Bounded H_\infty-calculus for pseudodifferential Douglis-Nirenberg systems of mild regularity

0.3 MB·English

by  R. Denk

Bounded H -calculus for pseudodifferential
∞
8
0 Douglis-Nirenberg systems of mild regularity
0
2
R. Denk, J. Saal, and J. Seiler
n
a
J
Abstract. We consider pseudodifferential Douglis-Nirenbergsystems on Rn withcomponents be-
4 longingtothestandardHo¨rmanderclassS∗ (Rn×Rn),0≤δ<1.Parameter-ellipticitywithrespect
2 1,δ
toasubsector Λ⊂Cisintroducedandshowntoimplythe existence ofabounded H∞-calculusin

] suitablescalesofSobolev,Besov,andHo¨lderspaces.Wealsoadmitnonpseudodifferentialperturba-
P
tions. Applications concern systems with coefficients of mild Ho¨lder regularity and the generalized
A
thermoelasticplateequations.
.
h
t
a
m
Contents
[

1 1. Introduction 1
v
8 2. Douglis-Nirenberg systems: Basic definitions and properties 2
4
3. Λ-elliptic Douglis-Nirenberg systems 3
7
3 4. Bounded H -calculus for perturbed Douglis-Nirenberg systems 15
∞
.
1
5. Systems with Ho¨lder continuous coefficients 19
0
8 6. The generalized thermoelastic plate equations 20
0
7. Further extensions 22
:
v
i References 22
X
r
a
1. Introduction
Theconceptofmaximalregularityisanimportanttoolinthemodernanalysisofnonlinear(parabolic)
evolution equations. For a densely defined closed operator A:D(A)⊂X →X in a Banach space X,
maximal L -regularity essentially means that the initial value problem u +Au(t) = f(t), u(0) = 0,
q t
for each right-hand side f ∈ L (R ,X) admits a unique solution with Au ∈ L (R ,X) (in case of
q + q +
invertibility of A this is equivalent to u ∈ W1(R ,X)∩L (R ,D(A))). In combination with fixed
q + q +
point arguments maximal L -regularity may be used to deduce existence and regularity results for
q
solutions of nonlinear problems.
ItisknownthatAisthegeneratorofananalyticsemi-groupinX,providedithasmaximalregularity.
The reverseimplication,however,isfalse.Thusitisnaturalto addressthe question,whichconditions
onAimplymaximalregularity.Onesuchconditionistheexistenceofaso-calledboundedH -calculus
∞
1
2 R.DENK,J.SAAL,ANDJ.SEILER
for A. This is a functional calculus, that allows to define f(A) ∈ L(X) for certain complex-valued
holomorphic functions f; for a short review see Subsection 4.2. This calculus was introduced by
McIntosh in [13] and recieved since then a lot of attention (cf. [3] and [11] for extensive expositions
and further literature). The existence of an H -calculus implies existence of bounded imaginary
∞
powers. Combining this with a classical result of Dore and Venni [6], maximal regularity follows. An
alternative approachto maximal regularity relies on the so-called R-boundedness of the resolvent.
The aim of the present paper is to establish conditions for perturbed Douglis-Nirenbergsystems that
ensure the existence of a bounded H -calculus, hence of maximal regularity. We consider pseudo-
∞
differential systems on Rn with components whose symbols belong to the standard Ho¨rmander class
S∗ (Rn ×Rn), 0 ≤ δ < 1 (the order ∗ is different for each component). The established condition
1,δ
is a condition of parameter-ellipticity with respect to a sector Λ ⊂ C containing the left half-plane,
called Λ-ellipticity throughout the paper. We give two, initially seemingly different, formulations of
Λ-ellipticity (see Definitions 3.1 and 3.2). The first is motivated by a notion of parameter-ellipticity
introduced by Denk, Menniken, and Volevich in [4], which is connected with the so-called Newton-
polygonassociatedwith the system.The secondformulationis modeledona conditionintroducedby
Kozhevnikov[8],[9],for classical(i.e. polyhomogeneous)Douglis-Nirenbergsystems.Althoughdiffer-
ent in appearance, we proof that both notions of ellipticity are equivalent. For Λ-elliptic systems we
constructinSection3aparametrixandshowthatsuchsystemsarediagonalizablemodulosmoothing
remainders. In Section 4 we establish the existence of a bounded H -calculus.
∞
Theperturbationsweadmitinouranalysisallowusnotonlytoconsidersystemswithsmoothsymbols
butalsowithsymbolsofamildHo¨lderregularity,seeSection5.Minimalregularityassumptionsonthe
symbols(i.e.,the coefficientsincaseofdifferentialsystems)areofparticularimportancewhenaiming
at nonlinear problems. As a further application, see Section 6, we consider the so-called generalized
thermoelastic plate equations introduced in [1], [14]. It has been shown in [5] that (for the involved
parameters belonging to the ‘parabolic region’) this equation can be seen as an evolution equation
withageneratorofananalyticsemi-group.Weimprovethisresult,showingtheexistenceofabounded
H -calculus.
∞
2. Douglis-Nirenberg systems: Basic definitions and properties
In this section we provide the basic notation and definitions that will be used throughout the paper.
Moreover,we recall some standard properties of pseudodifferential operators.
Definition 2.1. The symbol class Sµ(Rn ×Rn) with µ ∈ R and 0 ≤ δ < 1 consists of all smooth
δ
functions a=a(x,ξ):Rn×Rn →C satisfying
kakµ := sup |DαDβa(x,ξ)|hξi−µ+|α|−δ|β| <∞
δ,k ξ x
x,ξ∈Rn
|α|+|β|≤k
for any k ∈N . As usual, we use the notation hξi:=(1+|ξ|2)1/2 and D :=−i∂. Frequently, we shall
0
simply write Sµ. In case δ =0, we suppress δ from the notation.
δ
The system of norms k·kµ , k ∈N , defines a Fr´echet topology on Sµ. To a given symbol a∈Sµ we
δ,k 0 δ δ
associate a continuous operator a(x,D):S(Rn)→S(Rn) by
[a(x,D)u](x)= e−ixξa(x,ξ)u(ξ)d¯ξ,
ZRn
b
H∞-CALCULUS FOR DOUGLIS-NIRENBERG SYSTEMS OF MILD REGULARITY 3
where u is the Fourier transform of u and d¯ξ = (2π)−ndξ. By duality, we extend this operator to
a(x,D):S′(Rn)→S′(Rn). This operator restricts to Sobolev spaces in the following way:
b
Theorem 2.2. Let a∈Sµ(Rn×Rn) and 1<p<∞. Then a(x,D) restricts to a continuous map
δ
a(x,D):Hs(Rn)−→Hs−µ(Rn)
p p
for any real s. Moreover, we have continuity of the mappings
(2.1) a7→a(x,D):Sµ(Rn×Rn)−→L(Hs(Rn),Hs−µ(Rn)).
δ p p
Thecontinuityof (2.1)entailsthatthenormka(x,D)kasaboundedoperatorbetweenSobolevspaces
can be estimated from above by Ckakµ with suitable constants k and C that do not depend on a.
δ,k
Pseudodifferential operators behave well under composition: There exists a continuous map
(a ,a )7→a #a :Sµ1 ×Sµ2 −→Sµ1+µ2
1 2 1 2 δ δ δ
such that a (x,D)a (x,D)=(a #a )(x,D). For an explicit formula of the so-called Leibniz-product
1 2 1 2
a #a see, for example [10]. In the sense of an asymptotic expansion we have
1 2
1
a #a ∼ ∂αa Dαa ,
1 2 α! ξ 1 x 2
α∈Nn
X0
i.e., for any positive integer N,
N−1
1
a #a − ∂αa Dαa ∈ Sµ1+µ2−(1−δ)N.
1 2 α! ξ 1 x 2 δ
|αX|=0
Definition 2.3. ADouglis-Nirenbergsystemisa(q×q)-matrix,q ∈N,ofpseudodifferential operators
A(x,D)= a (x,D)
ij
1≤i,j≤q
(cid:16) (cid:17)
such that there exist real numbers m ,...,m and l ,...,l with the property that
1 q 1 q
a (x,ξ)∈Sli+mj(Rn×Rn) ∀i,j =1,...,q
ij δ
and the numbers r :=l +m satisfy r ≥r ≥...≥r ≥0.
i i i 1 2 q
A Douglis-Nirenberg system in the sense of the previous definition induces continuous operators
q q
A(x,D): ⊕ Hs+mj(Rn)−→ ⊕ Hs−li(Rn) ∀s∈R.
p p
j=1 i=1
Due to the requested nonnegativity of the r we have that s+m ≥ s−l . Therefore, we may (and
i i i
q q
will) consider A(x,D) as an unbounded operator in ⊕ Hps−li with domain ⊕ Hps+mj(Rn).
i=1 j=1
3. Λ-elliptic Douglis-Nirenberg systems
3.1. Parameter-ellipticity. FromnowonletΛdenoteaclosedsubsectorofthe complexplain,
i.e.
(3.1) Λ=Λ(θ)= reiϕ | r ≥0, θ ≤ϕ≤2π−θ , 0<θ <π.
WeletA(x,D)beasystemasinD(cid:8)efinition2.3.Moreover,forsim(cid:9)plicityofexposition,weshallassume
from now on that
(3.2) r >r >...>r ≥0, r :=l +m .
1 2 q i i i
4 R.DENK,J.SAAL,ANDJ.SEILER
Let us point out that this assumption is mainly made for notational convenience; the main results
of the present paper, i.e. parametrix construction, diagonalization, and existence of a bounded H -
∞
calculus,remainvalid(in anadaptedformulation)alsoin the generalcase when in(3.2)some (or all)
ofthe inequalitiesarereplacedbyequalities.Letusalsomentionthatweassumeneitheranyordering
nor positivity or negativity of the numbers l ,...,l , m ,...,m .
1 q 1 q
We shall now introduce two notions of parameter-ellipticity, where the parameter-space is just the
above sectorΛ, and then show that they are equivalent.These conditions are modeled onthose given
in [4] and [8], [9]. To this end let
(3.3) P(x,ξ;λ)=P (x,ξ;λ):=det A(x,ξ)−λ
A
(cid:0) (cid:1)
denote the characteristicpolynomialofA(x,ξ) (where we identify λ with λI andwhere I denotes the
identity matrix). It is straightforwardto verify (see also Lemma 3.5) that
|P(x,ξ;λ)|≤C(hξir1 +|λ|)·...·(hξirq +|λ|) ∀x,ξ ∈Rn ∀λ∈C
with a suitable constant C ≥0.
Definition 3.1. A(x,D) is said to be Λ-elliptic if, for some constants C >0 and R≥0,
(3.4) |P(x,ξ;λ)|≥C(hξir1 +|λ|)·...·(hξirq +|λ|) ∀x∈Rn ∀|ξ|≥R ∀λ∈Λ.
For the second definition let us introduce further notation. We call
A[κ](x,D)= a (x,D) , 1≤κ≤q,
ij
1≤i,j≤κ
(cid:16) (cid:17)
the κ-th principal minor of A(x,D) and let
E =diag(0,...,0,1)∈Cκ×κ.
κ
Definition 3.2. A(x,D) is called Λ-elliptic (with principal minors) if
(3.5) det A[κ](x,ξ)−λE ≥Chξir1+...+rκ−1(hξirκ +|λ|) ∀x∈R ∀|ξ|≥R ∀λ∈Λ,
κ
(cid:12) (cid:0) (cid:1)(cid:12)
with suita(cid:12)ble constants C >0 an(cid:12)d R≥0, for each 1≤κ≤q.
Theorem 3.3. The two notions of Λ-ellipticity given in Definition 3.1 and 3.2, respectively, are
equivalent.
Proof. That Λ-elliptic with principal minors implies Λ-ellipticity in the sense of Definition 3.1 we
shall prove in Corollary 3.10, below. For the other implication, we proceed in two steps:
Step 1: First we will show that the condition of Λ-ellipticity with principal minors is satisfied for
λ = 0. More precisely, we will show that there exist R ≥ 0 and C > 0 such that for all κ = 1,...,q
we have
|detA[κ](x,ξ)|≥Chξir1+···+rκ ∀x∈Rn ∀|ξ|≥R ∀λ∈Λ.
Assume this is not the case.Then there exists a κ∈{1,...,q} and a sequence (xk,ξk)k∈N ⊂Rn×Rn
with |ξ |→∞ and
k
(3.6) detA[κ](x ,ξ ) hξ i−r1−···−rκ −k−→−−∞→0.
k k k
(cid:12) (cid:12)
(cid:12) (cid:12)
H∞-CALCULUS FOR DOUGLIS-NIRENBERG SYSTEMS OF MILD REGULARITY 5
We define r := rκ+2rκ+1 ∈ (rκ+1,rκ) and choose the sequence (λk)k∈N ⊂ Λ by λk := hξkirλ0 with a
fixed λ ∈Λ, |λ |=1. We will consider the q×q-matrix
0 0
A[κ](x,ξ) 0
A˜[κ](x,ξ,λ):=
0 −λIq−κ!
where I stands for the (q−κ)-dimensional unit matrix. Due to (3.6) we have
q−κ
|detA˜[κ](x ,ξ ,λ )|
k k k k→∞
(3.7) −−−−→0.
hξkir1+···+rκ|λk|q−κ
For a rescaling of the matrix A˜[κ], we set ǫ := r−rj >0 for j =κ+1,...,q and
j 2
D (ξ):=diag hξi−l1,...,hξi−lκ,hξi−lκ+1−ǫκ+1,...,hξi−lq−ǫq ,
1
D2(ξ):=diag(cid:0)hξi−m1,...,hξi−mκ,hξi−mκ+1−ǫκ+1,...,hξi−m(cid:1)q−ǫq .
We will estimate the coefficient(cid:0)s b (x,ξ,λ) of the matrix (cid:1)
ij
B(x,ξ,λ):=D (ξ) (A(x,ξ)−λ)−A˜[κ](x,ξ,λ) D (ξ).
1 2
(cid:16) (cid:17)
For i,j ≤κ we get
|b (x ,ξ ,λ )|=δ hξ i−ri|λ |=δ hξ ir−ri −k−→−−∞→0.
ij k k k ij k k ij k
For i,j >κ one has
|b (x ,ξ ,λ )|=|a (x ,ξ )|hξ i−li−mj−ǫi−ǫj −k−→−−∞→0,
ij k k k ij k k k
since aij(x,ξ) ∈ Sli+mj(Rn ×Rn) and ǫi,ǫj > 0. In the same way, we get |bij| → 0 in the cases
i ≤ κ, j > κ and i > κ, j ≤ κ, where now only one factor of the form hξki−ǫi appears. Hence
k→∞
B(x ,ξ ,λ ) −−−−→ 0. By direct computation, D (ξ )(A(x ,ξ ) − λ )D (ξ ) can be shown to be
k k k 1 k k k k 1 k
bounded,uniformlyink.Sincethe determinantis uniformlycontinuousonboundedsets,wethuscan
conclude that
detD (ξ ) A(x ,ξ )−λ D (ξ )−detD (ξ )A˜[κ](x ,ξ ,λ )D (ξ )→0.
1 k k k k 2 k 1 k k k k 2 k
From this, the definition(cid:0)of |λ |, and (3(cid:1).7) we obtain
k
det A(x ,ξ )−λ
k k k k→∞
−−−−→0.
(cid:12)hξki(cid:0)r1+···+rκ|λk|q−κ(cid:1)(cid:12)
(cid:12) (cid:12)
By our choice of r and λ we have
k
q
(hξkirj +|λk|)
j=1 k→∞
−−−−→1.
hξQkir1+···+rκ|λk|q−κ
The last two statements yield
q
det A(x ,ξ )−λ · hξ irj +|λ | −1 −k−→−−∞→0
k k k k k
j=1
(cid:12) (cid:0) (cid:1)(cid:12) Q (cid:0) (cid:1)
which contradicts the Λ-(cid:12)ellipticity of A(x,D)(cid:12). Thus the conditions of Definition 3.2 are satisfied for
λ=0.
Step 2: Now we wantto show that condition (3.5) holds for λ6=0. If this is not the case there exists
a κ∈{1,...,q} and a sequence (xk,ξk,λk)k∈N ⊂Rn×Rn×Λ with |ξk|→∞ and
|det(A[κ](x ,ξ )−λ E )|
k k k κ k→∞
(3.8) −−−−→0.
hξkir1+···+rκ−1(hξkirκ +|λk|)
6 R.DENK,J.SAAL,ANDJ.SEILER
We shall use the equality
det A[κ](x,ξ)−λE =detA[κ](x,ξ)−λdetA[κ−1](x,ξ),
κ
which is valid due to the(cid:0)linearity of the d(cid:1)eterminant with respect to the κ-th column.
(i) First we show that liminf hξkirκ >0. If this is not the case we may assume, by passing
k→∞ |λk|
to a subsequence, that hξkirκ −k−→−−∞→0. Now we apply Step 1 of this proof to estimate
|λk|
|det(A[κ](x ,ξ )−λ E )|
k k k κ
hξkir1+···+rκ−1(hξkirκ +|λk|)
|detA[κ−1](x ,ξ )| |detA[κ](x ,ξ )|
k k k k
≥|λ | −
k hξkir1+···+rκ−1(hξkirκ +|λk|) hξkir1+···+rκ−1(hξkirκ +|λk|)
|λk| hξkirκ
≥C −C
1 hξkirκ +|λk| 2 hξkirκ +|λk|
with twopositive constantsC andC . Fork →∞ the right-handside ofthe lastinequality
1 2
tends to C >0 which contradicts (3.8).
1
(ii) In the same way we show liminf |λk| > 0. If this does not hold, we may assume
k→∞ hξkirκ
|λk| −k−→−−∞→0. Thus we obtain
hξkirκ
|det(A[κ](xk,ξk)−λkEκ)| hξkirκ |λk| k→∞
≥C −C −−−−→C >0,
hξkir1+···+rκ−1(hξkirκ +|λk|) 1 hξkirκ +|λk| 2 hξkirκ +|λk| 1
again a contradiction to (3.8).
(iii) Due to (i) and (ii), there exist positive constants C and C with
3 4
C hξ irκ ≤|λ |≤C hξ irκ
3 k k 4 k
for sufficiently large k. As in Step 1, we use the scaling matrices D (ξ) and D (ξ), now
1 2
setting r :=r . For the coefficients of the matrix
κ
A[κ](x,ξ)−λE 0
κ
B(x,ξ,λ):=D (ξ) A(x,ξ)−λI − D (ξ)
1 q 2
" 0 −λIq−κ!#
(cid:0) (cid:1)
we obtain the estimates
δijhξki−ri|λk| :i,j <κ
0 :i=κ,j <κ or i<κ,j =κ
|bij(xk,ξk,λk)|=||aaiijj((xxkk,,ξξkk))||hhξξkkii−−llii−−mmjj−−ǫǫji ::ii≤>κκ,,jj ≤>κκ .
In all cases |b (x ,ξ,|λaij)(|x−kk−,→−ξ−∞k→)|h0ξ.kiI−nlit−hmejs−aǫmi−eǫjway a:si,bje>forκe we obtain, using (3.8) and
ij k k k
the equality
q
hξkirj +|λk|
j=1 k→∞
−−−−→1,
hξkir1+···+Qrκ−(cid:0)1 hξkirκ +|λ(cid:1)k| |λk|q−κ
that
(cid:0) q (cid:1)
det A(x ,ξ )−λ · hξ irj +|λ | −1 −k−→−−∞→0.
k k k k k
j=1
This contradict(cid:12)(cid:12)s the(cid:0)Λ-ellipticity of(cid:1)(cid:12)(cid:12)A(Qx,D(cid:0)) and finishe(cid:1)s the proof. (cid:3)
H∞-CALCULUS FOR DOUGLIS-NIRENBERG SYSTEMS OF MILD REGULARITY 7
3.2. Construction of the parametrix. Throughoutthis subsectionletA(x,D) be aΛ-elliptic
Douglis-Nirenbergsystem.Forsimplicityweshallassumethat(3.4)holdswithR=0.Asthefollowing
lemma shows, for our purposes that is no restriction:
Lemma 3.4. Let A(x,D) be Λ-elliptic. Then there exists an α ≥0 such that the system A (x,D):=
0 α
A(x,D)+α satisfies
|P (x,ξ;λ)|≥C(hξir1 +|λ|)·...·(hξirq +|λ|) ∀x∈Rn ∀ξ ∈Rn ∀λ∈Λ,
Aα
whenever α≥α .
0
Proof. By definition, we have
P (x,ξ;λ)=P (x,ξ;λ−α)=det A(x,ξ)−(λ−α) .
Aα A
Obviously, there exist constants d≤1≤D such that (cid:0) (cid:1)
dhλi≤|λ−α|≤Dhλi ∀λ∈Λ.
As λ−α∈Λ for each λ∈λ, the Λ-ellipticity of A(x,D) thus yields that
|P (x,ξ;λ)|≥C (hξir1 +hλi)·...·(hξirq +hλi)
Aα α
uniformly in x∈Rn, |ξ|≥R and λ∈Λ. Let us consider those ξ with |ξ|≤R. Clearly,
sup kA(x,ξ)k<∞.
x∈Rn,|ξ|≤R
Thus, choosing α large enough, A (x,ξ) has no spectrum in Λ and
0 α
dhλiq ≤|P (x,ξ;λ)|≤Dhλiq ∀λ∈Λ
Aα
uniformly in x∈Rn and |ξ|≤R, for suitable constants d≤1≤D. This yields the result. (cid:3)
Lemma 3.5. Define
G(0)(x,ξ;λ)= g(0)(x,ξ;λ) := A(x,ξ)−λ −1.
ij
1≤i,j≤q
Then the following uniform in (x,ξ,λ(cid:16))∈Rn×Rn(cid:17)×Λ estimat(cid:0)es hold true(cid:1):
|DαDβg(0)(x,ξ;λ)|≤C (hξiri +|λ|)−1(hξirj +|λ|)−1hξili+mj−|α|+δ|β|
ξ x ij αβ
in case i6=j, and
|DαDβg(0)(x,ξ;λ)|≤C (hξiri +|λ|)−1hξi−|α|+δ|β|.
ξ x ii αβ
Proof. According to Cramer’s rule we have
1
g(0)(x,ξ;λ)= det(A(x,ξ)−λ)(i,j),
ij P(x,ξ,λ)
where B(i,j) denotes the matrix obtained by deleting the j-th row and i-th column of the matrix B.
Let us consider the case i6=j. Set Z(l) ={1,...,q}\{l}.Then, suppressing (x,ξ) from the notation,
det(A(x,ξ)−λ)(i,j) is a linear combination of terms
(a −λ)···(a −λ)·a ···a ,
i1,i1 ik,ik ik+1,πik+1 iq−1,πiq−1
where Z(j) ={i ,...,i },1≤k ≤q−2, andπ :Z(j) →Z(i) is a bijection. Eachof these terms can
1 q−1
be estimated from aboveby hξilihξimj (hξirl+|λ|). Together with the ellipticity assumption(3.4)
l=1
l6=Qi,j
8 R.DENK,J.SAAL,ANDJ.SEILER
this shows the desired estimate in case |α| = |β| = 0. The general case follows similarly using chain
and product rule. The case i=j is analogous. (cid:3)
Note also that the estimates of G(0) from the previous lemma for α=β =0 are easily seen to imply
the estimate (3.4). Thus this would yield another equivalent definition of Λ-ellipticity.
As adirect consequenceofthese estimates,we getthe naturalfact thatΛ-elliticity is preservedunder
perturbations by lower order terms:
Corollary 3.6. Let A(x,ξ) and A(x,ξ) be two Douglis-Nirenberg systems such that A(x,ξ) is Λ-
elliptic and for each 1 ≤ i,j ≤ q the (i,j)-th component of R(x,ξ) := A(x,ξ)−A(x,ξ) has order
e
l +m −ε for some ε>0. Then also A(x,ξ) is Λ-elliptic.
i j
e
Proof. For large enough |ξ| we haeve
det A(x,ξ)−λ =det(A(x,ξ)−λ)det 1+(A(x,ξ)−λ)−1R(x,ξ) .
Define M(ξ)=diag(cid:0)heξim1,...,h(cid:1)ξimq and L(ξ)=diag(cid:0)hξil1,...,hξilq . Conjugatio(cid:1)n with M yields
(3.9) det 1+(A((cid:0)x,ξ)−λ)−1R(x,(cid:1)ξ) =det 1+M((cid:0)ξ)G(0)(x,ξ;λ)L(cid:1)(ξ)L(ξ)−1R(x,ξ)M(ξ)−1 .
The (i,j)-th(cid:0)component of L−1RM−1 i(cid:1)s just (cid:0) (cid:1)
r (x,ξ)hξi−mjhξi−li ∈S−ε.
ij δ
Due to Proposition 3.5, the (i,j)-th component of MG(0)L can be estimated from above by
|g(0)(x,ξ;λ)hξimihξilj|≤C(hξiri +|λ|)−1(hξirj +|λ|)−1hξiri+rj ≤C
ij
for i6=j, and analogouslyfor i=j. Therefore the matrix on the right-handside of (3.9) tends to the
identity matrix for |ξ| → ∞, uniformly in (x,λ). Hence the absolute value of the determinant (3.9)
canbe estimatedfrombelowby 1/2forsufficiently large|ξ|andall(x,λ)∈Rn×Λ.Thuswith Aalso
A satisfies the ellipticity assumption given in Definition 3.1. (cid:3)
Peroceeding with G(0) from Lemma 3.5, we define recursively for ν ∈N
1
(3.10) G(ν)(x,ξ;λ)= (∂αG(m))(x,ξ;λ)(DαA)(x,ξ)G(0)(x,ξ;λ).
α! ξ x
m+|α|=ν
mP<ν
By induction, each ∂α∂βG(ν), ν ≥1, is a finite linear combination of terms
ξ x
(3.11) G(0)(∂α1∂β1A)·...·G(0)(∂αk∂βkA)G(0)
ξ x ξ x
with|α |+...+|α |=|α|+ν,|β |+...+|β |=|β|+ν,andk ≥2.Fromthiswededucethefollowing:
1 k 1 k
Proposition 3.7. Let A(x,D) be Λ-elliptic and G(ν)(x,ξ;λ)= g(ν)(x,ξ;λ) be defined as in
ij
1≤i,j≤q
(3.10). In case ν ≥1 we have (cid:16) (cid:17)
|∂α∂βg(ν)(x,ξ;λ)|≤C (hξiri +|λ|)−1(hξirj +|λ|)−1hξili+mj−(1−δ)ν−|α|+δ|β|
ξ x ij αβ
for all 1≤i,j ≤q, uniformly in (x,ξ,λ)∈Rn×Rn×Λ (note that the estimates are also valid for the
elements on the diagonal, i.e., i=j).
H∞-CALCULUS FOR DOUGLIS-NIRENBERG SYSTEMS OF MILD REGULARITY 9
Proof. For n∈N, let B(n)(x,ξ)= b(n)(x,ξ) be systems with b(n) ∈Sli+mj. The proof
ij ij
1≤i,j≤q
relies on two kinds of estimates. (cid:16) (cid:17)
First,let H =(B(3)G(0))·...·(B(N)G(0))for anarbitraryN ≥3.Then, byinduction onN,it is easy
to see that
(3.12) |h (x,ξ;λ)|≤C(hξirj +|λ|)−1hξili+mj.
ij
Second, consider H =G(0)B(1)G(0)B(2)G(0). We shall use the explicit formula
q
e h = g(0)b(1)g(0)b(2)g(0).
ij iα αβ βγ γδ δj
α,β,γ,δ=1
P
e
If in a summand β =γ, we can estimate it by
C g(0)(x,ξ;λ)hξilα+mβ(hξirβ +|λ|)−1hξilβ+mδg(0)(x,ξ;λ)
iα δj
(cid:12) (cid:12)
in view of Lemma 3.5. N(cid:12)ow (cid:12)
(cid:12) (cid:12)
(hξiri +|λ|)−1hξili :i=α
|g(0)(x,ξ;λ)hξilα|≤C
iα (hξiri +|λ|)−1hξili(hξirα +|λ|)−1hξilα+mα :i=α

≤C(hξiri +|λ|)−1hξili

and, analogously,
|g(0)(x,ξ;λ)hξimδ|≤C(hξirj +|λ|)−1hξimj.
δj
Thus we estimate the summand by
C(hξiri +|λ|)−1(hξirj +|λ|)−1(hξirβ +|λ|)−1hξirβ+li+mj
≤C(hξiri +|λ|)−1(hξirj +|λ|)−1hξili+mj.
Arguing analogously in the case β 6=γ we arrive at the estimate
(3.13) |h (x,ξ;λ)|≤C(hξiri +|λ|)−1(hξirj +|λ|)−1hξili+mj.
ij
Combining both estimatees (3.12) and (3.13) yields
(3.14) |(HH) (x,ξ;λ)|≤C(hξiri +|λ|)−1(hξirj +|λ|)−1hξili+mj.
ij
To finally prove the staetement of the proposition we set
B(n)(x,ξ):=hξi|αn|−δ|βn|∂αn∂βna(x,ξ).
ξ x
Then, according to (3.11), we can represent ∂α∂βG(ν) as a linear combination of terms
ξ x
G(0)B(1)·...·G(0)B(k)G(0)hξi−(1−δ)ν−|α|+δ|β|
with k ≥2. It remains to use the above estimate (3.14). (cid:3)
Using these estimates we are now in the position to construct a parametrix for A(x,D) − λ. For
standard systems this construction can be found in [10]. However, we deal with Douglis-Nirenberg
systems and also make precise the remainder estimate.
10 R.DENK,J.SAAL,ANDJ.SEILER
Theorem 3.8. There exists a G(x,ξ;λ)= g (x,ξ;λ) such that
ij
1≤i,j≤q
(cid:16) (cid:17)
(3.15) |∂α∂βg (x,ξ;λ)|≤C (hξiri +|λ|)−1hξi−|α|+δ|β|
ξ x ii αβ
and, for i6=j,
(3.16) |∂α∂βg (x,ξ;λ)|≤C (hξiri +|λ|)−1(hξirj +|λ|)−1hξili+mj−|α|+δ|β|
ξ x ij αβ
Moreover, for all 1≤i,j ≤q,
|∂α∂β{g (x,ξ;λ)−g(0)(x,ξ;λ))−1}|
ξ x ij ij
(3.17)
≤C (hξiri +|λ|)−1(hξirj +|λ|)−1hξili+mj−(1−δ)−|α|+δ|β|
αβ
All these estimates hold uniformly in (x,ξ,λ) ∈ Rn×Rn×Λ and for all α,β ∈ Nn. Passing to the
0
operator-level, we have
G(x,D;λ)(A(x,D)−λ)=1+R(0)(x,D;λ),
(3.18)
(A(x,D)−λ)G(x,D;λ)=1+R(1)(x,D;λ)
with remainders R(k)(x,ξ;λ)= r(k)(x,ξ;λ) satisfying
ij
1≤i,j≤q
(cid:16) (cid:17)
(3.19) |∂α∂βr(k)(x,ξ;λ)|≤C hλi−1hξi−N <∞.
ξ x ij αβN
for arbitrary N ∈N and all α,β ∈Nn.
0
Proof. The symbol G is defined by means of assymptotic summation as
∞
G(x,ξ;λ):=G(0)(x,ξ;λ)+ χ(ε |ξ|)G(ν)(x,ξ;λ),
ν
ν=1
P
where χ : R → [0,1] is a smooth 0-excision function1 and ε > ε > ... −j−→−∞→ 0 sufficiently fast. By
1 2
Lemma 3.5andProposition3.7the estimates(3.15),(3.16),and(3.17)then hold.Itremainstoverify
(3.18). To this end let us define
N−1
QN(x,ξ;λ)= G(ν)(x,ξ;λ),
ν=0
X
N−1
1
JN(x,ξ;λ)= ∂αQN(x,ξ;λ)Dα(A(x,ξ)−λ)
α! ξ x
|αX|=0
for N ∈N. A direct computation shows that
1
JN(x,ξ;λ)−1= ∂αG(ν)(x,ξ;λ)DαA(x,ξ).
α! ξ x
ν<NX,|α|<N
ν+|α|≥N
By Lemma 3.5 and Proposition 3.7 it is easily seen that then
(3.20) |∂α∂β(JN(x,ξ;λ)−1)|≤C (hξiri +|λ|)−1hξili+mj−(1−δ)N.
ξ x ij αβ
1i.e.χvanishesidenticallyinaneighborhoodof0and1−χisasmoothfunctionwithcompactsupport