Embedding conditions (4.16) and (4.17)

[L[℘](∇^(%)_+,µ)]_|H(%)= [Lµ[℘](∇^(%)₊ )]_|H(%)∈LIsom(H(%),F(%)), µ≥µ0. According to Theorem 4.12 and Theorem 2.47 (iii) we then derive

hL[℘](∇^(%+µ)₊ )i

|H(%+µ)=h

Mµ⁻¹Lµ[℘](∇^(%)₊ )Mµ

i

|H(%+µ). Thus, we obtain

hL[℘](∇^(%+µ)₊ )i

|H(%+µ) ∈ LIsom(H(%+µ),F(%+µ)), hL[℘](∇^(%+µ)₊ )i−1

|H(%+µ)

= h

Mµ⁻¹(Lµ⁻¹)[℘](∇^(%)₊ )Mµ

i

|F(%+µ)

due to Remark 2.21. In generalL⁻¹[℘](∇^(%+µ)₊ )is not well-defined because we only have detL[℘](λ, z)6= 0

for sufficiently large|λ|.

4.3 Embedding conditions (4.16) and (4.17)

The next pages are devoted to illuminate the requirements (4.16) and (4.17). In most cases this is only a condition on the position of the regularity tuples(ηji(`, κ, s), σji(`, κ, s))and(ηji(`, κ, t), σji(`, κ, t))for i, j= 1, . . . , mand`, κ= 0, . . . , M.

First, we investigate (4.16) and (4.17) for non-mixed scales. The next proposition clarifies this situa-tion.

Proposition 4.14 (Embedding condition I). If (F`,K`) = (F,K) for all ` ∈ {0, . . . , M} in the situation of Theorem 4.12, then the assumptions (4.16) and (4.17) are always fulfilled.

Proof. Lets_j+t_i be increasing. We defineO1(γ) :=O_Fi(γ) +β₁, O2(γ) := (s_j+t_i)(γ) +αγ+β₂ with β₁:=β_in(O_Fi),α:=α_in(s_j+t_i), andβ₂:=β_in(s_j+t_i). Note that we have

O_Hi(γ) = O_Fj(γ) + (sj+ti)(γ)

= (O1+O2)(γ)−αγ−(β1+β2).

For fixed`, κ∈ {0, . . . , M} we define

σ := σ_ji(`, κ, t) =m<sub>`</sub>(O_Fj) +m_κ(s_j+t_i), η := ηji(`, κ, t) =b`(O_Fj) +bκ(sj+ti).

Thenσ+α=m`(O1) +mκ(O2)and η+β1+β2 =b`(O1) +bκ(O2) hold. Lemma 3.41 directly yields (η+β1+β2, σ+α)∈N(O1+O2). Using Lemma 4.5 we get

\

`=0,...,M

W_F,K,%^{m`(O1+O2),b`(O1+O2)}(Rⁿ⁺¹+ , X),→W_F,K,%^{σ+α,η+β1+β2}(Rⁿ⁺¹+ , X).

Due to (4.15) we havem`(O1+O2)>0 for `= 0, . . . , M if F =Bp₀,q₀. LetN⁰ :=N({(α, β1+β2)}).

Then the isomorphismΦ_N⁰(∇^(%)₊ )yieldsHi,→W_F,K,%^σ,η (Rⁿ⁺¹+ , X)and thereforeHi,→Hij. The embeddingF^j ,→F^ij follows in the same way.

In contrast to Proposition 4.14 we are also interested in spaces like

0B_p,%^s (R⁺, Lp(Rⁿ))∩Lp,%(R⁺, B^2s_p (Rⁿ)), s >0, p∈(1,∞)

to handle boundary value problems where the mixed order system acts on the trace spaces. To get analog results on the assumptions (4.16) and (4.17) as in Proposition 4.14 we have to provide analog results as in Lemma 4.5 (i) of the form

W_F,K,%^s0+s,r0(Rⁿ⁺¹+ , X)∩W_L,M,%^s0,r0+r(Rⁿ⁺¹+ , X),→W_J^s0_,I,%^{+σs,r0+(1−σ)r}(Rⁿ⁺¹+ , X).

To obtain further embeddings as in Lemma 4.5 (i) we need the compatibility of the real interpolation method with the intersection of Banach spaces. There are several results for the so-called ’intersection problem’ if the underlying Banach spaces are quasi-linearizable, see in [Pee71], [Pee74], [Tri78]. The next result goes back to P. Grisvard and can be found in [Gri72]. J. Peetre showed in [Pee74] that Grisvard’s result is a special case of the more general result in [Pee74]. P. Grisvard’s result on the intersection problem works without the concrete assumption of quasi-linearizability.

Remark 4.15 (Intersection problem [Gri72]). Let Z andY be Banach spaces withY ,→Z and let A: D(A)⊆Z→Z

be a linear closed unbounded operator. Let the following conditions be fulfilled:

(i) It holds that(−∞,0)⊆ρ(A)and there exists aC₀>0 such that kt(t−A)⁻¹uk_Z ≤C₀kuk_Z, t <0, u∈Z.

(ii) The resolvent(t−A)⁻¹ isY-invariant for all t <0 and there existsC₁>0 such that kt(t−A)⁻¹uk_Y ≤C₁kuk_Y, t <0, u∈Y.

Then we obtain

(Z, Y ∩D(A))θ,p= (Z, Y)θ,p∩(Z, D(A))θ,p

with equivalent norms for allθ∈(0,1)and p∈(1,∞).

Lemma 4.16. Let X be a Banach space of class HT with property (α). For all s⁰, s, r ≥0, r⁰ ∈ R, p0, p1∈(1,∞), and%≥0we have

W_F,K,%^s0,r0 (Rⁿ⁺¹+ , X), W_F,K,%^s0+s,r0(Rⁿ⁺¹+ , X)∩W_F,K,%^s0,r0+r(Rⁿ⁺¹+ , X)

θ,p

=

W_F,K,%^s0,r0 (Rⁿ⁺¹+ , X), W_F,K,%^s0+s,r0(Rⁿ⁺¹+ , X)

θ,p∩

W_F,K,%^s0,r0 (Rⁿ⁺¹+ , X), W_F,K,%^s0,r0+r(Rⁿ⁺¹+ , X)

θ,p

where(F,K) = (Hp₀, Hp₁),θ∈(0,1), andp∈(1,∞).

Proof. We use the abbreviationF^α(K^β) :=W_F,K,%^α,β (Rⁿ⁺¹+ , X). Fors= 0orr= 0the assertion is trivial.

Lets, r >0,Z :=H_p^s0

0(H_p^r0

1),Y :=H_p^s0+s

0 (H_p^r0

1), andA:= Λ_r(∇^Z₊). According to Proposition 2.49 we have D(A) =H_p^s₀⁰(H_p^r₁^0+r)with equivalent norms. In the following we show that we can apply Remark 4.15.

(i) There exists δ0(r) > 0 and τ(r) > 0 such that for all t < 0, δ ≤ δ0(r), and z ∈ Σⁿ_δ we have t−Λr(z)∈ −(τ(r)+Sπ/2). In particular, this yieldsϕt:= (t−Λr)⁻¹∈H^∞(Σⁿ_δ). With Theorem 1.19 we then obtaint∈ρ(A)and(t−A)⁻¹=ϕ_t(∇^Z₊)for allt <0. Using a homogeneity argument we derive the boundedness of the function

χ(t, z) :=tϕ_t(z), (t, z)∈(−∞,0)×Σⁿ_δ.

Therefore, there exists C > 0such thatkχ(t,·)k_∞≤C for allt <0. We get kχ(t,∇^Z₊)k_L(Z)≤C for all t < 0. Altogether we obtain kt(t−A)⁻¹uk_Z =kχ(t,∇^Z₊)uk_Z ≤Ckuk_Z for all t < 0 and u∈Z.

4.3: Embedding conditions (4.16) and (4.17) 95 (ii) The assertion of (i) is also true for the operatorB := Λr(∇^Y₊)and due to the pointwise definition of the natural extension, cf. Definition 2.40, we easily get(t−A)⁻¹_|Y = (t−B)⁻¹∈L(Y). Altogether we getkt(t−A)⁻¹ukY ≤CkukY for allt <0and u∈Y.

Special cases of the next embedding result can be found in [EPS03, Remark 5.3] and [MS11, Proposi-tion 3.2], for example. However, the authors of [MS11] consider spaces with a type of polynomial weight in the time domain. In [MS11, Proposition 3.2] the authors have the additional restriction s ∈ (0,2), which appears because they consider powers of the time derivative(1−∂t)^s. These powers cannot be controlled for larges. Some arguments of the proofs given there are related to arguments used here but we use the abstract interpolation result of Remark 4.15, respectively Lemma 4.16, instead of powers of the time derivative. With this approach we can drop the restriction ons.

Proposition 4.17 (Embedding result II). Let X be a Banach space of class HT with property(α).

For alls⁰≥0,r⁰∈R,s, r >0,%≥0, andσ∈(0,1) we have the embedding Proposi-tion 2.24 and Remark 2.17 (iii) we obtain

h (cf. Remark A.9) and the reiteration theorem (cf. Theorem A.10) we get

h

Using Lemma 4.16, Remark 2.17 (iii), and Proposition 2.24 we get

With these embeddings and (4.19) we obtain which yields the first claimed embedding.

The second embedding can be obtained in the same way by using (4.19) and the embeddings H_p^s0+s−ε

It attracts attention that we have a linking between the outer scale L = Hp₀ and the inner scale M=Bp₁p₀ due to the occurrence of p0 in Bp₁p₀. This coupling always appears if we apply Proposi-tion 2.24.

Proposition 4.18 (Embedding result III). Let X be a Banach space of classHT with property (α).

For alls⁰ >0,r⁰∈R,s, r≥0,%≥0,1< p₀, p₁<∞, andσ∈(0,1) we have the embedding Using (4.21), Proposition 2.24, and the embeddings

H_p^s₀^0+s−ε(H_p^r₁⁰)∩H_p^s₀^0−ε(H_p^r₁^0+r) ,→ H_p^s₀^0+σs(H_p^r₁^{0+(1−σ)r−δ}), H_p^s₀^0+s+ε(H_p^r₁⁰)∩H_p^s0₀^+ε(H_p^r₁^0+r) ,→ H_p^s₀^0+σs(H_p^r₁^{0+(1−σ)r+δ})

4.3: Embedding conditions (4.16) and (4.17) 97 from Lemma 4.5 (i) we obtain

B_p^s0+s

0 (H_p^r0)∩B_p^s0

0(H_p^r0+r

1 ) ,→

H_p^s0+σs

0 (H_p^{r0+(1−σ)r−δ}

1 ), H_p^s0+σs

0 (H_p^{r0+(1−σ)r+δ}

1 )

1/2,p0

= H_p^s0₀^+σs(B^r_p⁰₁^+(1−σ)r_p0 ), which proves the first assertion.

(ii) We omit the proof of the second assertion because it follows the same idea as in (i).

The embeddings which we have proven in this section are very useful for the application of the main result on mixed order systems but they are also of their own interest. Therefore, we also want to state these results for spaces with more general domains in the space variable.

Remark 4.19 (Further embeddings). Let X be a Banach space of class HT with property (α). Let Ω⊆ Rⁿ be sufficiently smooth (e.g. Ω satisfies a strong local Lipschitz condition) such that there exist bounded extension and restriction operators in the Bessel-potential and Besov scale. All embeddings in Lemma 4.5, Proposition 4.17, and Proposition 4.18 remain valid if we replace Rⁿ by Ω, i.e. replace W_F,K,%^σ,η (R+×Rⁿ, X)by the spaceW_F,K,%^σ,η (R+×Ω, X). This can be obtained by Lemma 2.7 and Lemma 2.4.

Before we give a sufficient condition for the embeddings (4.16) and (4.17) we introduce a useful definition for this context.

Definition 4.20(Admissible scale). Let O1 andO2 be positive order functions such that O1− O2 is a monotone order function. Let the scale

(F`,K`)∈ {(Hp₀, Bp₁p₀),(Bp₀, Hp₁)}, `= 0, . . . , M be given such that there existτ ∈ {0, . . . , M−1} with

(F`,K`) = (H_p0, B_p1p0), `∈ {0, . . . , τ}, (F`,K`) = (Bp<sub>1</sub>p<sub>0</sub>, Hp<sub>0</sub>), `∈ {τ+ 1, . . . , M}.

The scale(F`,K`)`=0,...,M is then called (O1,O2)-admissible if we have

(b_τ(O₂), m_τ(O₂)) 6= (b_τ+1(O₂), m_τ+1(O₂)), if O₁− O₂ is increasing, (bτ(O1), mτ(O1)) 6= (bτ+1(O1), mτ+1(O1)), if O1− O2 is decreasing.

Note that this definition is also meaningful ifO1− O2 has trivial index, i.e. there exists α, β∈ Rsuch that(O1− O2)(γ) =αγ+β for all γ >0 and thereforeO1− O2 is increasing as well as decreasing.

Proposition 4.21 (Embedding condition II). Consider the situation of Theorem 4.12 with (F`,K`)∈ {(Hp₀, Bp₁p₀),(Bp₀, Hp₁)}, `= 0, . . . , M.

If the scale(F`,K`)`=0,...,M is(O_Hi,O_Fj)-admissible for alli, j= 0, . . . , M, then the embedding conditions (4.16) and (4.17) are fulfilled. In this sense the scales(Hp₀, Bp₁p₀)and(Bp₀, Hp₁)are compatible.

Proof. Let sj+ti be increasing. As in the proof of Proposition 4.14 we set O1(γ) := O_Fi(γ) +β1 and O2(γ) := (sj+ti)(γ) +αγ+β2whereβ1:=βin(O_Fi),α:=αin(sj+ti), andβ2:=βin(sj+ti). Note that we haveO_Hi(γ) =O_Fj(γ) + (sj+ti)(γ) = (O1+O2)(γ)−αγ−(β1+β2). For fixed`, κ∈ {0, . . . , M}we define

σ := σji(`, κ, t) =m`(O_Fj) +mκ(sj+ti), η := ηji(`, κ, t) =b`(O_Fj) +bκ(sj+ti)

and getσ+α=m_{`</sub>(O<sub>1</sub>) +m<sub>κ</sub>(O<sub>2</sub>)andη+β<sub>1</sub>+β<sub>2</sub>=b<sub>`}(O₁) +b_κ(O₂). Lemma 3.41 already yields that (η+β₁+β₂, σ+α)∈ N(O₁+O₂). Compared to the proof of Proposition 4.14 we have to determine the position of the tuple (η+β₁+β₂, σ+α) more precisely. Here we can benefit from the results of Lemma 3.42. LetΓ = Γ(O1+O2)have the same meaning as in Lemma 3.42 and defineN :=N(O1+O2).

(i) Let(η+β1+β2, σ+α)∈N\Γ. This case is clear by an application of Lemma 4.5 (i), Proposition 4.17, Proposition 4.18, and the trivial embeddings

0H_p^s

0(R+, Y), ₀B_p^s

0(R+, Y) ,→ 0H_p^t

0(R+, Y)∩0B_p^t

0(R+, Y), H^s_p1(Rⁿ, X), B^s_p1p0(Rⁿ, X) ,→ H^t_p1(Rⁿ, X)∩B^t_p1p0(Rⁿ, X) fors > t.

(ii) Let(η+β₁+β₂, σ+α)∈Γ\N_V andς ∈ {0, . . . , I(O₁)} with

i_ς(O₁) =i_ς(O_Fj)≤` < i_ς+1(O₁) =i_ς+1(O_Fj).

Due to Lemma 3.42 we may assume without loss of generality µ₁6= 0, i_µ1(O2) =i_ς(O1) =:j,

andi_µ1−1(O2)≤κ < i_µ1(O2). Then(b_j(O1+O2), m_j(O1+O2))and(b_j−1(O1+O2), m_j−1(O1+O2)) are the endpoints of the edge including (η+β₁+β₂, σ+α). The assumption on the scales then yields

M

\

µ=0

W_F^{mµ(O1+O2), bµ(O1+O2)}

µ,Kµ,% (Rⁿ⁺¹+ , X)

,→ W_F^{mj(O1+O2), bj(O1+O2)}

j,Kj,% (Rⁿ⁺¹+ , X)∩W_F^{mj−1(O1+O2),bj−1(O1+O2)}

j−1,Kj−1,% (Rⁿ⁺¹+ , X) ,→ W_F^{σ+α, η+β1+β2}

`,K`,% (Rⁿ⁺¹+ , X)

due to Lemma 4.5 (i), Proposition 4.17, Proposition 4.18, and (F`,K`) = (Fj,Kj). Note that τ /∈(iς(O1), iς+1(O1))forτas in Definition 4.20. DefiningN⁰:=N({(α, β1+β2)})the isomorphism Φ_N⁰(∇^(%)₊ )then yieldsHi,→W_F^σ,η

`,K<sub>`</sub>,%(Rⁿ⁺¹+ , X).

(iii) Let(η+β1+β2, σ+α)∈NV andς ∈ {0, . . . , I(O1)}with

iς(O1) =iς(O_Fj)≤` < iς+1(O1) =iς+1(O_Fj).

According to Lemma 3.42 we then have iµ₁(O2) ≤ κ < iµ₂(O2) where iµ₁(O2) and iµ₂(O2) are given as in Lemma 3.42. Due to Lemma 3.42 we derive

(η+β1+β2, σ+α) = (bj(O1+O2), mj(O1+O2))∈NV

where

j:=







i_ς(O1), κ < i_ς(O1),

κ, iς(O1)≤κ < iς+1(O1), iς+1(O1)−1, κ≥iς+1(O1).

Due to our assumptions we have(F`,K`) = (Fj,Kj). Note thatτ /∈(iς(O1), iς+1(O1)). This yields

M

\

µ=0

W_F^{mµ(O1+O2), bµ(O1+O2)}

µ,Kµ,% (Rⁿ⁺¹+ , X) ,→ W_F^{mj(O1+O2), bj(O1+O2)}

j,Kj,% (Rⁿ⁺¹+ , X)

= W_F^{σ+α, η+β1+β2}

`,K`,% (Rⁿ⁺¹+ , X).

As in part (ii) the operatorΦ_N⁰(∇^(%)₊ )then showsHi,→W_F,K,%^σ,η (Rⁿ⁺¹+ , X).

The embeddingsFj ,→Fij can be proved in exactly the same way if s_j+t_i is decreasing.

At last we want to state a corollary in which we present a condensed version of Theorem 4.12 which is sufficient for many applications.

4.3: Embedding conditions (4.16) and (4.17) 99 Corollary 4.22. LetXbe a Banach space of classHT with property(α). LetL ∈[HP[K](Sθ×Σⁿ_δ)]^m×m, θ > π/2, be an N-parabolic mixed order system such that for eachi, j= 1, . . . , mthe order functions_j+t_i is positive or negative. Let%≥0,s⁰_{`</sub>≥0,r<sub>`}⁰ ∈R,`= 0, . . . , M, such that

O_Hi(γ) := max

` {[s<sup>0</sup><sub>`</sub>+m_{`</sub>(t<sub>i</sub>)]γ+r<sub>`}⁰+b_`(t_i)}, γ >0, O_Fj(γ) := max

` {[s<sup>0</sup><sub>`</sub>−m_{`</sub>(s<sub>j</sub>)]γ+r<sub>`}⁰−b_`(s_j)}, γ >0, i, j= 1, . . . m are positive order functions. Furthermore, let the scale

(F`,K`)∈ {(Bp, H_p),(H_p, B_p)}, 1< p <∞, `= 0, . . . , M be(O_Hi,O_Fj)-admissible for all i, j= 1, . . . mand let

s⁰<sub>`</sub>>max{max{−m`(ti), m`(si)}:i= 1, . . . , m} (4.22) for all`∈ {0, . . . , τ}whereτis taken from Definition 4.20. Using the same notation as in Definition 2.25 we define fori, j= 1, . . . mthe spaces

Hi:=

M

\

`=0

W^s

0

`+m`(ti), r⁰<sub>`</sub>+b`(ti)

F_{`</sub>,K<sub>`},% (Rⁿ⁺¹+ , X), Fj:=

M

\

`=0

W^s

0

`−m`(sj), r⁰<sub>`</sub>−b`(sj)

F_{`</sub>,K<sub>`},% (Rⁿ⁺¹+ , X).

Then there existsµ0>0 such that for all µ≥µ0

[Lµ[℘](∇^(%)₊ )]_|H∈LIsom(H,F), ([Lµ[℘](∇^(%)₊ )]_|H)⁻¹= [Lµ⁻¹[℘](∇^(%)₊ )]_|F whereH:=Qm

i=1Hi andF:=Qm

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. The order functionssj+ti are positive or negative so we haveαin(sj+ti) = 0orαde(sj+ti) = 0, respectively. This yieldsδi,1 = δj,2 = 0 for all i, j = 1, . . . , m. The embeddings (4.16) and (4.17) are fulfilled due to Proposition 4.21. Assumption (4.22) ensures that the time-regularity is positive for the Besov scales. Hence the assertion follows from Theorem 4.12.

Remark 4.23(Possible extensions). It should be possible to extend the theory to general order func-tions such that we can drop the assumption that the order funcfunc-tions(sj+ti)ij have to be monotone. In this case we also have to introduce more general weight functions, i.e. we have to allow arbitrary quotients of the formΞν1/Ξν2 in Definition 3.25 (vi). Up to now, from the view of applications, there is no reason to generalize the theory in this direction.

Chapter5

Some results on Triebel-Lizorkin spaces

In this chapter we want to generalize the results of Chapter 4 on mixed order systems to the scale of Triebel-Lizorkin spaces. Triebel-Lizorkin spaces naturally appear in parabolic Lp-Lq-theory as the trace spaces of the solution. This fact was independently noted in [Ber87a], [Wei05], and [DHP07]. In Chapter 7 we will show that we are able to establish anLp-Lq-theory for free boundary value problems by using mixed order systems on Triebel-Lizorkin spaces. We explicitly consider the Lp-Lq two-phase Stefan problem with Gibbs-Thomson correction in Section 7.8.

Our philosophy to treat vector-valued Triebel-Lizorkin spaces is to present them as a complex inter-polation spaceF_pq^s = [H_p^s₀, B^s_p1]θ. This representation holds in the scalar-valued case but fails in general in the vector-valued case. Therefore we prove a related representation by ourselves in Corollary 5.11, which reads as follows

F_pq^s(R^m,[H, X]θ) = [H_p^s₀(R^m, H), B_p^s₁(R^m, X)]θ (5.1) for a Banach spaceX and a Hilbert space H. Using (5.1) we can transfer the results in Chapter 2 and Chapter 4 to vector-valued Triebel-Lizorkin spaces. So we explicitly derive that∇+ admits a bounded jointH^∞-calculus on vector-valued Triebel-Lizorkin spaces. Moreover, we get the analog result of Theo-rem 4.12 for vector-valued Triebel-Lizorkin spaces.

To establish compatibility embeddings as in Proposition 4.17 we need the concepts of scalar-valued anisotropic Triebel-Lizorkin spaces with mixed norms. Literature concerning these spaces is rare and therefore we only want to restrict ourselves on scalar-valued mixed order systems.

In the next sections we give the basic definition of several types of Triebel-Lizorkin spaces and state some results on them. The presented approach to Triebel-Lizorkin spaces on half spaces is rather the same as in Section 2.2, where we have used the concepts of retractions.

5.1 Vector-valued Triebel-Lizorkin spaces and interpolation

Here we follow the work of H. Triebel [Tri97], respectively H.J. Schmeißer and W. Sickel [SS05], for the definition of vector-valued Triebel-Lizorkin spaces by the so-called decomposition method.

Definition 5.1([Tri97, Definition 15.4], [SS05, Definition 1]). Letψ∈D(R^m)withψ(x)∈[0,1],x∈R^m, andψ(x) = 1if |x| ≤ A, and ψ(x) = 0 if |x| > B for0 < A < B <∞. We define the smooth dyadic decomposition of unity by

ϕ0:=ψ, ϕ1(x) :=ϕ0(x/2)−ϕ0(x), ϕj(x) :=ϕ1(2^−j+1x), j≥2.

For s ≥ 0, 1 < p, q < ∞, and any Banach space X we define the X-valued Triebel-Lizorkin space F_p,q^s (R^m, X)as the set of allu∈S⁰(R^m, X)with

kuk_Fs

p,q(R^m,X):=

(2^sjF⁻¹[ϕjFf])j∈N0

_`

q(X)

_L

p(R^m)<∞.

101

Remark 5.2. In [RS96, Proposition 6] one can find a characterization of vector-valued Triebel-Lizorkin spaces by differences.

It is astonishing that the Banach space valued Triebel-Lizorkin scale does not cover the Bessel-potential scale in general. This statement is based on some results in Littlewood-Paley theory and will be clarified with the next remark.

Remark 5.3. (i) LetX be a Banach space ands≥0,1< p <∞. We then have H_p^s(R^m, X) =F_p,2^s (R^m, X)

if and only ifX can be renormed as a Hilbert space. This result can be found in [RdFT87, p. 283], [SS05, Remark 5], [Ama09, Remark 4.5.3], and the references given therein.

(ii) For all Banach spacesX,s >0, and1< p <∞ we have B_p^s(R^m, X) =F_p,p^s (R^m, X).

This observation is obvious due to the parallel definitions of Besov and Triebel-Lizorkin spaces (cf.

Definition 2.16) and Fubini’s theorem.

In Section 2.2 we defined all Besov and Bessel-potential spaces on half spaces by retractions. In the next lines we want to generalize these concepts to vector-valued Triebel-Lizorkin spaces. As in Section 2.2 we use the mappingsr⁺,r⁺₀,e⁺, ande⁺₀, which were introduced in Definitions 2.11 and 2.13.

Definition 5.4. LetX be a Banach space of classHT ands≥0,1< p, q <∞. Following the approach of [Ama09] for Bessel-potential and Besov spaces, we defineX-valued Triebel-Lizorkin spaces on the half space by

F_p,q^s (R^m+, X) := r⁺ F_p,q^s (R^m, X) ,

0F_p,q^s (R^m+, X) := r⁺₀ F_p,q^s (R^m, X) with the quotient norms

kukF_p,q^s (R^m₊,X):= inf{kfkF_p,q^s (R^m,X): f ∈F_p,q^s (R^m, X)withr⁺(f) =u}, kuk₀F_p,q^s (R^m₊,X):= inf{kfkF_p,q^s (R^m,X):f ∈F_p,q^s (R^m, X)with r⁺₀(f) =u}.

The operatorsr⁺ andr⁺₀ were introduced in Definitions 2.11 and 2.13.

Similar to Definition 2.20 we define Triebel-Lizorkin spaces with exponential weights.

Definition 5.5 (Triebel-Lizorkin spaces with exponential weights). LetX be a Banach space of classHT and1< p, q <∞. Fors >0 and exponential weight%≥0 we define

0F_p,q,%^s (R+, X) := M−% 0F_p,q^s (R+, X) with norm

kfk_0Fs

p,q,%(R+,X) := kM%fk_0Fs

p,q(R+,X), f ∈0F_p,q,%^s (R+, X).

Remark 5.6 (Properties of Triebel-Lizorkin spaces with exponential weights). Let X be a Banach space of classHT,s >0,%≥0, and1< p, q <∞. From the definitions we already derive

[M%]_|0Fs

p,q,%(R+,X)∈LIsom 0F_p,q,%^s (R+, X),0F_p,q^s (R+, X) . Proposition 5.7. For each Banach space X of classHT we have the embeddings

F_pq^s0+s(R^m, X) ,→ H_p^s0(R^m, X),

0F_pq^s0+s(R^m+, X) ,→ 0H_p^s0(R^m+, X),

0F_pq,%^s0+s(R+, X) ,→ 0H_p,%^s0 (R+, X) fors⁰≥0,s >0,1< p, q <∞, and%≥0.

5.1: Vector-valued Triebel-Lizorkin spaces and interpolation 103 Proof. Using [SS01, Proposition 2 (ii)] we derive

F_pq^s0+s(R^m, X),→B_p,max{p,q}^s0+s (R^m, X).

Employing (2.6) there existsθ∈(0,1) such that B_p,max{p,q}^s0+s (R^m, X) =

H_p^s0(R^m, X), H_p^s0+s+1(R^m, X)

θ,max{p,q},→H_p^s0(R^m, X).

An usual retraction argument, cf. Lemma 2.7, and the exponential weight isomorphisms then also yield the other claimed embeddings.

Remark 5.8. (i) The vector-valued spaceF_p,q^s (R^m, X) (s >0) is a retract ofL_p(R^m, `^s_q(X))where

`^s_q(X) :=

(

(xk)_k∈N0 ⊆X:

∞

X

k=0

2^skq|xk|^q <∞ )

, k(xk)_k∈N0k_`s q(X):=

∞

X

k=0

2^skq|xk|^q

!1/q

. This is well-known in the scalar case (cf. [Tri78, Theorem 2.4.2]) and the proof can be lifted to the vector-valued case (cf. [SS01, Proof of Proposition 12]). See also [Ama00, Theorem 3.1] for a similar proof and a corresponding result for vector-valued Besov spaces.

(ii) For an arbitrary interpolation couple {X0, X1},s0, s1>0,1< p0, p1, q0, q1<∞, andθ∈(0,1)we obtain

F_p^s₀⁰_,q0(R^m, X0), F_p^s₁¹_,q1(R^m, X1)

θ = F_p^s_θ^θ_,qθ(R^m,[X0, X1]θ), (5.2) with1/pθ= (1−θ)/p0+θ/p1,1/qθ= (1−θ)/q0+θ/q1, andsθ= (1−θ)s0+θs1. The equality in (5.2) immediately follows from (i), Lemma 2.5, Theorem A.11, and Theorem A.12.

Remark 5.9. Fors >0,1< p, q <∞, and any Banach spaceX we have

D(R+, X),→^d 0F_pq^s(R+, X). (5.3) The embedding S(R, X) ,→^d F_pq^s(R, X) can be proved by a retraction argument similar to the proof for Besov spaces in [Ama09, p. 52]. Another retraction argument then yields0S(R+, X),→^d 0F_pq^s(R+, X).

UsingD(R, X),→^d S(R, X)(cf. [Ama03, Lemma 4.1.4]) we obtain (5.3).

The next proposition is our starting-point for a transfer of the results in Chapter 2 and Chapter 4 to the Triebel-Lizorkin scale. More precisely, it allows to transfer results by complex interpolation.

Proposition 5.10. Let {X, H} be an interpolation couple whereX is a Banach space of classHT and H is a Hilbert space. Let1< p <∞,s >0,%≥0, and

q∈ 2p

p+ 1,2p

(5.4) (cf. Figure 5.1). Then there exist1 < p0, p1<∞and θ∈(0,1) such that1/p= (1−θ)/p0+θ/p1 and 1/q= (1−θ)/2 +θ/p1. With this we obtain

F_p,q^s (R^m,[H, X]θ) =

H_p^s₀⁰(R^m, H), B^s_p¹₁(R^m, X)

θ, (5.5)

F_p,q^s (R^m+,[H, X]θ) =

H_p^s₀⁰(R^m+, H), B^s_p¹₁(R^m+, X)

θ, (5.6)

0F_p,q^s (R^m+,[H, X]θ) =

0H_p^s₀⁰(R^m+, H),0B_p^s₁¹(R^m+, X)

θ, (5.7)

0F_p,q,%^s (R+,[H, X]_θ) =

0H_p^s0

0,%(R+, H),₀B^s_p¹

1,%(R+, X)

θ (5.8)

for alls₀, s₁>0 withs= (1−θ)s₀+θs₁.

1

Figure 5.1: Illustration of the set of tuples(p, q)satisfying (5.4)

Proof. (I) The spaceH is a Hilbert space and therefore we obtain for allp0, p1∈(1,∞)ands0, s1>0 Using Remark 5.8 (ii) we directly obtain

H_p^s0

(II) According to Remark 2.19 we have the retraction respectively coretraction

r⁺ ∈ L(H_p^s₀⁰(R^m, H), H_p^s₀⁰(R^m+, H))∩L(B_p^s1₁(R^m, X), B_p^s1₁(R^m+, X)), e⁺ ∈ L(H_p^s₀⁰(R^m+, H), H_p^s₀⁰(R^m, H))∩L(B_p^s1₁(R^m+, X), B_p^s1₁(R^m, X)).

Thus, (5.5) and Lemma 2.5 yield

F_p,q^s (R^m+,[H, X]_θ) = r⁺F_p,q^s (R^m,[H, X]_θ) of (5.8) then follows from Proposition 5.6, Lemma 2.6, and (5.7).

5.1: Vector-valued Triebel-Lizorkin spaces and interpolation 105 The next corollary is very useful because it gives the possibility to obtain the Bessel-valued Triebel-Lizorkin spaces by complex interpolation of the tame pure Bessel-potential scale and the Bessel-valued Besov scale. If (5.4) is fulfilled, then we can lift results to the Bessel-valued Triebel-Lizorkin scale by simple interpolation arguments.

Corollary 5.11. Let 1< p <∞,s >0,r∈R, and%≥0. Then for all q∈

2p p+ 1,2p

there existθ∈(0,1) and1< p0, p1<∞such that 1/p= (1−θ)/p0+θ/p1 and1/q= (1−θ)/2 +θ/p1. With this we obtain

F_p,q^s (R^m, H_q^r(Rⁿ)) =

H_p^s₀⁰(R^m, H₂^r(Rⁿ)), B^s_p¹₁(R^m, H_p^r₁(Rⁿ))

θ, (5.9)

F_p,q^s (R^m+, H_q^r(Rⁿ)) = H_p^s0

0(R^m+, H₂^r(Rⁿ)), B^s_p¹

1(R^m+, H_p^r

1(Rⁿ))

θ, (5.10)

0F_p,q^s (R^m+, H_q^r(Rⁿ)) =

0H_p^s0

0(R^m+, H₂^r(Rⁿ)),₀B_p^s1

1(R^m+, H_p^r

1(Rⁿ))

θ, (5.11)

0F_p,q,%^s (R+, H_q^r(Rⁿ)) =

0H_p^s₀⁰_,%(R+, H₂^r(Rⁿ)),0B^s_p¹_1,%(R+, H_p^r₁(Rⁿ))

θ (5.12)

for alls0, s1>0 withs= (1−θ)s0+θs1.

Proof. Let p0, p1, and θ be as in Proposition 5.10. Defining H := H₂^r(Rⁿ) and X := H_p^r

1(Rⁿ) we get [H, X]θ=H_q^r(Rⁿ). Hence Proposition 5.10 yields the claim.

Corollary 5.12. Let 1< p <∞,s >0,r∈R, andq∈ _2p

p+1,2p

. Then we have

e⁺∈L(F_p,q^s (R^m+, H_q^r(Rⁿ)), F_p,q^s (R^m, H_q^r(Rⁿ))), r⁺∈L(F_p,q^s (R^m, H_q^r(Rⁿ)), F_p,q^s (R^m+, H_q^r(Rⁿ))), e⁺₀ ∈L(₀F_p,q^s (R^m+, H_q^r(Rⁿ)), F_p,q^s (R^m, H_q^r(Rⁿ))), r⁺₀ ∈L(F_p,q^s (R^m, H_q^r(Rⁿ)),₀F_p,q^s (R^m+, H_q^r(Rⁿ))).

Hence r⁺, r₀⁺, e⁺, and e⁺₀, which are introduced in Definitions 2.11 and 2.13, are retractions and co-retractions on the spaceF_p,q^s (R^m, H_q^r(Rⁿ)).

Proof. This follows immediately by an interpolation argument from Remark 2.19 and Corollary 5.11 with s0=s1=s.

Corollary 5.11 shows that we can reach all Bessel-potential space valued isotropic Triebel-Lizorkin spacesF_p,q^s (R^m, H_q^r(Rⁿ))by interpolation of the scalesH_p(H₂)andB_p(H_p)ifpand qfulfill (5.4). This representation is used frequently to derive results for the Bessel-valued Triebel-Lizorkin scale. For example we can show that the definition of₀F_p,q^s (R^m+, H_q^r(Rⁿ))is equivalent to the definition by vanishing traces.

In order to do this we need the following abstract interpolation result, which can be found in [LM68] or [KMM07, Theorem 7.10].

Remark 5.13. Let{X0, X1},{Z0, Z1}, and{Y0, Y1} be interpolation couples with X0∩X1

,→d Xi, Z0∩Z1

,→d Zi, Yi,→Zi, i= 0,1

and suppose that there exists an operatorD:X0+X1→Z0+Z1 such thatD_|Xi∈L(Xi, Zi)fori= 0,1.

Define the spaces

Xi(D) :={u∈Xi: Du∈Yi}, kukX_i(D):=kukX_i+kDukY_i, u∈Xi(D), i= 0,1.

If there exist operators G: Z0+Z1 → X0+X1 and K: Z0+Z1 → Y0+Y1 with G_|Zi ∈ L(Zi, Xi), K_|Zi ∈L(Zi, Yi)fori= 0,1 and

DGu=u+Ku, u∈Zi, i= 0,1, then we derive

[X0(D), X1(D)]_θ={u∈[X0, X1]_θ:Du∈[Y0, Y1]_θ}, θ∈(0,1).

Proposition 5.14. Let s >0,r∈R, andk∈N0 such that k+ 1/p < s < k+ 1 + 1/p. Then we have

0F_p,q^s (R+, H_q^r(Rⁿ)) =n

u∈F_p,q^s (R+, H_q^r(Rⁿ)) : u^(j)(0) = 0, for all 0≤j≤ko for all p∈(1,∞)andq∈(2p/(1 +p),2p). Due to Proposition 5.7 the traces are classical.

Proof. According to Corollary 5.11 there exist constantsp0, p1∈(1,∞)andθ∈(0,1)such that we have 1/p= (1−θ)/p0+θ/p1and1/q= (1−θ)/2 +θ/p1. Forsj :=s+ 1/pj−1/p,j = 0,1, we derive

s₀∈(k+ 1/p₀, k+ 1 + 1/p₀), s₁∈(k+ 1/p₁, k+ 1 + 1/p₁), ands= (1−θ)s0+θs1. Corollary 5.11 then yields

0F_p,q^s (R+, H_q^r(Rⁿ)) =

0H_p^s0₀(R+, H₂^r(Rⁿ)),0B_p^s1₁(R+, H_p^r₁(Rⁿ))

θ.

Here it is crucial that we can chooses₀ ands₁ such that the Bessel-potential space as well as the Besov space possess the same number of vanishing traces. This is necessary to apply Remark 5.13.

In order to apply Remark 5.13 we define X₀:=H_p^s0

0(R+, H₂^r(Rⁿ)), Z₀:=

k

Y

j=0

H₂^r(Rⁿ),

X1:=B^s_p¹₁(R+, H_p^r₁(Rⁿ)), Z1:=

k

Y

j=0

H_p^r₁(Rⁿ),

Y₀:=Y₁ :={0}, and the trace operatorDu := (u^(j)(0))_j=0,...,k with the corresponding extension oper-atorG. These operators are well-defined and possess the requested boundedness according to [Ama09, Theorem 4.5.4, Theorem 4.6.3]. Additionally, we getDGu =ufor all u∈X0+X1 why we can define K:= 0. The density assumptions of Remark 5.13 are obvious. We then have

X₀(D) = {u∈H_p^s0

0(R+, H₂^r(Rⁿ)) : Du= 0}

= 0H_p^s₀⁰(R+, H₂^r(Rⁿ)), kuk_X0(D)=kuk_0H^s0

p0(R+,H₂^r(Rⁿ)), X1(D) = {u∈B_p^s1₁(R+, H_p^r₁(Rⁿ)) :Du= 0}

= ₀B_p^s1

1(R+, H_p^r

1(Rⁿ)), kuk_X1(D)=kuk

0B^s_p¹₁(R+,H_p^r

1(Rⁿ))

by Remark 2.17 (iv). Hence we obtain

0F_p,q^s (R+, H_q^r(Rⁿ))) =

0H_p^s₀⁰(R+, H₂^r(Rⁿ))),0B^s_p¹₁(R+, H_p^r₁(Rⁿ))

θ

= {u∈[X₀, X₁]_θ:Du= 0}

= n

u∈F_p,q^s (R+, H_q^r(Rⁿ))) : (u^(j)(0))j=0,...,k= 0o by (5.10) and (5.11).

5.2 Anisotropic Triebel-Lizorkin spaces and representation by

Im Dokument General parabolic mixed order systems in Lp and applications (Seite 103-116)