Journal of Inequalities in Pure and
Applied Mathematics
BEURLING VECTORS OF QUASIELLIPTIC SYSTEMS
OF DIFFERENTIAL OPERATORS
volume 4, issue 5, article 85,
2003.
RACHID CHAILI
Dépatement de Mathématiques,
U.S.T.O. Oran, Algérie.
EMail: chaili@univ-usto.dz
Received 12 May, 2003;
accepted 27 June, 2003.
Communicated by: D. Hinton
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
061-03
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Abstract
We show the iterate property in Beurling classes for quasielliptic systems of
differential operators.
2000 Mathematics Subject Classification: 35H30, 35H10
Key words: Beurling vectors, Quasielliptic systems, Differential operators
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2
Preliminary Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3
Local Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
4
The Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
References
Beurling Vectors of
Quasielliptic Systems of
Differential Operators
Rachid Chaili
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1.
Introduction
The aim of this work is to show the iterate property in Beurling classes for
quasielliptic systems of differential operators. This property is proved for elliptic systems in [2]. A synthesis of results on the iterate problem is given in
[1].
Q
Let (m1 , . . . , mn ) ∈ Zn+ , mj ≥ 1, 1 ≤ j ≤ n , we set µ = nj=1 mj , m =
max{mj }, qj = mmj and q = (q1 , . . . , qn ). If α ∈ Zn+ and β ∈ Zn+ , we denote
|α| = α1 + α2 + · · · + αn , Dα = D1α1 ◦ · · · ◦ Dnαn , where Dj = 1i · ∂x∂ j ,
Q Pn
α
hα, qi = j=1 αj qj and αβ = nj=1 βjj .
Let (Mp )+∞
p=0 be a sequence of real positive numbers such that
(1.1)
M0 = 1, ∃a > 0, 1 ≤
Beurling Vectors of
Quasielliptic Systems of
Differential Operators
Rachid Chaili
Mp
Mp+1
≤
≤ ap , p ∈ Z∗+ ,
Mp−1
Mp
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(1.2) ∃b > 0, ∃c > 0,
c
p
j
Mp−j Mj
≤ Mp ≤ bp Mp−j Mj ,
p, j ∈ Z+ , j ≤ p,
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(1.3) ∀m ≥ 2, ∃d > 0, ∀p, h ∈ Z+ , h ≤ m;
m−h
(Mpm )
Close
h
m
(Mpm+m ) ≤ d (Mpm+h ) ,
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(1.4)
∀m ≥ 2, ∃H > 0, ∀p, h ∈ Z+ , h ≤ p;
Mpm
≤ H p−h
Mhm
Mp
Mh
m
.
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Let (Pj (x, D))N
j=1 be q−quasihomogeneous differential operators of order
∞
m with C coefficients in an open subset Ω of Rn , i.e.
X
Pj (x, D) =
ajα (x) Dα .
hα,qi≤m
We define the quasiprincipal symbol of the operator Pj (x, D) by
X
Pjm (x, ξ) =
ajα (x) ξ α .
hα,qi=m
Definition 1.1. The system (Pj )N
j=1 is said q−quasielliptic in Ω if for each x0 ∈
Ω we have
(1.5)
N
X
Beurling Vectors of
Quasielliptic Systems of
Differential Operators
Rachid Chaili
Title Page
|Pjm (x0 , ξ)| =
6 0,
n
∀ξ ∈ R \{0}.
Contents
j=1
Definition 1.2. Let M = (Mp ) be a sequence satisfying (1.1) – (1.4),
the space
N
of Beurling vectors of the system (Pj (x, D))N
j=1 in Ω, denoted BM Ω, (Pj )j=1 ,
is the space of u ∈ C ∞ (Ω) such that ∀K compact of Ω, ∀L > 0, ∃C > 0, ∀k ∈
Z+ ,
(1.6)
kPi1 . . . , Pik ukL2 (K) ≤ CLkm Mkm ,
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where 1 ≤ il ≤ N, l ≤ k.
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Definition 1.3. Let l = (l1 , . . . , ln ) ∈ Rn+ and M be a sequence satisfying (1.1)
l
– (1.4), we call anisotropic Beurling space in Ω, denoted BM
(Ω), the space of
∞
u ∈ C (Ω) such that ∀K compact of Ω, ∀L > 0, ∃C > 0, ∀α ∈ Z+n ,
(1.7)
kDα uk
L2 (K)
≤ CL<α,l>
n
Y
Mαj
lj
.
j=1
Remark 1.1. If lj = 1, j = 1, . . . , n, we obtain, thanks to (1.2) the definition of
isotropic Beurling space BM (Ω), (see [4]).
The principal result of this work is the following theorem:
Theorem 1.1. Let M and M 0 be two sequences satisfying (1.1) – (1.4) and
(1.8)
p
0
X
Mhm
Mpm+m
lim
= 0.
0
p→+∞
Mhm Mpm+m
h=0
q
Let (Pj )N
j=1 be q−quasielliptic system with BM (Ω) coefficients, then
BM 0
Ω, (Pj )N
j=1
q
⊂ BM
0 (Ω) .
Beurling Vectors of
Quasielliptic Systems of
Differential Operators
Rachid Chaili
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2.
Preliminary Lemmas
Let ω be an open neighbourhood of the origin, we set K = k = hα, qi , α ∈ Zn+
and we define
X
|u|k,ω =
kDα ukL2 (ω) , u ∈ C ∞ (ω) , k ∈ K.
hα,qi=k
If ρ > 0 we set


Bρ = x ∈ Rn ,

n
X
2
! 12
(xj ) qj
<ρ


.

j=1
The two following lemmas are in [6].
Rachid Chaili
Title Page
∞
Lemma 2.1. Let u ∈ C (Ω), r ∈ K and p ∈ Z+ , then
X
(2.1)
|u|pm+r,ω ≤
|Dα u|r,ω .
hα,qi=pm
Lemma 2.2. Let k = pm + r < pm + jm, where k, r ∈ K and p, j ∈ Z∗+ , then
∃c(j) > 0, ∀Bρ ⊂ ω, ∀ε ∈]0, 1[, ∀u ∈ C ∞ (ω) ,
(2.2)
Beurling Vectors of
Quasielliptic Systems of
Differential Operators
r
|u|k,Bρ ≤ ε |u|(p+j)m,Bρ + c(j)ε− jm−r |u|pm,Bρ .
If a ∈ C ∞ (ω) , we denote [a, Dα ] u = Dα (au) − aDα u and if P is a
differential operator, we define [P, Dα ] u = Dα (P u) − P (Dα u) .
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q
Lemma 2.3. Let B be a bounded subset of Rn and a ∈ BM
B , then ∀L >
0, ∃C > 0, ∀u ∈ C ∞ B , ∀p ∈ Z∗+ ,
(2.3)
X
X
α
|[a, D ] u|0,B ≤ C
hα,qi=pm
L
pm−k
k≤pm−1
k∈K
Mpmµ
Mkµ
µ1
|u|k,B .
q
Proof. Let L > 0, as a ∈ BM
B , there exists C1 > 0 such that
α
|D a| ≤ C1 L
hα,qi
n
Y
Mαj
qj
, ∀α ∈
Zn+ ,
Beurling Vectors of
Quasielliptic Systems of
Differential Operators
j=1
Rachid Chaili
therefore, with the Leibniz formula, we get
n
X α Y
q
α
β hα−β,qi
(2.4)
|[a, D ] u|0,B ≤
D u 0,B C1 L
Mαj −βj j .
β
We need the following easy inequality
(2.5)
α
β
≤
µ1
qj µ ! µ1
n Y
hα, qi µ
αj
≤
.
hβ, qi µ
βj
j=1
(2.6)
cl−1
Mhj ≤ MPl
j=1
j=1
hj
≤ b(l−1)
Pl
j=1
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It is easy to check that from condition (1.2) we have
l
Y
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j=1
β<α
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hj
l
Y
j=1
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M hj ,
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hence
n
Y
Mαj −βj
qj µ
j=1
Pnj=1 qj µ−1
1
≤
M.
c
This inequality with (1.2) and (2.5) imply
(2.7)
α
β
Y
n
Mαj −βj
j=1
qj
Plj=1 qj 1
Mhα,qiµ µ
1
≤
|u|k,B .
c
Mhβ,qiµ
Z∗+
As the number of α ∈
satisfying hα, qi = pm and α > β, is limited by
pm−hβ,qi
C2
, where C2 depends only of n, then (2.4) and (2.7) give
X
hα,qi=pm
|[a, Dα ] u|0,B
Plj=1 qj
1
X
1
Mpmµ µ
pm−k
≤
C1
(C2 L)
|u|k,B ,
c
M
kµ
k≤pm−1
k∈K
from which the desired estimate is obtained.
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Quasielliptic Systems of
Differential Operators
Rachid Chaili
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3.
Local Estimates
q
Let (Pj )N
be
a
q−quasielliptic
system
with
coefficients
in
B
B
, where B
M
j=1
is a neighbourhood of the origin. The following lemma is a light modification
of an analogous lemma in [6, Lemma 2.3].
Lemma 3.1. Let ω be a small neighbourhood of the origin, ρ > 0 and δ ∈]0, 1[,
such that B ρ+δ ⊂ ω. Then there exists C > 0, not depending on ρ and δ, such
that for any u ∈ C ∞ (ω) ,


N
X
X

(3.1)
|u|m,Bρ ≤ C 
|Pj u|0,Bρ+δ +
δ −m+k |u|k,Bρ+δ  .
j=1
k≤m−1
k∈K
Rachid Chaili
Lemma 3.2. Let ω, ρ and δ be as in Lemma 3.1, then ∃C > 0, ∀L > 0, ∃A >
0, ∀p ∈ Z∗+ , ∀u ∈ C ∞ (ω)
(3.2)
≤C
|Pj u|pm,Bρ+δ + δ
−m
|u|pm,Bρ+δ
j=1
JJ
J
1
+
|u|
(4e)m (p+1)m,Bρ+δ
p
X
Mpm+m
+A
L(p+1−h)m
|u|hm,Bρ+δ
Mhm
h=0
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and
(3.3)
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|u|(p+1)m,Bρ
N
X
Beurling Vectors of
Quasielliptic Systems of
Differential Operators
|u|m,Bρ ≤ C
N
X
j=1
|Pj u|0,Bρ+δ + δ
−m
|u|0,Bρ+δ
1
|u|
+
(4e)m m,Bρ+δ
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Proof. From (2.1) and (3.1) we obtain

(3.4)
|u|(p+1)m,Bρ
≤C
N
X
|Pj u|pm,Bρ+δ +
j=1
N
X
X
|[Pj , Dα ] u|0,Bρ+δ
j=1 hα,qi=pm

+
X

δ −m+k |u|pm+k,Bρ+δ  ,
k≤m−1
k∈K
Following the same idea as in the proof of Lemma 2.2 of [2], we get
Beurling Vectors of
Quasielliptic Systems of
Differential Operators
Rachid Chaili
(3.5)
X
|[Pj , Dα ] u|0,Bρ+δ
hα,qi=pm
≤ C0
X
Lpm+m−s
s≤pm+m−1
s∈K
M(pm+m)µ
Msµ
Title Page
µ1
|u|s,Bρ+δ .
On the other hand, there exists h ∈ Z+ and r ∈ K such that s = hm + r,
r < nm − n, (see [6, (1.3)]). As s ≤ pm + m − 1, then h ≤ p. From (2.2) we
have
(3.6)
|u|s,Bρ+δ ≤ ε |u|(h+n)m,Bρ+δ + C2 ε
r
− nm−r
|u|hm,Bρ+δ
if s = hm + r, where 0 ≤ h ≤ p − n + 1 and 0 ≤ r < nm − n, and
(3.7)
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r
|u|s,Bρ+δ ≤ ε |u|pm+m,Bρ+δ + C2 ε− jm−r |u|hm,Bρ+δ
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if s = hm + r where h = p + 1 − j, 1 ≤ j ≤ n − 1 and 0 ≤ r ≤ jm − 1.
Let ε0 ∈]0, 1[ and put
ε=ε
0
M(h+n)mµ
and
ε=ε
0
µ1
Msµ
Msµ
M(p+1)mµ
µ1
L−nm+r in (3.6)
L−jm+r in (3.7) .
Beurling Vectors of
Quasielliptic Systems of
Differential Operators
According to (1.3) we obtain for any s satisfying (3.6),
L−s
1
(Msµ ) µ
|u|s,Bρ+δ
Rachid Chaili
L−(h+n)m
≤ε
1 |u|(h+n)m,Bρ+δ
M(h+n)mµ µ
0
Title Page
+ C2 d0 ε0−m
L
−hm
(Mhmµ )
1
µ
|u|hm,Bρ+δ
and for any s satisfying (3.7) ,
L−s
1
(Msµ ) µ
|u|s,Bρ+δ
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L−(p+1)m
≤ε
1 |u|(p+1)m,Bρ+δ
M(p+1)mµ µ
0
00 0−nm
+ C2 d ε
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L−hm
1
(Mhmµ ) µ
|u|hm,Bρ+δ .
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These inequalities and (3.5) give
X
|[Pj , Dα ] u|0,Bρ+δ ≤ C 0 nε0 |u|(p+1)m,Bρ+δ
hα,qi=pm
+c (ε0 )
p
X
L(p+1−h)m
h=0
−1
Choosing ε0 = (2CC 0 N n (4e)m )
N
X
X
|[Pj , Dα ] u|0,Bρ+δ ≤
J=1 hα,qi=pm
M(pm+m)µ
Mhmµ
µ1
!
|u|hm,Bρ+δ
.
, then we obtain, with (1.4),
1
1
|u|
2C (4e)m (p+1)m,Bρ+δ
+A
p
X
(HL)(p+1−h)m
h=0
Mpm+m
|u|hm,Bρ+δ .
Mhm
It follows from this inequality: ∀L > 0, ∃A > 0,
(3.8)
N
X
X
J=1 hα,qi=pm
|[Pj , Dα ] u|0,Bρ+δ ≤
1
1
|u|
2C (4e)m (p+1)m,Bρ+δ
+A
p
X
L(p+1−h)m
h=0
Mpm+m
|u|hm,Bρ+δ .
Mhm
It remains the estimate of the third term of the right-hand side of (3.4) . From
(2.2) , we have
Beurling Vectors of
Quasielliptic Systems of
Differential Operators
Rachid Chaili
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k
|u|pm+k,Bρ+δ ≤ ε |u|pm+m,Bρ+δ + C2 ε− m−k |u|pm,Bρ+δ .
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−1
Setting ε = ε0 δ m−k and choosing ε0 = (2C1 C (4e)m )
(3.9)
X
, then we obtain
δ −m+k |u|pm+k,Bρ+δ
k≤m−1
k∈K
≤
1
1
0 −m
|u|pm,Bρ+δ .
m |u|(p+1)m,Bρ+δ + C2 δ
2C (4e)
The estimates (3.4) , (3.8) and (3.9) imply (3.2) . The estimate (3.3) is obtained
from (3.1) and (3.9) with p = 0.
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Differential Operators
Rachid Chaili
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4.
The Main Result
Let R > 0, to every sequence M satisfying (1.1) – (1.4) we define
p
σM
(u) =
1
Mpm
sup
R/2≤ρ<R
(R − ρ)pm |u|pm,Bρ .
The following lemma is in [2].
Lemma 4.1. Let ω be as in Lemma 3.1, R ∈]0, 1[ such that BR ⊂ ω, M, M 0 two
sequences satisfying (1.1) – (1.4) and u ∈ BM 0 ω, (Pj )N
j=1 , then for any L >
0, there exists an increasing positive sequence (Cp )+∞
p=0 such that ∀p, l ∈ Z+ ,
(4.1)
p
σM
0
Rachid Chaili
0
Mpm+lm
(Pi0 · · · Pil u) ≤ Cp
Lpm+lm .
0
Mpm
Title Page
where the sequence (Cp ) is constructed by recurrence,
Cp+1 = Cp
p
0
X
Mhm
Mpm+m
NC + A
0
Mhm Mpm+m
h=0
Beurling Vectors of
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,
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where C and A are the constants of Lemma 3.2 and C0 is the constant satisfying
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0
kPi0 · · · Pil ukL2 (BR ) ≤ C0 Llm Mlm
.
Close
Theorem 4.2. Let M and M 0 be two sequences satisfying (1.1) – (1.4) and
(4.2)
p
0
X
Mhm
Mpm+m
= 0.
lim
0
p→+∞
Mhm Mpm+m
h=0
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q
Let (Pj )N
j=1 be q-quasielliptic system with coefficients in BM (Ω) , then
q
BM 0 Ω, (Pj )N
j=1 ⊂ BM 0 (Ω) .
Proof. We must verify (1.7) near every point x of Ω. By a translation of x at
the origin, there exists a neighbourhood ω of the origin for which the precedent
lemmas are true. Let L > 0 and let (Cp )+∞
p=0 be as in Lemma 4.1, then from
(4.2) there exists p0 ∈ Z+ such that Cp+1 ≤ 2N CCp , p ≥ p0 , hence
Cp ≤ Cp0 (2N C)p−p0 ≤ Cp0 (2N C)pm+lm , ∀l ∈ Z+ .
For p ≤ p0 , this inequality is true because the sequence (Cp )+∞
p=0 is increasing.
Let R ∈]0.1[ such that B R ⊂ ω, from (4.1) we obtain
p
σM
0 (Pi0 · · · Pil u) ≤ Cp0
0
Mpm+lm
(2N CL)pm+lm , ∀p, l ∈ Z+ .
0
Mpm
In particular for l = 0,
pm
1
R
pm
p
|u|pm,BR/2 ≤ σM
,
0 (u) ≤ Cp0 (2N CL)
0
2
Mpm
hence
|u|pm,BR/2 ≤ Cp0
4N C
L
R
pm
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0
Mpm
,
which can be rewritten as
(4.3)
Beurling Vectors of
Quasielliptic Systems of
Differential Operators
0
∀L > 0, ∃C > 0, |u|pm,BR/2 ≤ CLpm Mpm
.
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The last inequality will allow us to conclude. In fact let k ∈ K, then there exists
p ∈ Z+ and r ∈ K, r < nm − n, such that k = pm + r. From (2.2) , (4.3) and
(2.6) , we obtain
r
0
0
|u|k,BR/2 ≤ εC 0 L(p+n)m M(p+n)m
+ C 0 C 00 ε− nm−r Lpm Mpm
µ1
µ1
r
1
1
0
0
≤ εC 0 L(p+n)m M(p+n)mµ
+ C 0 C 00 ε− nm−r Lpm Mpmµ
.
c
c
Setting
ε=
0
M(pm+r)µ
Beurling Vectors of
Quasielliptic Systems of
Differential Operators
! µ1
0
M(p+n)mµ
L−nm+r ,
Rachid Chaili
then from (1.3) we get
0
|u|k,BR/2 ≤ C1 Lk Mkµ
(4.4)
µ1
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.
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By an imbedding theorem of anisotropic Sobolev spaces (see [5]), from (4.4)
and (1.2) we obtain
0
sup |Dα u (x)| ≤ C2 (bL)hα,qi Mhα,qiµ
µ1
.
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BR/2
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The last estimate, with (2.6) gives
α
hα,qi
sup |D u (x)| ≤ C3 (bL)
BR/2
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b
hα,qinµ
n
Y
j=1
! µ1
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Mα0 j qj µ
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hα,qi
≤ C3 (bL)
bhα,qinµ
≤ C3 b(1+n+mµ) L
hα,qi
n
Y
bqj µ(αj qj µ) Mα0 j
j=1
n Y
Mα0 j
qj
qj µ
! µ1
,
j=1
q
from there u ∈ BM
0 BR/2 .
As a corollary we obtain from Theorem 1.1, the principal result of [2]. Theorem 1.1 also gives a result of regularity of solutions of differential equations in
Beurling classes.
Corollary 4.3. Under the assumptions of Theorem 1.1, the following assertions
are equivalent:
q
i) u ∈ D0 (Ω) and Pj u ∈ BM
0 (Ω) ,
q
ii) u ∈ BM
0 (Ω) .
q
For anisotropic projective Gevrey classes G{s},q (Ω) = BM
(Ω), Mp = (p!)s ,
s ≥ 1, we have the same result.
Corollary 4.4. Let s, s0 be such that s0 > s ≥ 1 and (Pj )N
j=1 q−quasielliptic
{s},q
system with coefficients in G
(Ω) , then
{s},q
G{s} Ω, (Pj )N
(Ω) .
j=1 ⊂ G
Beurling Vectors of
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Differential Operators
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Corollary 4.5. Let M and M 0 be two sequences satisfying (1.1) – (1.4) and
(4.5)
p
0
X
Mhm
Mpm+m
lim
= 0,
0
p→+∞
Mhm Mpm+m
h=0
and let (Pj )N
j=1 be an elliptic system with coefficients in BM (Ω) , then
0
BM 0 Ω, (Pj )N
j=1 ⊂ BM (Ω) .
Beurling Vectors of
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Differential Operators
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References
[1] P. BOLLEY, J. CAMUS AND L. RODINO, Hypoellipticité analytiqueGevrey et itérés d’opérateurs, Ren. Sem. Mat. Univers. Politec. Torino, 45(3)
(1989), 1–61.
[2] C. BOUZAR AND R. CHAILI, Régularité des vecteurs de Beurling de systèmes elliptiques, Maghreb Math. Rev., 9(1-2) (2000), 43–53.
[3] C. BOUZAR AND R. CHAILI, Vecteurs Gevrey d’opérateurs différentiels
qusihomogènes, Bull. Belg. Math. Soc., 9 (2002), 299–310.
[4] J.L. LIONS AND E. MAGENES, Problèmes aux Limites Non Homogènes
et Applications, t. 3, Dunod Paris, 1970.
[5] P.I. LIZORKIN, Nonisotropic Bessel potentials. Imbedding theorems for
(r ,...,r )
Sobolev spaces Lp 1 n with fractional derivatives, Soviet Math. Dokl.,
7(5) (1966), 1222–1226.
[6] L. ZANGHIRATI, Iterati di operatori quasi-ellittici e classi di Gevrey, Boll.
U.M.I., 5(18-B) (1981), 411–428.
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