Necessity of parameter-ellipticity for multi-order systems of differential equations

Universität Konstanz

Necessity of parameter-ellipticity for multi-order systems of differential equations

Robert Denk Melvin Faierman

Konstanzer Schriften in Mathematik Nr. 278, Februar 2011

ISSN 1430-3558

© Fachbereich Mathematik und Statistik Universität Konstanz

Fach D 197, 78457 Konstanz, Germany

Konstanzer Online-Publikations-System (KOPS) URN: http://nbn-resolving.de/urn:nbn:de:bsz:352-opus-130872

URL: http://kops.ub.uni-konstanz.de/volltexte/2011/13087/

MULTI-ORDER SYSTEMS OF DIFFERENTIAL EQUATIONS

R. DENK AND M. FAIERMAN

Abstract. In this paper we investigate parameter-ellipticity conditions for multi-order systems of differential equations on a bounded domain. Under suitable assumptions on smoothness and on the order structure of the system, it is shown that parameter-dependent a priori-estimates imply the conditions of parameter-ellipticity, i.e., interior ellipticity, conditions of Shapiro-Lopatinskii type, and conditions of Vishik-Lyusternik type. The mixed-order systems con- sidered here are of general form; in particular, it is not assumed that the diag- onal operators are of the same order. This paper is a continuation of an article by the same authors where the sufficiency was shown, i.e., a priori-estimates for the solutions of parameter-elliptic multi-order systems were established.

1. Introduction and main results

In this paper, we will study multi-order boundary value problems defined over a bounded domain in Rⁿ. Under rather general assumptions on the structure of the system, it was shown in the paper [DF] that parameter-ellipticity implies uniform a priori-estimates for the solutions. Now we will show that the conditions of parameter-ellipticity are also necessary.

Parameter-elliptic boundary value problems and a priori estimates for them were treated, e.g., in [ADF] (scalar problems), [DFM] (systems of homogeneous type), and [F] (multi-order systems). The notion of parameter-ellipticity for general multi- order systems was introduced by Kozhevnikov ([K1], [K2]) and by Denk, Mennicken, and Volevich ([DMV]). The mentioned papers had restrictions on the orders of the operators which excluded, for instance, boundary conditions of Dirichlet type (see [ADN, Section 2], [G, p. 448]). In the paper [DF], these restrictions were removed.

Let us consider in a bounded domain Ω ⊂ Rⁿ, n ≥ 2, with boundary Γ the boundary value problem

A(x, D)u(x)−λu(x) =f(x) in Ω,

B(x, D)u(x) =g(x) on Γ. (1.1)

HereA(x, D) = A_jk(x, D)

j,k=1,...,N is anN×N-matrix of linear differential op- erators,N ∈N,N ≥2,u(x) = (u₁(x), . . . , u_N(x))^T andf(x) = (f₁(x), . . . , f_N(x))^T are defined on Ω (^T denoting the transpose), whereasB(x, D) = B_jk(x, D)

j=1,...,Ne k=1,...,N

is anNe ×N-matrix of boundary operators, and g(x) = (g₁(x), . . . , g

Ne(x))^T is de- fined on Γ.

Date: February 24, 2011.

1991Mathematics Subject Classification. Primary 35J55; Secondary 35S15.

Key words and phrases. Parameter-ellipticity, multi-order systems, a priori estimates.

1

To describe the order structure of the boundary value problem (A, B), let{sj}^N_j=1 and{tj}^N_j=1 denote sequences of integers satisfyings1≥ · · · ≥sN,t1≥ · · · ≥tN ≥ 0, and putm_j :=s_j+t_j(j= 1, . . . , N). We assume

m1=· · ·=mk₁ > mk₁+1=· · ·=mkd−1 > mkd−1+1=· · ·=mk_d>0, wherekd=N. We setmej:=mk_j (j= 1, . . . , d), and assume that 2Nr:=Pk_r

j=1mj

is even for r = 1, . . . , d. We also set k₀ := 0 and N₀ := 0. Further, let {σj}^N_j=1^e , Ne :=N_d, be a sequence of integers satisfying max_jσ_j < s_N. It was shown in [DF, Section 2] that we may also assumesj≥0 (j= 1, . . . , N) andσj <0 (j= 1, . . . ,Ne).

Defineκ0:= max{t1,−σ1, . . . ,−σ

Ne}. Concerning (A, B), we will assume that ordA_jk≤s_j+t_k (j, k= 1, . . . , N),

ordBjk≤σj+tk (j= 1, . . . ,N , ke = 1, . . . , N).

Using the standard multi-index notationD^α=D₁^α1· · ·D^α_nⁿ,D_j =−i_∂x^∂

j, we write Ajk(x, D) =P

|α|≤sj+t_ka^jk_α(x)D^α forx∈Ω andj, k = 1, . . . , N and Bjk(x, D) = P

|α|≤σj+tkb^jk_α(x)D^α for x∈ Γ, j = 1, . . . ,Ne, and k= 1, . . . , N. With respect to the smoothness, we will suppose

(S) (1) Γ is of classC^κ0−1,1∩C^s1,

(2)a^jk_α ∈C^sj(Ω) (|α| ≤sj+tk) ifsj>0 and

a^jk_α ∈C⁰(Ω) (|α|=sj+tk), a^jk_α ∈L_∞(Ω) (|α|< sj+tk) ifsj= 0, (3)b^jk_α ∈C^−σj−1,1(Γ) (|α| ≤σ_j+t_k).

Let ˚Ajk(x, ξ) consist of all terms in Ajk(x, ξ) which are exactly of order sj+tk, and set

A(x, ξ) := ˚˚ Ajk(x, ξ)

j,k=1,...,N (x∈Ω, ξ∈Rⁿ).

Analogously, define ˚B(x, ξ) = ˚Bjk(x, ξ)

j=1,...,Ne k=1,...,N

forx∈Γ, ξ∈Rⁿ. The operator B(x, D) is said to be essentially upper triangular if ˚Bjk(x, D) = 0 for j =N`−1+ 1, . . . , N`, k= 1, . . . , k`−1,`= 2, . . . , d.

To formulate the ellipticity conditions, let A^(r)₁₁(x, ξ) := ˚Ajk(x, ξ)

j,k=1,...,k_r (r= 1, . . . , d).

LetI` denote the `×` unit matrix,Ie` :=Ik_{`</sub>−k`−1, andIe`,0 := diag 0·Ie1, . . . ,0· Ie<sub>`−1},Ie`

. In the following, let L ⊂Cbe a closed sector in the complex plane with vertex at the origin. The following condition is taken from [DF, Section 2] (cf. also [DMV, Section 3]).

(E) For eachx∈Ω,ξ∈Rⁿ\ {0},λ∈ L, andr= 1, . . . , dwe have det A^(r)₁₁(x, ξ)−λeI_r,0

6= 0.

If condition (E) holds, the operatorA(x, D)−λIN is said to be parameter-elliptic in L. In order to formulate conditions of Shapiro-Lopatinskii type, for x⁰ ∈Γ we rewrite the boundary value problem (1.1) in terms of local coordinates associated to x⁰. In these coordinates x⁰ = 0, and the positive x_n-axis coincides with the direction of the inner normal to Γ. We will keep the notation for Aand B in the

new coordinates. In local coordinates associated tox⁰∈Γ, let B^(r,r)_r,1 (0, ξ⁰, Dn) := ˚Bjk(0, ξ⁰, Dn)

j=1,...,Nr

k=1,...,k_r

(r= 1, . . . , d), B_r,1^(1,r)(0, ξ⁰, Dn) := ˚Bjk(0, ξ⁰, Dn)

j=Nr−1+1,...,N_r k=1,...,kr

(r= 2, . . . , d),

The following conditions (see [DF, Section 2]) are of Shapiro-Lopatinskii type and of Vishik-Lyusternik type, respectively (cf. also [DV, Section 2.3]).

(SL) For each x⁰ ∈Γ rewrite (1.1) in local coordinates associated to x⁰. Then forr= 1, . . . , d, the boundary value problem on the half-line,

A^(r)₁₁(0, ξ⁰, Dn)v(xn)−λeIr,0v(xn) = 0 (xn>0), B_r,1^(r,r)(0, ξ⁰, Dn)v(xn) = 0 (xn= 0),

|v(xn)| →0 (xn→ ∞)

(1.2)

has only the trivial solution forξ⁰ ∈Rⁿ⁻¹\ {0}, λ∈ L.

(VL) For each x⁰ ∈Γ rewrite (1.1) in local coordinates associated to x⁰. Then forr= 2, . . . , d, the boundary value problem on the half-line,

A^(r)₁₁(0,0, D_n)v(x_n)−λeI_r,0v(x_n) = 0 (x_n>0), B^(1,r)_r,1 (0,0, Dn)v(xn) = 0 (xn= 0),

|v(x_n)| →0 (x_n→ ∞)

(1.3)

has only the trivial solution forλ∈ L \ {0}.

We will show that (E), (SL), and (VL) are necessary for a priori estimates to hold. In order to formulate these estimates, we will introduce parameter-dependent norms.

ForG⊂R<sup>`</sup>open,`∈N,s∈Nand 1< p <∞, letkuks,p,G denote the norm in the standard Sobolev spaceW_p^s(G). For λ∈C\ {0} andj= 1, . . . , dset

|||u|||^(j)_s,p,G:=kuk_s,p,G+|λ|^s/mejkuk_0,p,G (u∈W_p^s(G)).

For s < 0, s ∈ Z, and j = 1, . . . , d, let H_p^s(Rⁿ) be the Bessel-potential space equipped with the parameter-dependent norm |||u|||^(j)_s,p,

Rⁿ := kF⁻¹hξ, λi^s_jF uk0,p,Rⁿ

whereF denotes the Fourier transform inRⁿ (x→ξ) and wherehξ, λi^s_j:= |ξ|²+

|λ|^2/mej1/2

. ForG⊂Rⁿ open, set |||u|||^(j)_s,p,G := inf{|||v|||^s_s,p,Rn :v ∈H_p^s(Rⁿ), v|G = u}. Finally, for s∈N we define the parameter-dependent norm on the boundary by

|||v|||^(j)_{s−1/p,p,∂G}:=kvk_{s−1/p,p,∂G}+|λ|^{(s−1/p)/mej}kvk0,p,∂G (v∈W_p^s−1/p(∂G)).

For j = 1, . . . , N, let π₁(j) := r if k_r−1 < j ≤ k_r. Similarly, for j = 1, . . . , N_d let π₂(j) := r if N_r−1 < j ≤ N_r. Note that, by definition, me_π1(j) = m_j for j= 1, . . . , N.

The aim of the paper is to show the following result.

Theorem 1.1. Let (S)hold, let 1< p <∞, and assume that there exist constants C0, C1>0 such that for allλ∈ L,|λ| ≥C0 and all u∈QN

j=1Wp^tj(Ω) the a priori

estimate

N

X

j=1

|||uj|||^(π_t ^1(j))

j,p,Ω ≤C1

X^N

j=1

|||fj|||^(π_−s^1(j))

j,p,Ω+

Ne

X

j=1

|||gj|||^(π_−σ^2(j))

j−1/p,p,Γ

(1.4) holds for f :=A(x, D)u−λu andg :=B(x, D)u. Assume further that B(x, D)is essentially upper triangular. Then the parameter-ellipticity conditions (E), (SL), and (VL)are satisfied.

Remark 1.2. In [DF], the following result was shown, where we refer to [DF] for the definitions of properly parameter-elliptic and compatible: Let (S), (E), (SL), and (VL) hold. Assume further that (A, B) is properly parameter-elliptic, thatB(x, D) is essentially upper triangular, and thatA(x⁰, D) andB(x⁰, D) are compatible at every x₀ ∈Γ. Then there existC₀, C₁ >0 such that for allλ∈ L, |λ| ≥C₀, the boundary value problem (1.1) has a unique solution u ∈ QN

j=1Wp^tj(Ω) for every f ∈ QN

j=1Hp^−sj(Ω) and every g ∈ QNe

j=1Wp^−σj−1/p(Γ), and the a priori estimate (1.4) holds.

In this sense, the sufficiency of parameter-ellipticity for the validity of the a priori estimate was shown in [DF] while Theorem 1.1 states the necessity of the conditions (E), (SL), and (VL).

2. Proof of the necessity

Throughout this section, we assume condition (S) to hold, and fix a closed sector L ⊂ C. In the following, C stands for a generic constant which may vary from inequality to inequality but which is independent of the functions appearing in the inequality and independent of λ. LetBδ(x⁰) :={x∈ Rⁿ :|x−x⁰|< δ}, and let Rⁿ+ := {x∈ Rⁿ : xn >0}, R+ := (0,∞). We start with some useful remarks on negative-order Sobolev spaces whereC₀^∞(Rⁿ+) stands for the set of all restrictions of functions inC₀^∞(Rⁿ) toRⁿ+.

Lemma 2.1. Lets∈N,1< p <∞,j∈ {1, . . . , N}. Then for allv∈L_p(Rⁿ)and allλ∈C,|λ| ≥1, we have:

a)|||v|||^(j)_−s,p,

Rⁿ≤Ckvk_−s,p,Rn,

b)|||D^αv|||^(j)_−s,p,Rn≤ |λ|^{(|α|−s)/mej}kvk0,p,Rⁿ for all |α| ≤s,

c) for each φ∈C₀^∞(Rⁿ) there exists a constant Cφ >0 independent of v such that kvφk−s,p,Rⁿ≤C_φkvk−s,p,Rⁿ and|||vφ|||^(j)_−s,p,Rn≤C_φ|||v|||^(j)_−s,p,Rn.

The same assertions hold if we replaceRⁿbyRⁿ+andC₀^∞(Rⁿ)in c) byC₀^∞(Rⁿ+).

Proof. a) We have

|||v|||^(j)_−s,p,Rn=kF⁻¹hξ, λi^−s_j F vk0,p,Rⁿ=

F⁻¹ hξi^s

hξ, λi^s_jhξi^−sF v _0,p,

Rⁿ

. Now the assertion follows immediately from the Mikhlin-Lizorkin multiplier theo- rem.

b) Similarly,

|λ|^{(s−|α|)/mej}|||D^αv|||^(j)_−s,p,

Rⁿ =kF⁻¹m(ξ, λ)F vk0,p,Rⁿ

withm(ξ, λ) :=|λ|^{(s−|α|)/mej}ξ^αhξ, λi^−s_j . Noting thatmis infinitely smooth inξand quasi-homogeneous in (ξ, λ) of degree 0 in the sense that m(ρξ, ρ^mejλ) = m(ξ, λ)

for ρ >0, we see that we may apply the Mikhlin-Lizorkin theorem to obtain the statement in b).

c) We make use of the dual pairing ofH_p^−s(Rⁿ) andW_q^s(Rⁿ), ¹_p+¹_q = 1, and get kvφk−s,p,Rⁿ= sup

ζ

|hvφ, ζi|= sup

ζ

Z

v(x)φ(x)ζ(x)dx = sup

ζ

|hv, φζi|, where the supremum is taken over allζ∈C₀^∞(Rⁿ) withkζks,q,Rⁿ≤1. Now we make use ofkφζks,q,Rⁿ ≤Cφkζks,q,Rⁿ withCφ :=Cs,qsup{|D^αφ(x)|: |α| ≤s, x∈ Rⁿ} where Cs,q is a constant depending on s and q only. We obtain sup_ζ|hv, φζi| ≤ Cφsup_ζ|hv, ζi|=Cφkvk_−s,p,Rⁿ.

For the parameter-dependent norms|||·|||^(j)_−s,p,Rnwe again consider the dual pairing betweenH_p^−s(Rⁿ) andW_q^s(Rⁿ), but now with respect to the parameter-dependent norm||| · |||^(j)_s,q,RnonW_q^s(Rⁿ). Then the result follows in exactly the same way, noting that

|||φζ|||^(j)_s,q,Rn =kφζks,q,Rⁿ+|λ|^s/mejkφζk0,p,Rⁿ ≤Cφ

kζks,q,Rⁿ+|λ|^s/mejkζk0,p,Rⁿ

. Finally, in the case of Rⁿ+ instead of Rⁿ the assertions of the lemma follow easily from the results inRⁿ and the fact that there exists an extension operator E:u 7→ Eu which is continuous as an operator from H_p^r(Rⁿ+) to H_p^r(Rⁿ) for all

|r| ≤s(see [T, p. 218]).

The following lemma will allow us to consider the model problem inRⁿ for the proof of the necessity.

Lemma 2.2. Assume that there exist constants C₀, C₁ >0 such that for all u∈ QN

j=1Wp^tj(Rⁿ) and all λ ∈ L, |λ| ≥ C₀, the a priori estimate (1.4) holds. Let x⁰∈Ω. Then there exist anx¹∈Ω, a δ >0 with Bδ(x¹)⊂Ω, and a eλ >0 such that for all λ∈ L with |λ| ≥eλ and allu∈QN

j=1Wp^tj(Rⁿ) with suppu⊂Bδ(x¹), we have

N

X

j=1

|||u|||^(π_t ^1(j))

j,p,Rⁿ≤C

N

X

j=1

|||f_j⁰|||^(π_−s^1(j))

j,p,Rⁿ, (2.1)

where we have setf⁰:= ( ˚A(x⁰, D)−λ)u.

Proof. In [DF, Prop. 4.1] it was shown that for any ε > 0 there exist a δ₀ > 0 and a λ0 > 0 such that for λ ∈ L, |λ| ≤ λ0, and all u ∈ QN

j=1Wp^tj(Rⁿ) with suppu⊂B_δ(x⁰)∩Ω we have

N

X

j=1

|||fj−f_j⁰|||^(π_−s^1(j))

j,p,Ω≤ε

N

X

j=1

kujkt_j,p,Ω

wheref := (A(x, D)−λ)u. Letεbe sufficiently small. Ifx⁰∈Ω we choosex¹:=x⁰ and δ := ¹₂min{δ0,dist(x⁰,Γ)}. If x⁰ ∈ Γ we choose x¹ ∈Bδ(x⁰)∩Ω andδ > 0 sufficiently small such that Bδ(x¹)⊂Bδ(x⁰)∩Ω. In both cases, the statement of the lemma follows easily by arguments similar to those used in the proof of [AV,

Lemma 4.2].

Proposition 2.3. Under the assumptions of Lemma 2.2, condition (E)is satisfied, i.e., forr= 1, . . . , d,x⁰∈Ω,ξ⁰∈Rⁿ\ {0}, andλ⁰∈ L we have

det A^(r)₁₁(x⁰, ξ⁰)−λ⁰Ier,0

6= 0.

Proof. Assume that (E) does not hold. Then there exist r ∈ {1, . . . , d}, x⁰ ∈ Ω, ξ⁰∈Rⁿ\{0},λ⁰∈ L, and a vectorh∈C^kr\{0}such that (A^(r)₁₁(x⁰, ξ⁰)−λ⁰Ier,0)h= 0.

Let us first consider the case λ⁰ = 0. We choose x¹ ∈ Ω, δ > 0 and eλ > 0 according to Lemma 2.1. Letφ∈C₀^∞(B_δ(x¹)) withφ6≡0, and forρ >1 set

uj(x) :=

(φ(x)e^iρξ0·xρ^−tjh_j, j= 1, . . . , k_r, 0, j=kr+ 1, . . . , N,

where ·denotes the inner product in Rⁿ. We are now going to use (2.1) to arrive at a contradiction. Indeed, we easily see that forj= 1, . . . , kr,

ku_jk_tj,p,Rn≥ |h_j| |ξ_{`</sub><sup>0</sup>|<sup>tj</sup>kφk<sub>0,p,R</sub>n−Cρ<sup>−1</sup> whereξ<sup>0</sup><sub>`} 6= 0. We further chooseµwith

mer+1< µ <mer if r < d and med/2< µ <med if r=d, (2.2) and chooseλ∈ L with|λ|=ρ^µ. Then it is clear that

|λ|^tj/mejku_jk_0,p,Rn=ρ^{−tj(1−µ/mej)}|h_j| kφk_0,p,Rn. Thus we have shown that

N

X

j=1

|||uj|||^(π_t ^1(j))

j,p,Rⁿ≥ 1 2

X^kr

j=1

|hj| |ξ_`⁰|^tj

kφk0,p,Rⁿ (2.3) for sufficiently largeρ.

Turning next to the right-hand side of (2.1), letj∈ {1, . . . , N}. Then

|||f_j⁰|||^(π_−s^1(j))

j,p,Rⁿ =

k_r

X

k=1

A˚jk(x⁰, D)uk−δjkλuk

(π1(j))

−sj,p,Rⁿ

≤

k_r

X

k=1

X

|α|=sj+tk

a^jk_α(x⁰)X

β

α β

ρ^−tkhkD^β(e^iρξ0·x)D^α−βφ

(π1(j))

−sj,p,Rⁿ

+

kr

X

k=1

δjkρ^−rk+µ|hk| |||e^iρξ0·xφ|||^(π_−s^1(j))

j,p,Rⁿ=:I1+I2, where δ_jk denotes the Kronecker delta and where P

β = P

β<α if j ≤ k_r and P

β=P

β≤α ifr < dandj > kr. (Here we used the fact thatA^(r)₁₁(x⁰, ξ⁰)h= 0.) It is clear thatI₂ →0 asρ→ ∞. Hence fixing our attention next uponI₁, we see thatI1≤Pk_r

k=1

P

|α|=sj+t_k

P

βI_1,k^α,β with I_1,k^α,β:=

α β

a^jk_α(x⁰)ρ^−tkD^β(e^iρξ0·x)D^α−βφ

(π1(j))

−sj,p,Rⁿ

.

To establishI_1,k^α,β →0 (ρ→ ∞) and, in consequence, a contradiction, it remains to show that for all appearing indices we have

ρ^−tk|||D^β(e^iρξ0·x)D^α−βφ|||^(π_−s^1(j))

j,p,Rⁿ→0 (ρ→ ∞).

Letj∈ {1, . . . , kr},|α|=sj+tk, andβ < α. If|β| ≤tk, we apply Lemma 2.1 b) to obtain

ρ^−tk|||D^β(e^iρξ0·x)D^α−βφ|||^(π_−s^1(j))

j,p,Rⁿ≤Cρ^{−tk−sjµ/mj}kD^β(e^iρξ0·x)D^α−βφk_0,p,Rn

≤Cρ^{−tk+|β|−sjµ/mj}|(ξ⁰)^β| kD^α−βφk0,p,Rⁿ→0 (ρ→ ∞).

Note here thats_j= 0 implies|β|< t_k.

If|β| ≥t_k, we write β=β₁+β₂ with|β₁|=t_k, |β₂|< s_j and fixψ∈C₀^∞(Rⁿ) withψ= 1 on suppφ. Then, using Lemma 2.1 b) and c),

ρ^−tk|||D^β(e^iρξ0·x)D^α−βφ|||^(π_−s^1(j))

j,p,Rⁿ =ρ^−tk|||D^β(e^iρξ0·xψ)D^α−βφ|||^(π_−s^1(j))

j,p,Rⁿ

≤Cρ^−tk|||D^β2(D^β1e^iρξ0·xψ)|||^(π_−s^1(j))

j,p,Rⁿ

≤Cρ^{−tk−(sj−|β2|)µ/mj}kD^β1e^iρξ0·xψk0,p,Rⁿ

≤Cρ^{−(sj−|β2|)µ/mj} kψk0,p,Rⁿ+Cρ⁻¹

→0 (ρ→ ∞).

Now letj∈ {k_r+ 1, . . . , N}. Again by Lemma 2.1 b), we have for|α|=s_j+t_k and|β| ≤ |α|

ρ^−tk|||D^β(e^iρξ0·x)D^α−βφ|||^(π_−s^1(j))

j,p,Rⁿ≤Cρ^{−sjµ/mj+sj}kD^α−βφk0,p,Rⁿ →0 (ρ→ ∞) asµ/mj>1.

Finally, the caseλ⁰6= 0 can be dealt with by arguing in a manner similar to that

above, except now we takeλ=λ⁰ρ^mer.

To prove the necessity of (SL) and (VL), we transform the problem to the half- space. For this let x⁰ ∈ Γ and assume that (A, B) is given in local coordinates associated tox⁰. Let{U,Φ} be a chart on Γ such thatx⁰= 0∈U, Φ(0) = 0, and Φ is a diffeomorphism of classC^κ0−1,1∩C^s1mappingUonto an open set inRⁿwith Φ(U∩Ω)⊂Rⁿ+, Φ(U ∩Γ)⊂Rⁿ⁻¹. We denote the push-forward of the operators A(x, D) andB(x, D) byA(y, D) ande Be(y, D), respectively, wherey= Φ(x).

Replacing Φ(x) by DΦ(0)⁻¹Φ(x), it is easily seen that we may assume the Jacobian DΦ(0) to be equal to In. Then we have ˚

Aejk(0, ξ) = ˚Ajk(0, ξ) and

˚

Be_jk(0, ξ) = ˚B_jk(0, ξ). In particular, (SL) and (VL) are satisfied for (A,e B) at 0e if only if this holds for (A, B) atx⁰= 0 (see also [DHP, p. 205]).

Lemma 2.4. Under the assumptions of Lemma 2.2, letx⁰∈Γand assume(A, B) to be written in coordinates associated tox⁰. Then there exist aδ >0and aeλ >0 such that for all u∈QN

j=1Wp^tj(Rⁿ+)with suppu⊂Bδ(0)∩Rⁿ+ and all λ∈ Lwith

|λ| ≥eλ, we have

N

X

j=1

|||uj|||^(π_t ^1(j))

j,p,Rⁿ+ ≤CX^N

j=1

|||f_j⁰|||^(π_−s^1(j))

j,p,Rⁿ++

Ne

X

j=1

|||g⁰_j|||^(π_−σ^2(j))

j−1/p,p,Rⁿ⁻¹

, (2.4)

wheref⁰:= ( ˚A(0, D)−λ)u,g⁰:= ˚B(0, D)u.

Proof. Let Φ be as above, and let A(y, D) ande B(y, D) be the push-forward ofe A(x, D) andB(x, D), respectively. Then

Φ_∗

(a^jk_α(x)−a^jk_α(0))D^αuk

= (ea^jk_α(y)−ea^jk_α(0))D_y^αuek+ X

|β|<|α|

ea^jk_α,β(y)D^β_yuek.

It was shown in the proof of [DF, Prop. 4.1], that for eachε >0 there exist aδ0>0 and aλ0>0 such that for allu∈QN

j=1Wp^tj(Ω) with suppu⊂Bδ0(0)∩Ω and all λ∈ L,|λ| ≥λ0, we have

|||Φ∗

(a^jk_α(x)−a^jk_α(0))D^αuk

|||^(π_−s^1(j))

j,p,Rⁿ₊ ≤εkeukkt_k,p,Rⁿ+

for|α|=s_j+t_k, and

|||Φ∗

a^jk_α(x)D^αuk

|||^(π_−s^1(j))

j,p,Rⁿ₊ ≤εkeukkt_k,p,Rⁿ₊

for|α|< sj+tk. From this we easily obtain that for allε >0 there existδ0, λ0>0 such that for allu∈QN

j=1Wp^tj(Ω) with suppu⊂Bδ₀(0),

N

X

j=1

|||fe_j|||_−sj,p,Rⁿ

+≤C

N

X

j=1

|||ff_j⁰|||_−sj,p,Rⁿ

++ε

N

X

j=1

ku_jk_tj,p,Rⁿ

+

where we have setf := (A(x, D)−λ)u,fe:= Φ∗f,ff⁰:= Φ∗f⁰.

To estimateg⁰, we first remark that we may assumeb^jk_α to be defined on Ω with b^jk_α ∈ C^−σj−1,1(Ω). We define the function h on Ω by hj := PNe

j=1b^jk_α(x)D^αuk, h⁰_j :=PNe

j=1b^jk_α(0)D^αuk and seteh:= Φ_∗h,hf⁰:= Φ_∗h⁰. In the same way as above, we obtain

Ne

X

j=1

|||egj|||^(π_−σ^2(j))

j−1/p,p,Rⁿ⁻¹≤

Ne

X

j=1

|||ehj|||^(π_−σ^2(j))

j−1/p,p,Rⁿ⁻¹

≤C

Ne

X

j=1

|||fh⁰_j|||^(π_−σ^2(j))

j−1/p,p,Rⁿ⁻¹+ε

N

X

j=1

kuke tj,p,Rⁿ+.

Finally, it was shown in [DF, p. 362-363] that there exist constantsc₁, c₂>0 such that for allu∈QN

j=1Wp^tj(Ω) with suppu⊂B_δ(x⁰),B_2δ(x⁰)⊂U, we have c1|||uj|||^(π_t ^1(j))

j,p,Ω ≤ |||euj|||^(π_t ^1(j))

j,p,Rⁿ+ ≤c2|||uj|||^(π_t ^1(j))

j,p,Ω , c1|||fj|||^(π_−s^1(j))

j,p,Ω≤ |||fej|||^(π_−s^1(j))

j,p,Rⁿ₊ ≤c2|||fj|||^(π_−s^1(j))

j,p,Ω, c₁|||g_j|||^(π_−σ^1(j))

j−1/p,p,Γ≤ |||eg_j|||^(π_−σ^1(j))

j−1/p,p,Rⁿ⁻¹≤c₂|||g_j|||^(π_−σ^1(j))

j−1/p,p,Γ.

(2.5)

Therefore, from the a priori-estimate (1.4) we obtain that for eachε >0 there exist δ,eλ >0 such that for u∈QN

j=1Wp^tj(Ω) with suppu⊂Bδ(0) andλ∈ L, |λ| ≥eλ, we have

N

X

j=1

|||eu_j|||^(π_t ^1(j))

j,p,Rⁿ+ ≤C

N

X

j=1

|||u_j|||^(π_t ^1(j))

j,p,Ω ≤CX^N

j=1

|||f_j|||^(π_−s^1(j))

j,p,Ω+

Ne

X

j=1

|||g_j|||^(π_−σ^2(j))

j−1/p,p,Γ

≤CX^N

j=1

|||fej|||^(π_−s^1(j))

j,p,Rⁿ₊+

Ne

X

j=1

|||gej|||^(π_−σ^2(j))

j−1/p,p,Rⁿ⁻¹

+ε

N

X

j=1

kuejk^(π_t ^1(j))

j,p,Rⁿ₊.

Takingεsmall enough andλlarge enough and noting (2.5) and˚

A(0, D) = ˚e A(0, D) and ˚

B(0, D) = ˚e B(0, D), we obtain the assertion of the Lemma.

Proposition 2.5. Assume that there exist constants C0, C1 > 0 such that for all u∈QN

j=1Wp^tj(Rⁿ)and all λ∈ L, |λ| ≥C0, the a priori estimate (1.4) holds.

Further, letx⁰∈Γ, and assume thatB(x⁰, D)is essentially upper triangular. Then condition (SL) holds atx⁰.

Proof. Let (A, B) be written in coordinates associated tox⁰ and assume that (SL) does not hold. Then there exist r ∈ {1, . . . , d}, λ⁰ ∈ L, ξ⁰₀ = (ξ⁰₁, . . . , ξ⁰_n−1) ∈ Rⁿ⁻¹\ {0}, and v 6≡ 0 satisfying (1.2). By Proposition 2.3, we know that the polynomial det(A^(r)₁₁(0, ξ⁰₀, τ)−λ⁰Ier,0) as a function ofτhas no real roots. Therefore, v=v(xn) is infinitely smooth and decays exponentially forxn → ∞, in particular, v∈Lp(R+).

Again, let us first consider the caseλ⁰= 0. We chooseφ⁰∈C₀^∞(Rⁿ⁻¹) such that φ⁰6≡0 and suppφ⁰⊂Bδ(0) withδfrom Lemma 2.4,ψ∈C₀^∞([0, δ)) with 0≤ψ≤1 andψ(xn) = 1 for 0≤xn ≤δ/2, and λ∈ Lwith |λ|=ρ^µ whereµ satisfies (2.2).

Forx∈Rⁿ+, we set w(x) :=e^iξ00·x0v(xn),φ(x) :=φ⁰(x⁰)ψ(xn), and uj(x) :=

(ρ^−tj+1/pwj(ρx)φ(x), j= 1, . . . , kr,

0, j=k_r+ 1, . . . , N. (2.6)

We will show that (2.4) leads to a contradiction for largeρ. For this we first remark that forj= 1, . . . , kr

ρkvj(ρxn)ψ(xn)k^p_0,p,R

+ =ρ Z ∞

0

|vj(ρxn)ψ(xn)|^pdxn

= Z ∞

0

|v_j(y_n)ψ(y_n/ρ)|^pdy_n% kv_jk^p_0,p,

R+ (ρ→ ∞).

Therefore, forρ≥ρ0,ρ0being sufficiently large, we have 1

2ρ^−1/pkvjk0,p,R+≤ kvj(ρxn)ψ(xn)k0,p,R+≤ρ^−1/pkvjk0,p,R+. In the same way, we see that for anyζ∈C₀^∞(Rⁿ+) andα∈Nⁿ0 we have

kD^αwj(ρx)ζ(x)k0,p,Rⁿ₊ ≤Cζρ^|α|−1/pkvjk_|α|,p,R+ with a constantCζ depending onζbut not onv orρ.

Turning now to the left-hand side of (2.4), the above considerations show that forρsufficiently large,

|||uj|||^(π_t ^1(j))

j,p,Rⁿ+≥ kujkt_j,p,Rⁿ+ ≥1

2|ξ_`⁰|^tjkφ⁰k_0,p,Rn−1kvjk0,p,R+. (2.7) On the right-hand side of (2.4), the terms|||f_j⁰|||^(π_−s^1(j))

j,p,Rⁿ₊ can be estimated in the same way as in the proof of Proposition 2.3. Indeed, we have

f_j⁰(x) =

k_r

X

k=1

X

|α|=sj+tk

X

β

a^jk_α(0) α

β

ρ^−tk+1/p(D^βw)(ρx)(D^α−βφ)(x) +δjkλuk

where P

β =P

β<α ifj ≤kr and P

β =P

β≤α ifj > kr. Here we used the fact A(0, D)w(x) =˚ e^iξ00·x0A(0, ξ˚ ₀⁰, D_n)v(x_n) = 0. From this we obtain in the same way as in the proof of Proposition 2.3

N

X

j=1

|||f_j⁰|||^(π_−s^1(j))

j,p,Rⁿ+→0 (ρ→ ∞). (2.8)

To estimateg⁰_j, we first remark that B˚_jk(0, ρξ⁰₀, D_n)v_k(ρx_n)ψ(x_n)

_x

n=0=ρ^σj+tkB˚_jk(0, ξ₀⁰, D_n)v_k(x_n) _x

n=0

by homogeneity and as ψ(xn) = 1 near xn = 0. Therefore, for j = 1, . . . , Nr we have

g_j=

kr

X

k=1

ρ^−tk+1/pB˚_jk(0, D)w_k(ρx)φ(x) _x

n=0

=

kr

X

k=1

X

|α|=σ_j+t_k

X

β<α

b^jk_α(0) α

β

ρ^−tk+1/pD^βw_k(ρx)D^α−βφ(x) _x

n=0

due toPk_r

k=1B˚jk(0, ξ⁰₀, Dn)vk(xn)|x_n=0 = 0. For j= 1, . . . , Nr,|α|=σj+tk, and β < αwe can estimate

ρ^−tk+1/p

D^βw(ρx)D^α−βφ(x) _x

n=0

_−σ

j−1/p,p,Rⁿ⁻¹

≤ρ^−tk+1/p

D^βw(ρx)D^α−βφ(x)

−σj,p,Rⁿ+

≤Cρ^−tk+1/p X

|γ|≤−σj

D^γ

D^βw(ρx)D^α−βφ(x) _0,p,

Rⁿ+

≤Cφρ^{−tk−σj+|β|}kvk0,p,R+ →0 (ρ→ ∞). (2.9) Further, for|λ|=ρ^µ we obtain

|λ|^{(−σj−1/p)/meπ2 (j)}kg_jk_0,p,Rn−1

=|λ|^{(−σj−1/p)/meπ2 (j)}

k_r

X

k=1

X

|α|=σj+tk

b^jk_α(0)D^αuk(x) _x

n=0

_0,p,

Rⁿ⁻¹

≤Cρ^{(σj+1/p)(1−µ/meπ2 (j))}→0 (ρ→ ∞)

asµ/me_π2(j)<1 andσj ≤ −1. From this and (2.9) we see that forj= 1, . . . , Nr

|||g⁰_j|||^(π_−σ^2(j))

j−1/p,p,Rⁿ⁻¹ →0 (ρ→ ∞). (2.10) Finally, for j > Nr we haveg_j⁰ = 0 asB(0, D) is assumed to be essentially upper triangular. From (2.7), (2.8), and (2.10) we obtain a contradiction to the a priori- estimate (2.4).

In the caseλ⁰6= 0, the result follows from similar considerations where we now

setλ=λ⁰ρ^mer again.

Proposition 2.6. Under the assumptions of Proposition 2.5, condition(VL)holds atx⁰.

Proof. The proof is similar to the proof of Proposition 2.5, and we only indicate some changes and additional remarks. Assuming v to be a nontrivial solution of (1.3), defineλ:=ρ^merλ⁰anduas in (2.6), but now settingξ⁰₀= 0, i.e., we set

u(x) :=

(ρ^−tj+1/pφ(x)vj(ρxn), j = 1, . . . , kr, 0, j =kr+ 1, . . . , N.

Now the left-hand side of (2.4) can be estimated from below by

|||u_j|||^(π_t ^2(j))

j,p,Rⁿ+ ≥ ku_jk_tj,p,Rⁿ

+ ≥ kD_n^tju_jk_0,p,Rⁿ

+≥1

2kv_jk_0,p,R+kφ⁰k_0,p,Rn−1