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) Gca,b=X2X1+i
c(x)|x|−4+(a(x)−b(x))|x|−3e−|1x| ∂ ∂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−|1x| ∂ ∂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
Gca,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, whereϕ ∈ [0, π/2), λ+, λ− ∈ C. For those cases in [1], [2], [4] and [5], we have given another proof
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 Gca,bis not hypoelliptic. Here we shall use the method of constructing parametrixes. Since Gca,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
Gca,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 Gca,bbecome
ˆ
Gca,b
x, d
d x, η
=
d
d x +ax
−2sign(x)e−|1x|η d
d x +bx
−2sign(x)e−|1x|η
− (c|x|−4+(a−b)|x|−3)e−|1x|η=
= d
2
d x2+(a+b)x−
2sign(x)e−|1x|η d
d x
+ abx−4e−|2x|η2+(b−c)|x|−4e−|1x|η−(a+b)|x|−3e−|1x|η. We would like to study solutions of the equation
(2) Gˆca,b
x, d
d x, η
ˆ
Putuˆ(x, η) = x e−be−
1
|x|ηf(z), where z = (b−a)e−|1x|η. Then we will have the following confluent hypergeometric equation for f(z)
zd
The equation has two linearly independent solutions
f1(z)=9
where9(α, γ ,z)is the Tricomi function (see [10], p. 255). Therefore we have the following pair of solutions of the equation (2)
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)= continously differentiable at x=0
Forη <0 we define the following pair of solutions of (2) are continously differentiable at x=0
By noting that |cv2,++(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,x′, η)|d x≤C; Z 1
−1|
G+(x,x′, η)|d x′≤C,
Z 1
−1|
G−(x,x′, η)|d x≤C; Z 1
−1|
G−(x,x′, η)|d x′≤C,
Z 1
−1|
∂G+(x,x′, η)
∂x |d x≤C; Z 1
−1|
∂G+(x,x′, η) ∂x |d x
′≤C,
Z 1
−1|
∂G−(x,x′, η)
∂x |d x≤C; Z 1
−1|
∂G−(x,x′, η) ∂x |d x
′≤C.
More general, for an arbitrary natural number n we will have
Z 1
−1|
DηnG+(x,x′, η)|d x≤Cη−n;
Z 1
−1|
DnηG+(x,x′, η)|d x′≤Cη−n,
Z 1
−1|
DηnG−(x,x′, η)|d x≤C|η|−n;
Z 1
−1|
DηnG−(x,x′, η)|d x′≤C|η|−n.
Finally if we define the operator Q
Qu(x,y)=
Z 1
−1 Z 1
−1
Z ∞
−∞
eiη(y−y′)φ(η)G(x,x′, η)u(x′,y′)d y′d x′dη,
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 Gca,b, and Q∗will serve a left parametrix for Gca∗,b= −Gc¯¯
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 Gca,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
|DαxDβy f| X α+β≤m
|DαxDβyGca∗,bv|
for all f, v∈C0∞(ω). Functions which violate this inequality will be constructed. For largeλlet fλ=F(λ2x, λ2y)λ5, where function F(x,y)∈C0∞(R)and
(10)
Z ∞
−∞
Z ∞
−∞
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 L0k(z)= n!1ezDzn(e−zzn)are the Laguerre polynomials, solve the equation (2). Put
vλ=χ (x)χ (y)
Z ∞
0
g(λρ)eiλ2yρU(x, λ2ρ)dρ, where g∈ C0∞(0,∞), χ ∈C0∞(−∞,∞),R
g(ρ)dρ=1, χ (x)=1, when|x| ≤ǫ, with a fixed small enough positive numberǫ. Now we will show that the inequality (9) does not hold for f =∂xfλandv =vλwith the parameterλlarge positive enough.
Indeed, forλlarge enough, fλ∈C0∞(ω)and
(11) −
Z Z ∂f λ
∂xvλ(x,y)d x d y=
Z Z ∂v
λ
∂x fλ(x,y)d x d y=A+B,
where
A=
Z Z Z
fλ(x,y)χ (y)χx(x)g(λρ)U(x, λ2ρ)eiλ
2yρ
dρd x d y,
B =
Z Z Z
fλ(x,y)χ (y)χ (x)g(λρ)Ux(x, λ2ρ)eiλ
2yρ
dρd x d y.
It is not difficult to show that limλ→∞B =1 and for every positive number N , there exists a number CN such that|A| ≤ CN(1+λ)−N. Next it is easy to check that the function
wλ(x,y)=
Z ∞
0
g(λρ)eiλ2yρU(x, λ2ρ)dρ solves the equation Gc¯¯
b,a¯wλ(x,y)=0. Hence
Gca∗,bχ (x)χ (y)wλ(x,y)=Q(x,y,Dx,Dy)wλ(x,y)
where Q(x,y,Dx,Dy)is a first order differential operator with coefficients vanishing in Sǫ =[−ǫ, ǫ]×[−ǫ, ǫ]. Next by similar argument in [2], [5] it is shown that
|DαxDβywλ(x,y)| ≤CN,m(1+λ)−N for any(x,y) /∈Sǫ. Therefore we deduce that
(12) sup X
α+β≤m
|DαxDβyGca∗,bvλ| ≤Cn(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,
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
Gca,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.
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−
Next forη >0 let us define the following solution of (2)
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
Gca,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)=
(
ˆ
u+1(x, η) if x≥0,
c1,U−+(η)uˆ−1(x, η)+c2,U+−(η)uˆ−2(x, η) if x≤0,
(14) V+(x, η)=
(
ˆ
u+2(x, η) if x≥0,
c1,V+−(η)uˆ−1(x, η)+c2,V+−(η)uˆ−2(x, η) if x ≤0,
where the coefficients cU1,+−,cU2,+−,c1,V+−,c2,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)=
(
c1,U+−(η)uˆ+1(x, η)+c2,U−+(η)uˆ+2(x, η) if x≥0,
ˆ
u−1(x, η) if x≤0,
(16) V−(x, η)=
(
c1,V−+(η)uˆ+1(x, η)+c2,V−+(η)uˆ+2(x, η) if x ≥0,
ˆ
u−2(x, η) if x≤0,
where the coefficients c1,U−+,cU2,−+,c1,V−+,c2,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,x′, η)=
(
G+(x,x′, η) if η≥C,
where, forη≥C:
G+(x,x′, η)=
V+(x, η)U+(x′, η)−V+(x′, η)U+(x, η)
D+(x′, η) if x≤x ′,
0 if x′≤x, and forη≤ −C:
G−(x,x′, η)=
V−(x, η)U−(x′, η)−V−(x′, η)U−(x, η)
D−(x′, η) if x≤x ′,
0 if x′≤x.
Now the rest of the proof goes through as in section 2. We leave the details to the readers.
REMARK3. 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
THEOREM 4. Assume that Re a+ <0,Re a− > 0,Re b+ > 0,Re b− >0. Then
Gca,bis not hypoelliptic nor local solvable at the origin.
Proof. We argue only for the non-resonance case a+ 6= b+,a− 6= b−. The half-resonance case can be treated analogously. Put
˜
U(x, η)=
(
˜
u+1(x, η) if x≥0,
cU1,˜−(η)u˜−1(x, η)+c2,U˜−(η)u˜−2(x, η) if x≤0,
where cU1,˜−(η),c2,U˜−(η) are chosen as in section 2. Since Re b+ > 0,Re a− > 0,Re b− > 0 the solutionU˜(x, η)exponentially decreases whenη → +∞. Now the rest of the proof goes as in section 2.
REMARK 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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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
