INTRODUCTION The aim of this work is to show the iterate property in Beurling classes for quasielliptic systems of differential operators

http://jipam.vu.edu.au/

Volume 4, Issue 5, Article 85, 2003

BEURLING VECTORS OF QUASIELLIPTIC SYSTEMS OF DIFFERENTIAL OPERATORS

RACHID CHAILI

DÉPATEMENT DEMATHÉMATIQUES, U.S.T.O. ORAN, ALGÉRIE.

[email protected]

Received 12 May, 2003; accepted 27 June, 2003 Communicated by D. Hinton

ABSTRACT. We show the iterate property in Beurling classes for quasielliptic systems of differ- ential operators.

Key words and phrases: Beurling vectors, Quasielliptic systems, Differential operators.

2000 Mathematics Subject Classification. 35H30, 35H10.

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 set µ = 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 +· · ·+α_n, D^α =D₁^α1 ◦ · · · ◦D_n^αn,whereD_j = 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

M_p−1 ≤ M_p+1

M_p ≤a^p, p∈Z^∗+,

(1.2) ∃b >0,∃c >0, c p

j

Mp−jM_j ≤M_p ≤b^pMp−jM_j, p, j ∈Z+, j ≤p,

(1.3) ∀m ≥2,∃d >0,∀p, h∈Z⁺, h≤m; (Mpm)^m−h(Mpm+m)^h ≤d(Mpm+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

.

ISSN (electronic): 1443-5756 c

2003 Victoria University. All rights reserved.

061-03

Let(P_j(x, D))^N_j=1beq−quasihomogeneous differential operators of ordermwithC^∞coef- ficients in an open subsetΩofRⁿ, i.e.

P_j(x, D) = X

hα,qi≤m

a_jα(x)D^α.

We define the quasiprincipal symbol of the operatorPj(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 = (Mp)be a sequence satisfying (1.1) – (1.4), the space of Beurling vec- tors of the system(P_j(x, D))^N_j=1 inΩ,denotedB_M

Ω,(P_j)^N_j=1

,is the space ofu∈ C^∞(Ω) such that∀Kcompact of Ω,∀L >0,∃C > 0,∀k ∈Z+,

(1.6) kP_i1. . . , P_ikuk_L2(K) ≤CL^kmM_km, where1≤i_l ≤N, l≤k.

Definition 1.3. Let l = (l₁, . . . , l_n) ∈ Rⁿ+ and M be a sequence satisfying (1.1) – (1.4), we call anisotropic Beurling space in Ω, denoted B_M^l (Ω), the space of u ∈ C^∞(Ω) such that

∀K compact of Ω,∀L >0,∃C > 0,∀α∈Z₊ⁿ,

(1.7) kD^αuk

L2(K) ≤CL<α,l>

n

Y

j=1

M_αjlj

.

Remark 1.1. If l_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 1.2. LetM andM⁰ be 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=1beq−quasielliptic system withB_M^q (Ω)coefficients, then

B_M⁰

Ω,(P_j)^N_j=1

⊂B^q_M0(Ω).

2. PRELIMINARY LEMMAS

Letω be 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,where k, r ∈ Kand p, j ∈ Z^∗+,then∃c(j) > 0,

∀B_ρ ⊂ω,∀ε ∈]0,1[,∀u∈C^∞(ω),

(2.2) |u|_k,B

ρ ≤ε|u|_(p+j)m,B

ρ+c(j)ε^−jm−rr |u|_pm,B

ρ.

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

Lemma 2.3. LetB be a bounded subset ofRⁿanda ∈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_αjqj

, ∀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_αj−β_j

qj

.

We need the following easy inequality (2.5)

α β

≤

n

Y

j=1

α_j β_j

qjµ!¹_µ

≤

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

Mhj ≤M^Pl

j=1hj ≤b^{(l−1)Plj=1hj}

l

Y

j=1

Mhj,

hence

n

Y

j=1

Mαj−βj

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_αj−β_j

qj

≤ 1

c

^Pl_j=1qjM_hα,qiµ M_hβ,qiµ

¹_µ

|u|_k,B.

As the number ofα ∈Z^∗+satisfyinghα, qi=pmandα > β,is limited byC₂^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

^Pl_j=1qj

(C₂L)^pm−k

M_pmµ M_kµ

_µ¹

|u|_k,B,

from which the desired estimate is obtained.

3. LOCAL ESTIMATES

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

,where B is a neigh- bourhood 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 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. Letω, ρandδbe as in Lemma 3.1, 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

|Pju|_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)mMpm+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

! .

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

|[Pj, 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 existsh∈Z+andr∈ Ksuch thats=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−rr |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−rr |u|_hm,B

ρ+δ

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

¹_µ

L^−nm+r in (3.6) and

ε =ε⁰

M_sµ M_(p+1)mµ

¹_µ

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

L^−s (Msµ)^µ1

|u|_s,B

ρ+δ ≤ε⁰ L^−(h+n)m M_(h+n)mµµ¹

|u|_(h+n)m,B

ρ+δ +C₂d⁰ε^0−m L^−hm (Mhmµ)^1µ

|u|_hm,B

ρ+δ

and for anyssatisfying(3.7), L^−s

(Msµ)^µ1

|u|_s,B

ρ+δ ≤ε⁰ L^−(p+1)m M_(p+1)mµ¹_µ

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

ρ+δ +C₂d⁰⁰ε^0−nm L^−hm (Mhmµ)^µ1

|u|_hm,B

ρ+δ.

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

_µ¹

|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 Mhm

|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)mMpm+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−kk |u|_pm,B

ρ+δ. 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.

4. THEMAINRESULT

LetR >0,to every sequenceM satisfying (1.1) – (1.4) we define σ^p_M(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. Let ω be as in Lemma 3.1, R ∈]0,1[such that B_R ⊂ ω, M, M⁰ two 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^p_M0(P_i0· · ·P_ilu)≤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

M_pm+m M_pm+m⁰

! ,

whereCandAare the constants of Lemma 3.2 andC₀is the constant satisfying kPi0· · ·Pi_luk_L2(BR) ≤C0L^lmM_lm⁰ .

Theorem 4.2. LetM andM⁰ be 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.

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

BM⁰

Ω,(Pj)^N_j=1

⊂B^q_M0(Ω).

Proof. We must verify (1.7) near every pointx ofΩ.By a translation ofxat the origin, there exists a neighbourhood ω of the origin for which the precedent lemmas are true. Let L > 0 and let(C_p)^+∞_p=0 be as in Lemma 4.1, then from(4.2)there exists p₀ ∈ Z+ such thatC_p+1 ≤ 2N CCp, p≥p0,hence

Cp ≤Cp0(2N C)^p−p0 ≤Cp0(2N C)^pm+lm, ∀l ∈Z⁺. Forp≤p0,this inequality is true because the sequence(Cp)^+∞_p=0 is increasing.

LetR ∈]0.1[such thatB_R ⊂ω,from(4.1)we obtain σ^p_M0(Pi0· · ·Pi_lu)≤Cp0

M_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 .

The last inequality will allow us to conclude. In fact letk ∈ K,then there existsp ∈ Z+and r∈ 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−rr L^pmM_pm⁰

≤εC⁰L^(p+n)m1

c M_(p+n)mµ^{0 1}_µ

+C⁰C⁰⁰ε^−nm−rr L^pm1

c M_pmµ⁰ _µ¹ .

Setting

ε= M_(pm+r)µ⁰ M_(p+n)mµ⁰

!_µ¹

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

!¹_µ

≤C3(bL)^hα,qi b^hα,qinµ

n

Y

j=1

b^{qjµ(αjqjµ)}

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 Theorem 1.2, the principal result of [2]. Theorem 1.2 also gives a result of regularity of solutions of differential equations in Beurling classes.

Corollary 4.3. Under the assumptions of Theorem 1.2, the following assertions are equivalent:

i) u∈D⁰(Ω)andP_ju∈B_M^q 0(Ω), ii) u∈B^q_M0(Ω).

For anisotropic projective Gevrey classesG^{s},q(Ω) =B_M^q (Ω),M_p = (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).

Corollary 4.5. LetM andM⁰ be 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=1 be an elliptic system with coefficients inB_M(Ω),then

B_M⁰

Ω,(P_j)^N_j=1

⊂B_M⁰(Ω).

REFERENCES

[1] P. BOLLEY, J. CAMUSANDL. 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 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 Sobolev spacesL^(rp^1,...,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.