O(γ) := max
1, γ−1 2
, γ >0.
Then we haveα:=αin(O) = 0andβ:=βin(O) = 1/2. IfQ∈HP(Sθ×Σnδ),θ > π/2, is a function such thatO is an upper order function, i.e.
|Q(λ, z)| ≤C·ΞO(λ, z) =C·1 +|λ|+|z|3/2
1 +|z|1/2 , |λ| ≥λ0
(for exampleQ(λ, z) =z+λ(1−z2)−1/4,(λ, z)∈Sθ×Σδ), then Theorem 4.6 yields
W =Lp,%(R+, Lq(Rn)), W−=Lp,%(R+, Hq−1/2(Rn)), V =Lp,%(R+, Hq1(Rn))∩0Hp,%1 (R+, Hq−1/2(Rn)), h
Qµ(∇(%)+ )i
|V ∈L(V,W) withs0:=r0 := 0,F:=Hp,K:=Hq,1< p, q <∞,%≥0, andµ≥λ0(Q,O).
For a special class of symbols which only depend onλand|ξ|the next result can be found in [DSS08, Theorem 3.2]. In particular, the quasi-homogenous coefficients are more concrete in [DSS08] as in our situation. Additionally, we obtain the vector-valued result.
Corollary 4.9. If P ∈SN[K](Sθ×Σnδ),θ > π/2,δ >0, then there exists µ >0 such that Pµ[℘](∇W+)∈LIsom
\
(r,s)∈NV(P)
WF,K,%s0+s,r0+r(Rn+1+ , X), WF,K,%s0,r0 (Rn+1+ , X)
, ℘∈K and the norm ofPµ[℘](∇W+)can be estimated from above by a constant independent of ℘∈K.
Proof. From the N-parabolicity we directly obtain 1/Pµ[℘]∈ H∞(Sθ×Σnδ)for sufficiently largeµ >0 and℘∈K. Theorem 1.19 yields thatPµ[℘](∇W+)is invertible with[Pµ[℘](∇W+)]−1=Pµ−1[℘](∇W+). Now we can apply Theorem 4.7 and Lemma 3.33. We obtain
1 Pµ[℘]
(∇W+)∈L
W, \
(r,s)∈NV(P)
WF,K,%s0+s,r0+r(Rn+1+ , X)
and therefore the assertion follows by Theorem 4.6.
4.2 Mixed order systems on spaces of mixed scales
In this thesis we consider mixed order systems onRn+1+ . We interpret mixed order systems as operators which are defined by the jointH∞-calculus of∇+ = (∂t,∇), cf. Chapter 2. These systems occur in the treatment of parabolic boundary value problems, for example. The intent of this section is to establish conditions which make enables us to determine the mapping properties of such a system on spaces of mixed scales (cf. Definition 2.25). It turns out that N-parabolicity of the determinant of the system is a crucial condition for this purpose.
For other approaches to mixed order systems we refer to [DMV98], [DV02a], [DV02b], [DSS09], [DD11], and [DS11] for example. These references include approaches by pseudo-differential operators and parameter-ellipticity. We also want to refer to the discussion in the introduction of this thesis. Our procedure can mainly be compared with [DSS08] and [DV08].
Definition 4.10 (Douglis-Nirenberg system). Let L ∈ [HP[K](Sθ×Σnδ)]m×m, m ∈ N, θ > π/2 be a matrix of holomorphic functions which are polynomially bounded. The systemL is called a mixed order system in the sense of Douglis-Nirenberg if there exist order functionssj :=Ojrow and tk :=Ocolk , j, k= 1, . . . , m, such thatsj+tk is a monotone upper order function of Lj,k for allj, k= 1, . . . , m, i.e.
the upper order structure of each component splits into rows and columns.
Similar to [Vol01, Definition 3.3], respectively [DV08, Definition 3.2], we introduce the concept of N-parabolic matrices. This useful property enables us to prove that a mixed order system gives rise to an isomorphism on appropriate spaces, cf. Theorem 4.12.
Definition 4.11(N-parabolic mixed order system). LetL ∈[HP[K](Sθ×Σnδ)]m×m,θ > π/2, be a mixed order system in the sense of Douglis-Nirenberg. Then the systemL is called an N-parabolic mixed order system if
(i) detL is N-parabolic with compact parameter, (ii) [O(detL)](γ) =Pm
j=1(sj(γ) +tj(γ))for allγ >0.
Note that our definition of an order function is stricter as in [DV08] because we assume a kind of monotonicity in the definition of increasing/decreasing order functions.
In Chapter 7 we present a selection of N-parabolic mixed order systems, which occur in the treatment of parabolic differential equations. For the existence of suitable order functions we refer to [Vol63], [Vol01], and [DV08]. Presently, it is not clear in which cases there exist increasing or decreasing order functions fulfilling Definition 4.10 and Definition 4.11. In many applications, however, it is transparent how to choose them.
For fixed1< p0, p1<∞we define theLp0 Hp−∞1
realization ofLµ[℘] :=L[℘](µ+·,·),µ≥0, by Lµ[℘](∇(%)+ ) := ((Lµ)j,k[℘](∇(%)+ ))j,k=1,...,m
with℘∈K, cf. Definition 2.48. In particular, we define Lµ[℘](∇+) =Lµ[℘](∇(0)+ ). In general, it is not clear how to choose appropriate spaces H:=Qm
i=1Hi and F:=Qm
i=1Fi such thatLµ[℘](∇+)acts as a bounded operator between them. Here the point of interest is thatLµ[℘](∇+) is even an isomorphism betweenHandF.
Before stating the main result of this section we want to motivate the embedding assumptions (4.16) and (4.17) appearing in Theorem 4.12. Up to now we have only considered the mapping properties of operatorsQ(∇(%)+ )on ground spaces of the form0F%α(R+,Kβ(Rn)). If we want to considerQ(∇(%)+ )as an operator between intersections of spaces with different scales like
0Bp,%s1(R+, Hpr1(Rn))∩0Hp,%s2(R+, Bp,r2(Rn)), s1, s2>0, r1, r2∈R,1< p <∞,
the scales(F1,K1) := (Bp, Hp)and(F0,K0) := (Hp, Bp)have to provide some compatibility attributes.
The next example demonstrates the necessity of this compatibility assumption:
Let N := N({(1,0),(0,2)}) and Q := Φ−1N . ThenQ has the upper strictly negative order function O(γ) =−max{γ,2}due to Proposition 4.3. Moreover, we define the spaces
H := H(1)∩H(0):=0Bp,%s (R+, Lp(Rn))∩Lp,%(R+, Bp2s(Rn)),
F := F(1)∩F(0):=0Bp,%s+1(R+, Lp(Rn))∩Lp,%(R+, B2(s+1)p (Rn)), s >0.
Considering the order structure of Qit seems to be natural that we want to have that Q(∇(%)+ ) acts as an isomorphism betweenHandF. According to Proposition 4.3 we already have
hQ(∇(%)+ )i
|H(1)
∈ LIsom
H(1),F(1)∩0Bp,%s (R+, Hp2(Rn)) , hQ(∇(%)+ )i
|H(0)
∈ LIsom
H(0),0Hp,%1 (R+, Bp2s(Rn))∩F(0) . So we deduce
hQ(∇(%)+ )i
|H∈LIsom H,F∩0Hp,%1 (R+, Bp2s(Rn))∩0Bp,%s (R+, Hp2(Rn)) . Therefore, we need the embedding
F,→0Hp,%1 (R+, Bp2s(Rn))∩0Bp,%s (R+, Hp2(Rn))
4.2: Mixed order systems on spaces of mixed scales 89
to deduceh
Q(∇(%)+ )i
|H∈LIsom(H,F).
In the next section we state some embeddings in Propositions 4.17 and 4.18, which help us to formulate conditions on the scales (F`,K`) such that these compatibilities are given. In Proposition 4.14 and Proposition 4.21 we then state some sufficient conditions on these scales. For further information on the assumptions of Theorem 4.12 see in Remark 4.13. In the following we state our main result on N-parabolic mixed order systems.
Theorem 4.12(Main Theorem on N-parabolic mixed order systems). LetX be a Banach space of class HT with property (α). Let L ∈ [HP[K](Sθ×Σnδ)]m×m, θ > π/2, δ > 0, be an N-parabolic mixed order system. Using the same notation as in Definition 4.10 and Definition 2.25 we define for i, j= 1, . . . mthe spaces
i=1Fi. In particular, we have the boundedness of the mappings K→L(H,F), ℘7→[Lµ[℘](∇(%)+ )]|H, K→L(F,H), ℘7→([Lµ[℘](∇(%)+ )]|H)−1.
Proof. (I) First, we consider the H∞-calculus of the inverse matrix symbol. According to
According to the definition of a Douglis-Nirenberg system the function sk +tp is an upper order function ofLk,p for everyk, p∈ {1, . . . , m}. LetSij be the set of all bijective functions
In the following we have to distinguish two cases:
(a1) LetObe a decreasing order function. For `∈ {0, . . . , M} we define
4.2: Mixed order systems on spaces of mixed scales 91 (b1) LetObe an increasing order function. If we define
s0(`) := s0`+m`(ti)≥αde(sj+ti), uniformly in℘∈K. Here assumption (4.17) comes into play. Due to (4.17) it is obvious that
M
\
`=0
V`(F) =Fj∩Fij=Fj. From (4.18) we deduce for largeµ
h
(Lµ−1)ij[℘](∇(%)+ )i
|Fj
∈L(Fj,Hi).
uniformly in℘∈K.
In both cases we derive for largeµh
Lµ−1[℘](∇(%)+ )i
(b2) LetO1 be a decreasing order function. Let
s(`) := m`(sj) +m`(ti), r(`) := b`(sj) +b`(ti),
O`(γ) := s(`)·γ+r(`)≥ O1(γ), γ >0.
ThenO`is also a decreasing upper order function ofLji, cf. Lemma 3.38 (i). Now Theorem 4.7 withs0(`),r0(`), andW`as in (b1) yields
Altogether we derive for largeµ hLµ−1[℘](∇(%)+ )i
uniformly in℘∈K. Therefore, we can compose these operators. Using Theorem 1.19 we get Lµ−1[℘](∇(%)+ )Lµ[℘](∇(%)+ )u =
|F, which ends the proof.
Remark 4.13. (i) In the next section, especially in Proposition 4.14 and Proposition 4.21, we will give sufficient conditions for the assumptions (4.17) and (4.16) in case of a “tame choice” of scales (F`,K`)`=0,...,M.
(ii) Assumption (4.15) avoids the appearance of negative regularities in the time domain.
(iii) If we only have positive and negative order functions sj +ti in the situation of Theorem 4.12, then the assumptions become simpler. If we additionally have a “tame choice” of scales (F`,K`), the assumptions (4.17) and (4.16) can also be dropped. This special case is sufficient for many applications and for the sake of clearness we will state this in Corollary 4.22.
(iv) In the situation of Theorem 4.12 we can eliminate the translation µ in the time co-variable by exponential weights. For this we define the spaces
Hi(σ) :=
4.3: Embedding conditions (4.16) and (4.17) 93