ON LOCAL PROPERTIES OF SOME CLASSES OF INFINITELY DEGENERATE ELLIPTIC DIFFERENTIAL

(1)

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^c_a,b=X₂X₁+i

c(x)|x|⁻⁴+(a(x)−b(x))|x|⁻³ e⁻^|¹^x^| ∂

∂y, where(x,y)∈R², i=√

−1, the functions a(x),b(x),c(x)satisfy a(x) =

a₊∈C if x>0 a₋∈C if x<0

=: a, b(x) =

b₊∈C if x>0 b₋∈C if x<0

=: b, c(x) =

c₊∈C if x>0 c₋∈C if x<0

=: c and

X₁ = ∂

∂x −i b(x)sign(x)|x|⁻²e⁻^|¹^x^| ∂

∂y, X₂ = ∂

∂x −i a(x)sign(x)|x|⁻²e⁻^|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₋ =

−eîϕ, b₊ = b₋ =eîϕ, c₊ =(1−λ₊)eîϕ, c₋ = (1−λ₋)eîϕ, whereϕ ∈ [0, π/2), λ₊, λ₋ ∈ C. For those cases in [1], [2], [4] and [5], we have given another proof

277

(2)

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,bis 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₋>0

THEOREM1. Assume that Re a₊<0, Re a₋<0, Re b₊>0, Re b₋>0. Then G^c_a,bis 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

ˆ

u(x, η)= Z _+∞

−∞

e⁻^iyηu(x,y)d y.

By this transform G^c_a,bbecome ˆ

G^c_a,b

x, d d x, η

= d

d x +ax⁻²sign(x)e⁻^|x|¹ η d

d x +bx⁻²sign(x)e⁻^|x|¹ η

− (c|x|⁻⁴+(a−b)|x|⁻³)e⁻^|^x¹^|η=

= d²

d x²+(a+b)x⁻²sign(x)e⁻^|x|¹ η d d x

+ abx⁻⁴e⁻^|x|² η²+(b−c)|x|⁻⁴e⁻^|x|¹ η−(a+b)|x|⁻³e⁻^|x|¹ η.

We would like to study solutions of the equation

(2) Gˆ^c_a,b

x, d

d x, η

ˆ

u(x, η)=0.

(3)

Putuˆ(x, η) = x e⁻^be⁻

1

|x|ηf(z), where z = (b−a)e⁻^|¹^x^|η. Then we will have the following confluent hypergeometric equation for f(z)

zd²f(z)

d z² +(1−z)d f(z) d z − c

b−a f(z)=0.

The equation has two linearly independent solutions f₁(z)=9

c b−a,1,z

, f₂(z)=e^z9

1− c

b−a,1,−z

where9(α, 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 : uˆ⁺₁(x, η):=x e⁻^b⁺^ηe⁻

1

|x|

9 c₊

b₊−a₊,1, (b₊−a₊)ηe⁻^|¹^x^|

,

u⁺₂(x, η):=x e⁻^a⁺^ηe⁻

|x|1

9

1− c₊

b₊−a₊,1, (a₊−b₊)ηe⁻^|x|¹

, (3)

if x≤0 : uˆ⁻₁(x, η):=x e⁻^b⁻^ηe⁻

1

|x|

9 c₋

b₋−a₋,1, (b₋−a₋)ηe⁻^|x|¹

,

ˆ

u⁻₂(x, η):=x e⁻^a⁻^ηe⁻

1

|x|

9

1− c₋

b₋−a₋,1, (a₋−b₋)ηe⁻^|^x¹^|

. As x →0 we have the following asymptotics foruˆ^±₁(x, η),uˆ^±₂(x, η)

ˆ

u⁺₁(x, η)≈ 1−x(log[(b₊−a₊)η]+8(c₊/(b₊−a₊))−28(1))

0 (c₊/(b₊−a₊)) ,

ˆ

u⁺₂(x, η)≈ 1−x(log[(a₊−b₊)η]+8(1−c₊/(b₊−a₊))−28(1))

0 (1−c₊/(b₊−a₊)) ,

(4) ˆ

u⁻₁(x, η)≈ −1−x(log[(b₋−a₋)η]+8(c₋/(b₋−a₋))−28(1))

0 (c₋/(b₋−a₋)) ,

ˆ

u⁻₂(x, η)≈ −1−x(log[(a₋−b₋)η]+8(1−c₋/(b₋−a₋))−28(1))

0 (1−c₋/(b₋−a₋)) ,

where8(z)is the Gauss polygramma function(8(z)=0(z)/ 0⁰(z)). Define D⁺(x, η) (respectively D⁻(x, η)) as the Wronskian of uˆ⁺₁(x, η), uˆ⁺₂(x, η) (resp. uˆ⁻₁(x, η),

ˆ

u⁻₂(x, η)). Similarly (as in [4], Proposition 2) it is not difficult to see that

D⁺(0, η) =







−

ctg _b^πc⁺

+−a₊ −i

sin_b^πc⁺

+−a₊ if arg(b₊−a₊)η∈(0, π]

−

ctg _b^πc⁺

+−a₊ +i

sin_b^πc⁺

+−a₊ if arg(b₊−a₊)η∈(−π,0]

D⁻(0, η) =







−

ctg _b^πc⁻

−−a₋ −i

sin_b^πc⁻

−−a₋ if arg(b₋−a₋)η∈(0, π]

−

ctg _b^πc⁻

−−a₋ +i

sin_b^πc⁻

−−a₋ if arg(b₋−a₋)η∈(−π,0].

(4)

Note that D⁺(0, η)· D⁻(0, η) 6= 0 when η 6= 0. Thereforuˆ⁺₁(x, η),uˆ⁺₂(x, η)are linearly independent solutions of (2) when x ≥0, η6=0, anduˆ⁻₁(x, η),uˆ⁻₂(x, η)are linearly independent solutions of (2) when x ≤ 0, η6= 0. Next forη > 0 we define the following pair of solutions of (2)

(5) u⁺(x, η)=

( uˆ⁺₁(x, η) if x≥0

c_1,^u⁺₋(η)uˆ⁻₁(x, η)+c_2,^u⁺₋(η)uˆ⁻₂(x, η) if x≤0

(6) v⁺(x, η)=

(

c^v_1,⁺₊(η)uˆ⁺₁(x, η)+c^v_2,⁺₊(η)uˆ⁺₂(x, η) if x≥0 ˆ

u⁻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−_b₋^c₋⁻_a₋

0 _c

+

b₊−a₊

0

1−_b₋^c₋⁻_a₋

D⁻(0, η)

+ −8 _c

b₊−+a₊

+48(1) 0

c₊ b₊−a₊

0

1−b₋^c−⁻a₋

D⁻(0, η) ,

c^u_2,⁺₋(η) = log

(b₊−a₊)η +log

(b₋−a₋)η

+8 _c

+

b₊−a₊

0 _c

−

b₋−a₋

0 _c

+

b₊−a₊

D⁻(0, η)

+

8 _c

−

b₋−a₋

−48(1)

0 _c

−

b₋−a₋

0 _c

+

b₊−a₊

D⁻(0, η) ,

c^v_1,⁺₊(η) = log

(a₊−b₊)η +log

(b₋−a₋)η +8

1−_b₊^c₋⁺_a₊

0 _c

−

b₋−a₋

0

1−_b₊^c₋⁺_a₊

D⁺(0, η)

+

8 _c

b₋−−a₋

−48(1) 0

c₋ b₋−a₋

0

1−b₊^c−⁺a₊

D⁺(0, η) ,

c^v_2,⁺₊(η) = −log

(b₊−a₊)η

−log

(b₋−a₋)η

−8 _c

+

b₊−a₊

0 _c

−

b₋−a₋

0 _c

+

b₊−a₊

D⁺(0, η)

+ −8 _c

−

b₋−a₋

+48(1)

0 _c

−

b₋−a₋

0 _c

+

b₊−a₊

D⁺(0, η) .

(5)

Forη <0 we define the following pair of solutions of (2)

(7) u⁻(x, η)=

( uˆ⁺₂(x, η) if x≥0

c_1,^u⁻₋(η)uˆ⁻₁(x, η)+c_2,^u⁻₋(η)uˆ⁻₂(x, η) if x≤0

(8) v⁻(x, η)=

(

c^v_1,⁻₊(η)uˆ⁺₁(x, η)+c^v_2,⁻₊(η)uˆ⁺₂(x, η) if x≥0 ˆ

u⁻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−_b₋^c₋⁻_a₋ 0

1−b₊^c−⁺a₊

0

1−b₋^c−⁻a₋

D⁻(0, η)

+ −8

1−b₊^c−⁺a₊

+48(1) 0

1−_b₊^c₋⁺_a₊ 0

1−_b₋^c₋⁻_a₋

D⁻(0, η) ,

c^u_2,⁻₋(η) = log

(b₋−a₋)η +8

1−_b₊^c₋⁺_a₊

0 _c

b₋−−a₋

0

1−b₊^c−⁺a₊

D⁻(0, η)

+

8 _c

b₋−−a₋

−48(1)

0 _c

−

b₋−a₋

0

1−_b₊^c₋⁺_a₊

D⁻(0, η) ,

c^v_1,⁻₊(η) = log

(a₋−b₋)η +8

1−_b₊^c₋⁺_a₊ 0

1−_b₋^c₋⁻_a₋ 0

1−_b₊^c₋⁺_a₊

D⁺(0, η)

+

8

1−b₋^c−⁻a₋

−48(1) 0

1−b₋^c−⁻a₋

0

1−b₊^c−⁺a₊

D⁺(0, η) ,

c^v_2,⁻₊(η) = −log

(b₊−a₊)η

−log

(a₋−b₋)η

−8 _c

+

b₊−a₊

0

1−_b₋^c₋⁻_a₋

0 _c

+

b₊−a₊

D⁺(0, η)

+ −8

1−_b₋^c₋⁻_a₋

+48(1) 0

1−b₋^c−⁻a₋

0 _c

+

b₊−a₊

D⁺(0, η) .

(6)

Put W^±(0, η) = u^±(0, η)v^±_x(0, η)−v^±(0, η)u^±_x(0, η). From (5), (6), (7), (8) we deduce that

W⁺(0, η)=c_2,^v⁺₊(η)D⁺(0, η), W⁻(0, η)=c_1,^u⁻₋(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₊

a₊−b₊ = −k−1, _a^c⁻

−−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 _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

W^±(x, η)=W^±(0, η)exp

− Z x

0

p(s, η)ds

, where

p(s, η)=

(a₊+b₊)x⁻²si gn(x)e⁻^1/^|^x^|ηif x≥0, (a₋+b₋)x⁻²si gn(x)e⁻^1/^|^x^|ηif x≤0.

Hence

W⁺(x, η)=











W⁺(0, η)e⁻^η(a⁺⁺^b⁺^)e^−1/|x|=c_2,^v⁺₊(η)D⁺(0, η)e⁻^η(a⁺⁺^b⁺^)e^−1/|x|

if x≥0, W⁺(0, η)e⁻^η(a⁻⁺^b⁻^)e^−1/|x|=c_2,^v⁺₊(η)D⁺(0, η)e⁻^η(a⁻⁺^b⁻^)e^−1/|x|

if x≤0,

W⁻(x, η)=











W⁻(0, η)e⁻^η(a⁺⁺^b⁺^)e⁻^1/^|^x^|=c_1,^u⁻₋(η)D⁻(0, η)e⁻^η(a⁺⁺^b⁺^)e⁻^1/^|^x^| if x≥0, W⁻(0, η)e⁻^η(a⁻⁺^b⁻^)e⁻^1/^|^x^|=c_1,^u⁻₋(η)D⁻(0, η)e⁻^η(a⁻⁺^b⁻^)e⁻^1/^|^x^| 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 or_a ^c⁺

+−b₊ 6= −k−1, _a ^c⁻

−−b₋ 6= −l−1 where k and l are non-negative integers, then u⁺(−1, η), v⁺(1, η)exponentially increase whenη→ +∞, and u⁻(−1, η), v⁻(1, η) exponentially increase whenη → −∞. Hence we construct the Green function as follows

G(x,x⁰, η)=

G⁺(x,x⁰, η) if η≥C, G⁻(x,x⁰, η) if η≤ −C, where

for η≥C : G⁺(x,x⁰, η)=

( _v+(x,η)u⁺(x⁰,η)

W⁺(x⁰,η) if x≤x⁰,

v⁺(x⁰,η)u⁺(x,η)

W⁺(x⁰,η) if x⁰≤x, for η≤ −C : G⁻(x,x⁰, η)=

( _v−(x,η)u⁻(x⁰,η)

W⁻(x⁰,η) if x≤x⁰,

v⁻(x⁰,η)u⁻(x,η)

W⁻(x⁰,η) if x⁰≤x.

(7)

By noting that |c^v_2,⁺₊(η)/W⁺(0, η)| < C and using the asymptotic behaviors of 9(α, γ ,z)at zero and at infinity we can show that (in the similar way as in [4])

Z 1

−1|G⁺(x,x⁰, η)|d x≤C; Z 1

−1|G⁺(x,x⁰, η)|d x⁰≤C, Z 1

−1|G⁻(x,x⁰, η)|d x≤C; Z 1

−1|G⁻(x,x⁰, η)|d x⁰≤C, Z 1

−1|∂G⁺(x,x⁰, η)

∂x |d x≤C; Z 1

−1|∂G⁺(x,x⁰, η)

∂x |d x⁰≤C, Z 1

−1|∂G⁻(x,x⁰, η)

∂x |d x≤C; Z 1

−1|∂G⁻(x,x⁰, η)

∂x |d x⁰≤C.

More general, for an arbitrary natural number n we will have Z 1

−1|D_ηⁿG⁺(x,x⁰, η)|d x≤Cη⁻ⁿ; Z 1

−1|Dⁿ_ηG⁺(x,x⁰, η)|d x⁰≤Cη⁻ⁿ, Z 1

−1|D_ηⁿG⁻(x,x⁰, η)|d x≤C|η|⁻ⁿ; Z 1

−1|D_ηⁿG⁻(x,x⁰, η)|d x⁰≤C|η|⁻ⁿ. Finally if we define the operator Q

Qu(x,y)= Z 1

−1

Z 1

−1

Z _∞

−∞

e^iη(y⁻^y⁰⁾φ(η)G(x,x⁰, η)u(x⁰,y⁰)d y⁰d x⁰dη,

whereφ(η) is a cut-off functionφ(η) ∈ C^∞(R), φ(η) = 0 if|η| ≤ C, φ(η) = 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 solvable 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 H¨ormander if G^c_a,bis local solvable at the origin for some its neighborhoodω, there exist constants C,m such that

(9) |

Z

fvd x d y| ≤C sup X

α+β≤m

|D^α_xD^β_y f| X

α+β≤m

|D^α_xD^β_yG^c_a,b^∗ v|

for all f, v∈C₀^∞(ω). Functions which violate this inequality will be constructed. For largeλlet f_λ=F(λ²x, λ²y)λ⁵, where function F(x,y)∈C₀^∞(R)and

(10)

Z _∞

−∞

Z _∞

−∞

F(x,y)d x d y=1.

(8)

Next it is easy to see that the function U(x, η)=

(

x e^{− ¯}^b⁺^ηe^−1/|x|L⁰_k (¯b₊− ¯a₊)ηe⁻^1/^|^x^|

if x≥0, x e^{− ¯}^b⁻^ηe^−1/|x|L⁰_l (¯b₋− ¯a₋)ηe⁻^1/^|^x^|

if x≤0,

where L⁰_k(z)= _n!¹e^zD_zⁿ(e⁻^zzⁿ)are the Laguerre polynomials, solve the equation (2).

Put

vλ=χ (x)χ (y) Z _∞

0

g(λρ)e^iλ²^yρU(x, λ²ρ)dρ, where g∈ C₀^∞(0,∞), χ ∈C₀^∞(−∞,∞),R

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 =∂_xf_λandv =v_λwith the parameterλlarge positive enough.

Indeed, forλlarge enough, f_λ∈C₀^∞(ω)and

(11) −

Z Z ∂f_λ

∂xv_λ(x,y)d x d y=

Z Z ∂vλ

∂x f_λ(x,y)d x d y=A+B, where

A= Z Z Z

f_λ(x,y)χ (y)χ_x(x)g(λρ)U(x, λ²ρ)e^iλ²^yρdρd x d y,

B = Z Z Z

f_λ(x,y)χ (y)χ (x)g(λρ)U_x(x, λ²ρ)e^iλ²^yρdρd 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 _∞

0

g(λρ)e^iλ²^yρU(x, λ²ρ)dρ solves the equation G^c^¯_¯

b,a¯wλ(x,y)=0. Hence

G^c_a,b^∗χ (x)χ (y)w_λ(x,y)=Q(x,y,D_x,D_y)w_λ(x,y)

where Q(x,y,Dx,Dy)is a first order differential operator with coefficients vanishing in S =[−, ]×[−, ]. Next by similar argument in [2], [5] it is shown that

|D^α_xD^β_yw_λ(x,y)| ≤C_N,m(1+λ)⁻^N for any(x,y) /∈S. Therefore we deduce that

(12) sup X

α+β≤m

|D^α_xD^β_yG^c_a,b^∗ v_λ| ≤C_n(1+λ)⁻^N.

Finally we see that (10), (11), (12) contradict (9).

REMARK1. 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.

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3. The case Re a₊>0, Re a₋>0, Re b₊>0, Re b₋>0

THEOREM2. Assume that Re a₊>0, Re a₋>0, Re b₊>0, Re b₋>0. Then G^c_a,bis 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.

Forη > 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 solutionsuˆ⁻₁(x, η),uˆ⁻₂(x, η)exponentially decrease whenη→ +∞.

A) If _b^c⁺

+−a₊ ∈/ Z₋∪0 then we set f = f_λ, v = vλ as in section 2, with U(x, η) replaced by V(x, η). B) If _b^c⁺

+−a₊ ∈Z₋∪0 then we set f =∂_xf_λ, v=v_λas in section 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, vwith large enoughλ.

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)

when x≥0 : u˜⁺₁(x, η):=x e⁻^a⁺^ηe⁻

|x|1 −2(c₊ηe⁻

|x|1 )¹29

1

2,1,4 c₊ηe⁻^|x|¹ ¹₂ ,

˜

u⁺₂(x, η):=x e⁻^a⁺^ηe⁻

1

|x|+2(c₊ηe⁻

1

|x|)¹2

9 1

2,1,−4 c₊ηe⁻^|¹^x^|¹₂ , when x≤0 : u˜⁻₁(x, η):=x e⁻^a⁻^ηe⁻

1

|x|−2(c₋ηe⁻

1

|x|)¹²9

1

2,1,4 c₋ηe⁻^|¹^x^|¹₂ ,

˜

u⁻₂(x, η):=x e⁻^a⁻^ηe⁻

1

|x|+2(c₋ηe⁻

1

|x|)¹²9

1

2,1,−4 c₋ηe⁻^|¹^x^|¹₂ . Next forη >0 let us define the following solution of (2)

˜

V(x, η)=

(u˜⁺₁(x, η) if x≥0,

c_1,^V^˜₋(η)u˜⁻₁(x, η)+c_2,^V^˜₋(η)u˜⁻₂(x, η) if x≤0,

where c_1,^V^˜₋(η),c^V_2,^˜₋(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 solutionsu˜⁺₁(x, η),u˜⁺₂(x, η)when x ≥0, and e⁻^a⁻^ηe

1

|x|

,x e⁻^a⁻^ηe

1

|x| when x ≤0.

C) When c₊ =0,c₋ 6=0 then we have solutions e⁻^a⁺^ηe

1

|x|

,x e⁻^a⁺^ηe

1

|x|

when x ≥0, andu˜⁻₁(x, η),u˜⁻₂(x, η)when x ≤0.

D) When c₊ =0,c₋ =0 then we have solutions e⁻^a⁺^ηe

1

|x|

,x e⁻^a⁺^ηe

1

|x|

when x ≥0, and e⁻^a⁻^ηe

1

|x|,x e⁻^a⁻^ηe

1

|x| 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.

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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.

REMARK2. 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

THEOREM 3. Assume that Re a₊ >0,Re a₋ < 0,Re b₊ > 0,Re b₋ <0. Then G^c_a,bis 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 forη >0 we define the following pair of solutions of (2)

(13) U⁺(x, η)=

(uˆ⁺₁(x, η) if x≥0,

c^U_1,₋⁺(η)uˆ⁻₁(x, η)+c^U_2,⁺₋(η)uˆ⁻₂(x, η) if x≤0,

(14) V⁺(x, η)=

(uˆ⁺₂(x, η) if x≥0,

c^V_1,⁺₋(η)uˆ⁻₁(x, η)+c_2,^V⁺₋(η)uˆ⁻₂(x, η) if x ≤0,

where the coefficients c^U_1,⁺₋,c^U_2,⁺₋,c_1,^V⁺₋,c_2,^V⁺₋ are chosen as in section 2 such that U⁺(x, η), V⁺(x, η)are continuously differentiable at x =0.

Forη <0 we define the following pair of solutions of (2)

(15) U⁻(x, η)=

(c^U_1,₊⁻(η)uˆ⁺₁(x, η)+c^U_2,⁻₊(η)uˆ⁺₂(x, η) if x≥0, ˆ

u⁻₁(x, η) if x≤0,

(16) V⁻(x, η)=

(c^V_1,⁻₊(η)uˆ⁺₁(x, η)+c_2,^V⁻₊(η)uˆ⁺₂(x, η) if x ≥0, ˆ

u⁻₂(x, η) if x≤0,

where the coefficients c^U_1,⁻₊,c^U_2,₊⁻,c_1,^V⁻₊,c_2,^V⁻₊are chosen such that U⁻(x, η),

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 whenη→ −∞. Therefore we construct the Green function as follows

G(x,x⁰, η)=

(G⁺(x,x⁰, η) if η≥C, G⁻(x,x⁰, η) if η≤ −C,