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volume 4, issue 5, article 85, 2003.

Received 12 May, 2003;

accepted 27 June, 2003.

Communicated by:D. Hinton

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Journal of Inequalities in Pure and Applied Mathematics

BEURLING VECTORS OF QUASIELLIPTIC SYSTEMS OF DIFFERENTIAL OPERATORS

RACHID CHAILI

Dépatement de Mathématiques, U.S.T.O. Oran, Algérie.

EMail:chaili@univ-usto.dz

c

2000Victoria University ISSN (electronic): 1443-5756 061-03

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Beurling Vectors of Quasielliptic Systems of

Differential Operators Rachid Chaili

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J. Ineq. Pure and Appl. Math. 4(5) Art. 85, 2003

http://jipam.vu.edu.au

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

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].

Let(m₁, . . . , m_n) ∈ Zⁿ+, m_j ≥ 1,1 ≤ j ≤ n , we setµ =Qn

j=1m_j, m = max{m_j}, q_j = _m^m

j andq = (q₁, . . . , q_n).Ifα ∈ Zⁿ+ andβ ∈ Zⁿ+ , we denote

|α| = α1 +α2 +· · · +αn, D^α = D₁^α¹ ◦ · · · ◦ D^α_nⁿ, where Dj = ¹_i · _∂x^∂

j, hα, qi=Pn

j=1α_jq_j and α

β

=Qn j=1

_α

j

βj

.

Let(M_p)^+∞_p=0 be a sequence of real positive numbers such that (1.1) M₀ = 1, ∃a >0, 1≤ M_p

Mp−1

≤ M_p+1 Mp

≤a^p, p∈Z^∗+,

(1.2) ∃b >0,∃c > 0, c p

j

Mp−jM_j

≤Mp ≤b^pMp−jMj, p, j ∈Z⁺, j ≤p, (1.3) ∀m≥2, ∃d >0, ∀p, h∈Z+, h≤m;

(M_pm)^m−h(M_pm+m)^h ≤d(M_pm+h)^m,

(1.4) ∀m≥2,∃H >0,∀p, h∈Z+, h≤p; M_pm

M_hm ≤H^p−h M_p

M_h m

.

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Let (P_j(x, D))^N_j=1 be q−quasihomogeneous differential operators of order mwithC^∞coefficients in an open subsetΩofRⁿ, i.e.

Pj(x, D) = X

hα,qi≤m

ajα(x)D^α.

We define the quasiprincipal symbol of the operatorP_j(x, D)by P_jm(x, ξ) = X

hα,qi=m

a_jα(x)ξ^α.

Definition 1.1. The system(P_j)^N_j=1is saidq−quasielliptic inΩif for eachx₀ ∈ Ωwe have

(1.5)

N

X

j=1

|P_jm(x₀, ξ)| 6= 0, ∀ξ∈Rⁿ\{0}.

Definition 1.2. LetM = (M_p)be a sequence satisfying (1.1) – (1.4), the space of Beurling vectors of the system(P_j(x, D))^N_j=1inΩ,denotedB_M

Ω,(P_j)^N_j=1 , is the space ofu∈C^∞(Ω)such that∀Kcompact ofΩ,∀L >0,∃C >0,∀k∈ Z+,

(1.6) kP_i₁. . . , P_i_kuk_L2(K) ≤CL^kmM_km, where1≤i_l≤N, l ≤k.

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Definition 1.3. Letl = (l₁, . . . , l_n)∈Rⁿ+andM be a sequence satisfying (1.1) – (1.4), we call anisotropic Beurling space inΩ,denotedB_M^l (Ω), the space of u∈C^∞(Ω)such that∀K compact ofΩ,∀L >0,∃C > 0,∀α∈Z₊ⁿ,

(1.7) kD^αuk

L2(K) ≤CL<α,l>

n

Y

j=1

M_α_jlj

.

Remark 1.1. Ifl_j = 1, j = 1, . . . , n,we obtain, thanks to (1.2) the definition of isotropic Beurling spaceB_M(Ω), (see [4]).

The principal result of this work is the following theorem:

Theorem 1.1. LetM andM⁰be two sequences satisfying (1.1) – (1.4) and

(1.8) lim

p→+∞

p

X

h=0

M_hm⁰ M_hm

M_pm+m M_pm+m⁰ = 0.

Let(P_j)^N_j=1 beq−quasielliptic system withB^q_M(Ω)coefficients, then B_M⁰

Ω,(P_j)^N_j=1

⊂B_M^q 0(Ω).

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2. Preliminary Lemmas

Letωbe an open neighbourhood of the origin, we setK=

k =hα, qi, α∈Zⁿ+

and we define

|u|_k,ω = X

hα,qi=k

kD^αuk_L2(ω), u∈C^∞(ω), k ∈ K.

Ifρ >0we set

B_ρ=







x∈Rⁿ,

n

X

j=1

(x_j)

2 qj

!¹₂

< ρ





 .

The two following lemmas are in [6].

Lemma 2.1. Letu∈C^∞(Ω),r∈ Kandp∈Z+,then

(2.1) |u|_pm+r,ω ≤ X

hα,qi=pm

|D^αu|_r,ω.

Lemma 2.2. Letk =pm+r < pm+jm,wherek, r∈ Kandp, j ∈Z^∗+,then

∃c(j)>0,∀Bρ⊂ω,∀ε∈]0,1[,∀u∈C^∞(ω),

(2.2) |u|_k,B

ρ ≤ε|u|_(p+j)m,B

ρ+c(j)ε⁻^jm−r^r |u|_pm,B

ρ.

If a ∈ C^∞(ω), we denote [a, D^α]u = D^α(au) −aD^αu and if P is a differential operator, we define[P, D^α]u=D^α(P u)−P (D^αu).

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Lemma 2.3. Let B be a bounded subset of Rⁿ anda ∈ B_M^q B

,then∀L >

0,∃C >0,∀u∈C^∞ B

,∀p∈Z^∗+,

(2.3) X

hα,qi=pm

|[a, D^α]u|_0,B ≤C X

k≤pm−1 k∈K

L^pm−k

M_pmµ M_kµ

_µ¹

|u|_k,B.

Proof. LetL >0,asa∈B_M^q B

,there existsC₁ >0such that

|D^αa| ≤C₁L^hα,qi

n

Y

j=1

M_α_jqj

, ∀α∈Zⁿ+,

therefore, with the Leibniz formula, we get (2.4) |[a, D^α]u|_0,B ≤X

β<α

α β

D^βu

0,BC₁L^hα−β,qi

n

Y

j=1

M_α_j_−β_jqj

.

We need the following easy inequality

(2.5)

α β

≤

n

Y

j=1

α_j βj

qjµ!_µ¹

≤

hα, qiµ hβ, qiµ

_µ¹ .

It is easy to check that from condition (1.2) we have

(2.6) c^l−1

l

Y

j=1

M_h_j ≤M^P^l

j=1hj ≤b^(l−1)^P^l^j=1^h^j

l

Y

j=1

M_h_j,

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hence

n

Y

j=1

M_α_j−β_j

qjµ

≤ 1

c

^Pⁿ_j=1qjµ−1

M.

This inequality with (1.2) and(2.5)imply (2.7)

α β

ⁿ Y

j=1

M_α_j−β_j

qj

≤ 1

c

^P^l_j=1qjM_hα,qiµ M_hβ,qiµ

¹_µ

|u|_k,B.

As the number of α ∈ Z^∗+ satisfyinghα, qi = pm and α > β, is limited by C₂^pm−hβ,qi,whereC₂ depends only ofn,then(2.4)and(2.7)give

X

hα,qi=pm

|[a, D^α]u|_0,B ≤ X

k≤pm−1 k∈K

C₁ 1

c

^P^l_j=1qj

(C₂L)^pm−k

M_pmµ M_kµ

_µ¹

|u|_k,B,

from which the desired estimate is obtained.

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3. Local Estimates

Let(Pj)^N_j=1 be aq−quasielliptic system with coefficients inB_M^q B

,whereB 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. Letωbe a small neighbourhood of the origin,ρ >0andδ∈]0,1[, such thatBρ+δ ⊂ ω. Then there existsC >0, not depending onρandδ, such that for anyu∈C^∞(ω),

(3.1) |u|_m,B

ρ ≤C







N

X

j=1

|P_ju|_0,B

ρ+δ + X

k≤m−1 k∈K

δ^−m+k|u|_k,B

ρ+δ





.

Lemma 3.2. Let ω, ρandδ be as in Lemma3.1, then∃C > 0,∀L > 0,∃A >

0,∀p∈Z^∗+,∀u∈C^∞(ω) (3.2) |u|_(p+1)m,B

ρ

≤C

N

X

j=1

|P_ju|_pm,B

ρ+δ +δ^−m|u|_pm,B

ρ+δ + 1

(4e)^m |u|_(p+1)m,B

ρ+δ

+A

p

X

h=0

L^(p+1−h)mM_pm+m

M_hm |u|_hm,B

ρ+δ

! , and

(3.3) |u|_m,B

ρ ≤C

N

X

j=1

|P_ju|_0,B

ρ+δ +δ^−m|u|_0,B

ρ+δ + 1

(4e)^m|u|_m,B

ρ+δ

! .

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Proof. From(2.1)and(3.1)we obtain

(3.4) |u|_(p+1)m,B

ρ ≤C





N

X

j=1

|P_ju|_pm,B

ρ+δ +

N

X

j=1

X

hα,qi=pm

|[P_j, D^α]u|_0,B

ρ+δ

+ X

k≤m−1 k∈K

δ^−m+k|u|_pm+k,B

ρ+δ





,

Following the same idea as in the proof of Lemma 2.2 of [2], we get

(3.5) X

hα,qi=pm

|[P_j, D^α]u|_0,B

ρ+δ

≤C⁰ X

s≤pm+m−1 s∈K

L^pm+m−s

M_(pm+m)µ M_sµ

_µ¹

|u|_s,B

ρ+δ. On the other hand, there exists h ∈ Z⁺ and r ∈ K such that s = hm +r, r < nm−n, (see [6, (1.3)]). Ass ≤pm+m−1,thenh ≤p.From(2.2)we have

(3.6) |u|_s,B

ρ+δ ≤ε|u|_(h+n)m,B

ρ+δ+C2ε⁻^nm−r^r |u|_hm,B

ρ+δ

ifs=hm+r,where0≤h≤p−n+ 1and0≤r < nm−n,and (3.7) |u|_s,B

ρ+δ ≤ε|u|_pm+m,B

ρ+δ +C₂ε⁻^jm−r^r |u|_hm,B

ρ+δ

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ifs=hm+rwhereh=p+ 1−j, 1≤j ≤n−1and0≤r≤jm−1.

Letε⁰ ∈]0,1[and put ε=ε⁰

M_sµ M_(h+n)mµ

_µ¹

L^−nm+rin (3.6) and

ε=ε⁰

M_sµ M_(p+1)mµ

_µ¹

L^−jm+r in (3.7). According to (1.3) we obtain for anyssatisfying(3.6),

L^−s (M_sµ)¹^µ

|u|_s,B

ρ+δ ≤ε⁰ L^−(h+n)m

M_(h+n)mµ¹_µ |u|_(h+n)m,B

ρ+δ

+C₂d⁰ε^0−m L^−hm (M_hmµ)¹^µ

|u|_hm,B

ρ+δ

and for anyssatisfying(3.7), L^−s

(M_sµ)¹^µ

|u|_s,B

ρ+δ ≤ε⁰ L^−(p+1)m

M_(p+1)mµ_µ¹ |u|_(p+1)m,B

ρ+δ

+C₂d⁰⁰ε^0−nm L^−hm (M_hmµ)^µ¹

|u|_hm,B

ρ+δ.

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These inequalities and(3.5)give X

hα,qi=pm

|[P_j, D^α]u|_0,B

ρ+δ ≤C⁰

nε⁰|u|_(p+1)m,B

ρ+δ

+c(ε⁰)

p

X

h=0

L^(p+1−h)m

M_(pm+m)µ M_hmµ

¹_µ

|u|_hm,B

ρ+δ

! . Choosingε⁰ = (2CC⁰N n(4e)^m)⁻¹,then we obtain, with (1.4),

N

X

J=1

X

hα,qi=pm

|[P_j, D^α]u|_0,B

ρ+δ ≤ 1 2C

1

(4e)^m |u|_(p+1)m,B

ρ+δ

+A

p

X

h=0

(HL)^(p+1−h)m M_pm+m

M_hm |u|_hm,B

ρ+δ. It follows from this inequality: ∀L >0, ∃A >0,

(3.8)

N

X

J=1

X

hα,qi=pm

|[P_j, D^α]u|_0,B

ρ+δ ≤ 1 2C

1

(4e)^m |u|_(p+1)m,B

ρ+δ

+A

p

X

h=0

L^(p+1−h)mM_pm+m

M_hm |u|_hm,B

ρ+δ. It remains the estimate of the third term of the right-hand side of (3.4). From (2.2),we have

|u|_pm+k,B

ρ+δ ≤ε|u|_pm+m,B

ρ+δ +C₂ε⁻^m−k^k |u|_pm,B

ρ+δ.

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Settingε=ε⁰δ^m−kand choosingε⁰ = (2C₁C(4e)^m)⁻¹,then we obtain

(3.9) X

k≤m−1 k∈K

δ^−m+k|u|_pm+k,B

ρ+δ

≤ 1 2C

1

(4e)^m |u|_(p+1)m,B

ρ+δ +C₂⁰δ^−m|u|_pm,B

ρ+δ. The estimates(3.4), (3.8)and(3.9)imply(3.2).The estimate(3.3)is obtained from(3.1)and(3.9)withp= 0.

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4. The Main Result

LetR >0,to every sequenceM satisfying (1.1) – (1.4) we define σ_M^p (u) = 1

M_pm sup

R/2≤ρ<R

(R−ρ)^pm|u|_pm,B

ρ. The following lemma is in [2].

Lemma 4.1. Letωbe as in Lemma3.1,R ∈]0,1[such thatB_R⊂ω, M, M⁰two sequences satisfying (1.1) – (1.4) andu∈B_M⁰

ω,(P_j)^N_j=1

, then for anyL >

0,there exists an increasing positive sequence(C_p)^+∞_p=0 such that ∀p, l ∈Z+, (4.1) σ_M^p 0(P_i₀· · ·P_i_lu)≤C_pM_pm+lm⁰

M_pm⁰ L^pm+lm. where the sequence(C_p)is constructed by recurrence,

C_p+1 =C_p N C+A

p

X

h=0

M_hm⁰ M_hm

Mpm+m

M_pm+m⁰

! ,

whereCandAare the constants of Lemma3.2andC₀is the constant satisfying kP_i₀· · ·P_i_luk_L2(BR)≤C₀L^lmM_lm⁰ .

Theorem 4.2. LetM andM⁰be two sequences satisfying (1.1) – (1.4) and

(4.2) lim

p→+∞

p

X

h=0

M_hm⁰ M_hm

M_pm+m M_pm+m⁰ = 0.

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Let(P_j)^N_j=1 beq-quasielliptic system with coefficients inB_M^q (Ω),then BM⁰

Ω,(Pj)^N_j=1

⊂B_M^q 0(Ω).

Proof. We must verify (1.7) near every point x of Ω. By a translation of xat the origin, there exists a neighbourhoodωof the origin for which the precedent lemmas are true. Let L > 0 and let (C_p)^+∞_p=0 be as in Lemma4.1, then from (4.2)there existsp₀ ∈Z+such thatC_p+1 ≤2N CC_p, p≥p₀,hence

C_p ≤C_p₀(2N C)^p−p⁰ ≤C_p₀(2N C)^pm+lm, ∀l∈Z+.

Forp≤p₀,this inequality is true because the sequence(C_p)^+∞_p=0is increasing.

LetR∈]0.1[such thatBR⊂ω,from(4.1)we obtain σ_M^p 0(P_i₀· · ·P_i_lu)≤C_p₀M_pm+lm⁰

M_pm⁰ (2N CL)^pm+lm, ∀p, l∈Z+. In particular forl = 0,

R 2

pm

1

M_pm⁰ |u|_pm,B

R/2 ≤σ_M^p 0(u)≤C_p₀(2N CL)^pm, hence

|u|_pm,B

R/2 ≤C_p₀

4N C R L

pm

M_pm⁰ , which can be rewritten as

(4.3) ∀L >0, ∃C >0, |u|_pm,B

R/2 ≤CL^pmM_pm⁰ .

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The last inequality will allow us to conclude. In fact letk ∈ K,then there exists p∈Z⁺andr∈ K, r < nm−n,such thatk=pm+r.From(2.2),(4.3)and (2.6),we obtain

|u|_k,B

R/2 ≤εC⁰L^(p+n)mM_(p+n)m⁰ +C⁰C⁰⁰ε⁻^nm−r^r L^pmM_pm⁰

≤εC⁰L^(p+n)m1

c M_(p+n)mµ⁰ ¹_µ

+C⁰C⁰⁰ε⁻^nm−r^r L^pm1

c M_pmµ⁰ _µ¹ . Setting

ε= M_(pm+r)µ⁰ M_(p+n)mµ⁰

!¹_µ

L^−nm+r, then from (1.3) we get

(4.4) |u|_k,B

R/2 ≤C₁L^k M_kµ⁰ _µ¹ .

By an imbedding theorem of anisotropic Sobolev spaces (see [5]), from (4.4) and (1.2) we obtain

sup

BR/2

|D^αu(x)| ≤C₂(bL)^hα,qi M_hα,qiµ⁰ _µ¹ .

The last estimate, with(2.6)gives

sup

B_R/2

|D^αu(x)| ≤C₃(bL)^hα,qi b^hα,qinµ

n

Y

j=1

M_α⁰

jqjµ

!_µ¹

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≤C₃(bL)^hα,qi b^hα,qinµ

n

Y

j=1

b^q^j^µ(α^j^q^j^µ) M_α⁰

j

qjµ!_µ¹

≤C₃ b^(1+n+mµ)L^hα,qi

n

Y

j=1

M_α⁰

j

qj

,

from thereu∈B_M^q 0 B_R/2 .

As a corollary we obtain from Theorem1.1, the principal result of [2]. The- orem1.1also gives a result of regularity of solutions of differential equations in Beurling classes.

Corollary 4.3. Under the assumptions of Theorem1.1, the following assertions are equivalent:

i) u∈D⁰(Ω)andP_ju∈B_M^q 0(Ω), ii) u∈B_M^q 0(Ω).

For anisotropic projective Gevrey classesG^{s},q(Ω) =B_M^q (Ω),Mp = (p!)^s, s ≥1,we have the same result.

Corollary 4.4. Let s, s⁰ be such that s⁰ > s ≥ 1 and(P_j)^N_j=1 q−quasielliptic system with coefficients inG^{s},q(Ω),then

G^{s}

Ω,(P_j)^N_j=1

⊂G^{s},q(Ω).

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Corollary 4.5. LetM andM⁰be two sequences satisfying (1.1) – (1.4) and

(4.5) lim

p→+∞

p

X

h=0

M_hm⁰ M_hm

M_pm+m M_pm+m⁰ = 0,

and let(P_j)^N_j=1be an elliptic system with coefficients inB_M (Ω),then B_M⁰

Ω,(P_j)^N_j=1

⊂B_M⁰(Ω).

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References

[1] P. BOLLEY, J. CAMUS AND L. RODINO, Hypoellipticité analytique- Gevrey et itérés d’opérateurs, Ren. Sem. Mat. Univers. Politec. Torino, 45(3) (1989), 1–61.

[2] C. BOUZARANDR. CHAILI, Régularité des vecteurs de Beurling de sys- tèmes 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 Sobolev spaces L^(rp¹^,...,rⁿ⁾ 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.