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 differ-
ential operators.
1. Introduction
We deal with local properties of differential operators 1
G
c a,b
= X
2
X
1
+ i cx
|x|
−4
+ ax − bx|x|
−3
e
−
1 |x|
∂ ∂
y ,
where x , y ∈ R
2
, i =
√ −1, the functions ax, bx, cx satisfy
ax =
a
+
∈ C if x 0 a
−
∈ C if x 0 =: a,
bx =
b
+
∈ C if x 0 b
−
∈ C if x 0 =: b,
cx =
c
+
∈ C if x 0 c
−
∈ C if x 0 =: c
and X
1
= ∂
∂ x
− ibxsignx|x|
−2
e
−
1 |x|
∂ ∂
y ,
X
2
= ∂
∂ x
− iaxsignx|x|
−2
e
−
1 |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
c a,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
−
= −e
iϕ
, b
+
= b
−
= e
iϕ
, c
+
= 1 − λ
+
e
iϕ
, c
−
= 1 − λ
−
e
iϕ
, where ϕ ∈ [0, π2,
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
c a,b
is not hypoelliptic. Here we shall use the method of constructing parametrixes. Since G
c a,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
−
T
HEOREM
1. Assume that Re a
+
0, Re a
−
0, Re b
+
0, Re b
−
0. Then G
c a,b
is not hypoelliptic nor local solvable if and only if c
+
a
+
− b
+
= k, c
−
a
−
− b
−
= l or
c
+
a
+
− b
+
= −k − 1, c
−
a
−
− b
−
= −l − 1 where k and l are non-negative integers.
Proof. I In this part we prove the hypoellipticity and the local solvability by construct- ing right and left parametrices. Let us consider the Fouruer transform of u with respect
to y
ˆux, η = Z
+∞ −∞
e
−iyη
ux , yd y. By this transform G
c a,b
become ˆ
G
c a,b
x , d
d x , η
= d
d x + ax
−2
signx e
−
1 |x|
η d
d x + bx
−2
signx e
−
1 |x|
η − c|x|
−4
+ a − b|x|
−3
e
−
1 |x|
η =
= d
2
d x
2
+ a + bx
−2
signx e
−
1 |x|
η d
d x + abx
−4
e
−
2 |x|
η
2
+ b − c|x|
−4
e
−
1 |x|
η − a + b|x|
−3
e
−
1 |x|
η. We would like to study solutions of the equation
2 ˆ
G
c a,b
x , d
d x , η
ˆux, η = 0.
On local properties 279
Put ˆux, η = xe
−be
− 1
|x|
η
f z, where z = b − ae
−
1 |x|
η . Then we will have the
following confluent hypergeometric equation for f z z
d
2
f z d z
2
+ 1 − z d f z
d z −
c b
− a f z
= 0. The equation has two linearly independent solutions
f
1
z = 9
c b
− a ,
1, z ,
f
2
z = e
z
9 1
− c
b − a
, 1,
−z where 9α, γ , z is the Tricomi function see [10], p. 255. Therefore we have the
following pair of solutions of the equation 2 if x
≥ 0 : ˆu
+ 1
x , η : = xe
−b
+
η e
− 1
|x|
9 c
+
b
+
− a
+
, 1, b
+
− a
+
η e
−
1 |x|
, u
+ 2
x , η : = xe
−a
+
η e
− 1
|x|
9 1
− c
+
b
+
− a
+
, 1, a
+
− b
+
η e
−
1 |x|
, 3
if x ≤ 0 :
ˆu
− 1
x , η : = xe
−b
−
η e
− 1
|x|
9 c
−
b
−
− a
−
, 1, b
−
− a
−
η e
−
1 |x|
, ˆu
− 2
x , η : = xe
−a
−
η e
− 1
|x|
9 1
− c
−
b
−
− a
−
, 1, a
−
− b
−
η e
−
1 |x|
. As x
→ 0 we have the following asymptotics for ˆu
± 1
x , η, ˆu
± 2
x , η ˆu
+ 1
x , η ≈
1 − x log[b
+
− a
+
η ]
+ 8c
+
b
+
− a
+
− 281 Ŵ c
+
b
+
− a
+
, ˆu
+ 2
x , η ≈
1 − x log[a
+
− b
+
η ]
+ 81 − c
+
b
+
− a
+
− 281 Ŵ 1
− c
+
b
+
− a
+
, 4
ˆu
− 1
x , η ≈
−1 − x log[b
−
− a
−
η ]
+ 8c
−
b
−
− a
−
− 281 Ŵ c
−
b
−
− a
−
, ˆu
− 2
x , η ≈
−1 − x log[a
−
− b
−
η ]
+ 81 − c
−
b
−
− a
−
− 281 Ŵ 1
− c
−
b
−
− a
−
, where 8z is the Gauss polygramma function 8z
= Ŵz Ŵ
′
z. Define D
+
x , η respectively D
−
x , η as the Wronskian of ˆu
+ 1
x , η, ˆu
+ 2
x , η resp. ˆu
− 1
x , η, ˆu
− 2
x , η. Similarly as in [4], Proposition 2 it is not difficult to see that D
+
0, η =
−
ctg
π c
+
b
+
−a
+
− i sin
π c
+
b
+
−a
+
if argb
+
− a
+
η ∈ 0, π]
− ctg
π c
+
b
+
−a
+
+ i sin
π c
+
b
+
−a
+
if argb
+
− a
+
η ∈ −π, 0]
D
−
0, η =
−
ctg
π c
−
b
−
−a
−
− i sin
π c
−
b
−
−a
−
if argb
−
− a
−
η ∈ 0, π]
− ctg
π c
−
b
−
−a
−
+ i sin
π c
−
b
−
−a
−
if argb
−
− a
−
η ∈ −π, 0].
280 N.M. Tri
Note that D
+
0, η · D
−
0, η 6= 0 when η 6= 0. Therefor ˆu
+ 1
x , η, ˆu
+ 2
x , η are linearly independent solutions of 2 when x
≥ 0, η 6= 0, and ˆu
− 1
x , η, ˆu
− 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
5 u
+
x , η =
ˆu
+ 1
x , η if x ≥ 0
c
u
+
1, −
η ˆu
− 1
x , η + c
u
+
2, −
η ˆu
− 2
x , η if x ≤ 0
6 v
+
x , η =
c
v
+
1, +
η ˆu
+ 1
x , η + c
v
+
2, +
η ˆu
+ 2
x , η if x ≥ 0
ˆu
− 1
x , η if x ≤ 0
where coefficients c
u
+
1, −
, c
u
+
2, −
, c
v
+
1, +
, c
v
+
2, +
are chosen such that u
+
x , η, v
+
x , η are continously differentiable at x
= 0 c
u
+
1, −
η =
− log a
−
− b
−
η − log b
+
− a
+
η − 8
1 −
c
−
b
−
−a
−
Ŵ
c
+
b
+
−a
+
Ŵ 1
−
c
−
b
−
−a
−
D
−
0, η +
−8
c
+
b
+
−a
+
+ 481 Ŵ
c
+
b
+
−a
+
Ŵ 1
−
c
−
b
−
−a
−
D
−
0, η ,
c
u
+
2, −
η =
log b
+
− a
+
η + log b
−
− a
−
η + 8
c
+
b
+
−a
+
Ŵ
c
−
b
−
−a
−
Ŵ
c
+
b
+
−a
+
D
−
0, η +
8
c
−
b
−
−a
−
− 481 Ŵ
c
−
b
−
−a
−
Ŵ
c
+
b
+
−a
+
D
−
0, η ,
c
v
+
1, +
η =
log a
+
− b
+
η + log b
−
− a
−
η + 8
1 −
c
+
b
+
−a
+
Ŵ
c
−
b
−
−a
−
Ŵ 1
−
c
+
b
+
−a
+
D
+
0, η +
8
c
−
b
−
−a
−
− 481 Ŵ
c
−
b
−
−a
−
Ŵ 1
−
c
+
b
+
−a
+
D
+
0, η ,
c
v
+
2, +
η =
− log b
+
− a
+
η − log b
−
− a
−
η − 8
c
+
b
+
−a
+
Ŵ
c
−
b
−
−a
−
Ŵ
c
+
b
+
−a
+
D
+
0, η +
−8
c
−
b
−
−a
−
+ 481 Ŵ
c
−
b
−
−a
−
Ŵ
c
+
b
+
−a
+
D
+
0, η .
On local properties 281
For η 0 we define the following pair of solutions of 2 7
u
−
x , η =
ˆu
+ 2
x , η if x ≥ 0
c
u
−
1, −
η ˆu
− 1
x , η + c
u
−
2, −
η ˆu
− 2
x , η if x ≤ 0
8 v
−
x , η =
c
v
−
1, +
η ˆu
+ 1
x , η + c
v
−
2, +
η ˆu
+ 2
x , η if x ≥ 0
ˆu
− 2
x , η if x ≤ 0
where the coefficients c
u
−
1, −
, c
u
−
2, −
, c
v
−
1, +
, c
v
−
2, +
are chosen such that u
−
x , η, v
−
x , η are continously differentiable at x
= 0 c
u
−
1, −
η =
− log a
−
− b
−
η − log a
+
− b
+
η − 8
1 −
c
−
b
−
−a
−
Ŵ 1
−
c
+
b
+
−a
+
Ŵ 1
−
c
−
b
−
−a
−
D
−
0, η +
−8 1
−
c
+
b
+
−a
+
+ 481 Ŵ
1 −
c
+
b
+
−a
+
Ŵ 1
−
c
−
b
−
−a
−
D
−
0, η ,
c
u
−
2, −
η =
log a
+
− b
+
η + log b
−
− a
−
η + 8
1 −
c
+
b
+
−a
+
Ŵ
c
−
b
−
−a
−
Ŵ 1
−
c
+
b
+
−a
+
D
−
0, η +
8
c
−
b
−
−a
−
− 481 Ŵ
c
−
b
−
−a
−
Ŵ 1
−
c
+
b
+
−a
+
D
−
0, η ,
c
v
−
1, +
η =
log a
+
− b
+
η + log a
−
− b
−
η + 8
1 −
c
+
b
+
−a
+
Ŵ 1
−
c
−
b
−
−a
−
Ŵ 1
−
c
+
b
+
−a
+
D
+
0, η +
8 1
−
c
−
b
−
−a
−
− 481 Ŵ
1 −
c
−
b
−
−a
−
Ŵ 1
−
c
+
b
+
−a
+
D
+
0, η ,
c
v
−
2, +
η =
− log b
+
− a
+
η − log a
−
− b
−
η − 8
c
+
b
+
−a
+
Ŵ 1
−
c
−
b
−
−a
−
Ŵ
c
+
b
+
−a
+
D
+
0, η +
−8 1
−
c
−
b
−
−a
−
+ 481 Ŵ
1 −
c
−
b
−
−a
−
Ŵ
c
+
b
+
−a
+
D
+
0, η .
282 N.M. Tri
Put W
±
0, η = u
±
0, ηv
± x
0, η − v
±
0, ηu
± x
0, η. From 5, 6, 7, 8 we deduce that
W
+
0, η = c
v
+
2, +
η D
+
0, η, W
−
0, η = c
u
−
1, −
η D
−
0, η. We see that W
±
0, η = 0 for |η| C if and only if
c
+
a
+
−b
+
= k,
c
−
a
−
−b
−
= l or
c
+
a
+
−b
+
= −k − 1,
c
−
a
−
−b
−
= −l − 1 where k and l are non-negative integers. Hence u
±
x , η, v
±
x , η are two linearly independent solutions of 2 for |η| C if and
only if
c
+
a
+
−b
+
6= k,
c
−
a
−
−b
−
6= l or
c
+
a
+
−b
+
6= −k − 1,
c
−
a
−
−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 W
±
x , η = W
±
0, η exp −
Z
x
ps, ηds ,
where ps, η
= a
+
+ b
+
x
−2
si gnx e
−1|x|
η if x
≥ 0, a
−
+ b
−
x
−2
si gnx e
−1|x|
η if x
≤ 0. Hence
W
+
x , η =
W
+
0, ηe
−ηa
+
+b
+
e
−1|x|
= c
v
+
2, +
η D
+
0, ηe
−ηa
+
+b
+
e
−1|x|
if x ≥ 0,
W
+
0, ηe
−ηa
−
+b
−
e
−1|x|
= c
v
+
2, +
η D
+
0, ηe
−ηa
−
+b
−
e
−1|x|
if x ≤ 0,
W
−
x , η =
W
−
0, ηe
−ηa
+
+b
+
e
−1|x|
= c
u
−
1, −
η D
−
0, ηe
−ηa
+
+b
+
e
−1|x|
if x ≥ 0,
W
−
0, ηe
−ηa
−
+b
−
e
−1|x|
= c
u
−
1, −
η D
−
0, ηe
−ηa
−
+b
−
e
−1|x|
if x ≤ 0.
Since Re a
+
0, Re a
−
0, Re b
+
0, Re b
−
0, if
c
+
a
+
−b
+
6= k,
c
−
a
−
−b
−
6= l or
c
+
a
+
−b
+
6= −k − 1,
c
−
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
Gx , x
′
, η =
G
+
x , x
′
, η if η
≥ C, G
−
x , x
′
, η if η
≤ −C, where
for η ≥ C : G
+
x , x
′
, η =
v
+
x ,ηu
+
x
′
,η W
+
x
′
,η
if x ≤ x
′
,
v
+
x
′
,η u
+
x ,η W
+
x
′
,η
if x
′
≤ x, for η
≤ −C : G
−
x , x
′
, η =
v
−
x ,ηu
−
x
′
,η W
−
x
′
,η
if x ≤ x
′
,
v
−
x
′
,η u
−
x ,η W
−
x
′
,η
if x
′
≤ x.
On local properties 283
By noting that |c
v
+
2, +
η W
+
0, η | C and using the asymptotic behaviors of
9α, γ , z at zero and at infinity we can show that in the similar way as in [4]
Z
1 −1
|G
+
x , x
′
, η |dx ≤ C ;
Z
1 −1
|G
+
x , x
′
, η |dx
′
≤ C , Z
1 −1
|G
−
x , x
′
, η |dx ≤ C ;
Z
1 −1
|G
−
x , x
′
, η |dx
′
≤ C , Z
1 −1
| ∂
G
+
x , x
′
, η ∂
x |dx ≤ C ;
Z
1 −1
| ∂
G
+
x , x
′
, η ∂
x |dx
′
≤ C , Z
1 −1
| ∂
G
−
x , x
′
, η ∂
x |dx ≤ C ;
Z
1 −1
| ∂
G
−
x , x
′
, η ∂
x |dx
′
≤ C. More general, for an arbitrary natural number n we will have
Z
1 −1
|D
n η
G
+
x , x
′
, η |dx ≤ Cη
−n
; Z
1 −1
|D
n η
G
+
x , x
′
, η |dx
′
≤ Cη
−n
, Z
1 −1
|D
n η
G
−
x , x
′
, η |dx ≤ C|η|
−n
; Z
1 −1
|D
n η
G
−
x , x
′
, η |dx
′
≤ C|η|
−n
. Finally if we define the operator Q
Qux , y =
Z
1 −1
Z
1 −1
Z
∞ −∞
e
iηy −y
′
φη Gx , x
′
, η ux
′
, y
′
d y
′
d x
′
dη, where φη is a cut-off function φη
∈ C
∞
R, φη = 0 if |η| ≤ C, φη = 1
if |η| ≥ 2C, then Q will serve a right parametrix for G
c a,b
, and Q
∗
will serve a left parametrix for G
c ∗
a,b
= −G
¯c ¯b,¯a
. 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
c
+
a
+
−b
+
= k,
c
−
a
−
−b
−
= l, where k, l are non-negative numbers. The other case can be treated similarly. By the theorem of
H¨ormander if G
c a,b
is local solvable at the origin for some its neighborhood ω, there exist constants C, m such that
9 |
Z f vd x d y
| ≤ C sup X
α +β≤m
|D
α x
D
β y
f |
X
α +β≤m
|D
α x
D
β y
G
c ∗
a,b
v |
for all f, v ∈ C
∞
ω . Functions which violate this inequality will be constructed. For
large λ let f
λ
= Fλ
2
x , λ
2
yλ
5
, where function Fx , y ∈ C
∞
R and
10 Z
∞ −∞
Z
∞ −∞
Fx , yd x d y = 1.
284 N.M. Tri
Next it is easy to see that the function U x , η
= x e
− ¯b
+
η e
−1|x|
L
k
¯ b
+
− ¯a
+
η e
−1|x|
if x ≥ 0,
x e
− ¯b
−
η e
−1|x|
L
l
¯ b
−
− ¯a
−
η e
−1|x|
if x ≤ 0,
where L
k
z =
1 n
e
z
D
n z
e
−z
z
n
are the Laguerre polynomials, solve the equation 2. Put
v
λ
= χxχy Z
∞
gλρe
iλ
2
yρ
U x , λ
2
ρ dρ,
where g ∈ C
∞
0, ∞, χ ∈ C
∞
−∞, ∞, R gρdρ = 1, χx = 1, when |x| ≤ ǫ,
with a fixed small enough positive number ǫ. Now we will show that the inequality 9 does not hold for f
= ∂
x
f
λ
and v = v
λ
with the parameter λ large positive enough. Indeed, for λ large enough, f
λ
∈ C
∞
ω and
11 −
Z Z
∂ f
λ
∂ x
v
λ
x , yd x d y =
Z Z
∂v
λ
∂ x
f
λ
x , yd x d y = A + B,
where A
= Z
Z Z
f
λ
x , yχ yχ
x
x gλρU x , λ
2
ρ e
iλ
2
yρ
dρd x d y, B
= Z
Z Z
f
λ
x , yχ yχ x gλρU
x
x , λ
2
ρ e
iλ
2
yρ
dρd x d y. It is not difficult to show that lim
λ →∞
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
w
λ
x , y =
Z
∞
gλρe
iλ
2
yρ
U x , λ
2
ρ dρ
solves the equation G
¯c ¯b,¯a
w
λ
x , y = 0. Hence
G
c ∗
a,b
χ x χ yw
λ
x , y = Qx, y, D
x
, D
y
w
λ
x , y where Qx , y, D
x
, D
y
is a first order differential operator with coefficients vanishing in S
ǫ
= [−ǫ, ǫ] × [−ǫ, ǫ]. Next by similar argument in [2], [5] it is shown that |D
α x
D
β y
w
λ
x , y | ≤ C
N,m
1 + λ
−N
for any x , y ∈ S
ǫ
. Therefore we deduce that 12
sup X
α +β≤m
|D
α x
D
β y
G
c ∗
a,b
v
λ
| ≤ C
n
1 + λ
−N
. 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.
On local properties 285
3. The case Re a
