
1 (x, η) + c2,− (η)ũ 2 (x, η) if x ≤ 0,
Ũ
Ũ
where c1,−
(η), c2,−
(η) are chosen as in section 2. Since Re b+ > 0, Re a− >
0, Re b− > 0 the solution Ũ (x, η) exponentially decreases when η → +∞. Now
the rest of the proof goes as in section 2.
R EMARK 4. The seven following cases Re a+ < 0, Re a− < 0, Re b+
0, Re b− < 0; Re a+ > 0, Re a− < 0, Re b+ > 0, Re b− > 0; Re a+ > 0, Re a−
0, Re b+ > 0, Re b− < 0; Re a+ < 0, Re a− < 0, Re b+ < 0, Re b− > 0; Re a+
0, Re a− > 0, Re b+ < 0, Re b− > 0; Re a+ > 0, Re a− < 0, Re b+ < 0, Re b−
0; Re a+ < 0, Re a− > 0, Re b+ < 0, Re b− < 0; can be considered analogously.
>
>
>
<
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288
N.M. Tri
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AMS Subject Classification: 35H05, 35D10, 35B65.
Nguyen Minh TRI
Institute of Mathematics
P. O. Box 631, Boho 10000
Hanoi, VIETNAM
e-mail: triminh@thevinh.ncst.ac.vn
Lavoro pervenuto in redazione il 02.11.1999 e, in forma definitiva, il 30.05.2000.