Auxiliary results on Bessel-valued Triebel-Lizorkin spaces

by Remark 5.24 and Remark B.15.

Forf ∈F_~p,q^0,~a(Rⁿ)we deduce with corresponding coretractionS. This implies

h

by Lemma 2.5, Theorem A.11, and Theorem A.12. The general case then follows by Remark 5.20 and the compatibility of interpolation and isomorphisms stated in Lemma 2.6.

5.3 Auxiliary results on Bessel-valued Triebel-Lizorkin spaces

In this section we want to lift the definitions, shifts, and other analog results of Chapter 2 and Chapter 4 to the setting of Bessel-valued Triebel-Lizorkin spaces.

Similar to Definition 2.25 we define an abbreviation for Bessel-valued Triebel-Lizorkin spaces.

Definition 5.27 (Spaces of mixed scales). For%≥0,1 < p <∞, and q ∈(2p/(1 +p),2p),s >0, r∈R, and

F=Fpq, K=Hq

we define

W_F,K,%^s,r (Rⁿ⁺¹+ ) :=0F_pq,%^s (R+, H_q^r(Rⁿ)).

Remark 5.28. Note that the restriction on q always comes from the fact that we obtain Bessel-valued Triebel-Lizorkin spaces by interpolation of a pure Bessel-potential scale and the Bessel-valued Besov scale, cf. Corollary 5.11. The Besov spaces can also be obtained by interpolation of Bessel spaces of course. In this sense our approach to Triebel-Lizorkin spaces is purely Bessel oriented.

It might be possible to drop the restriction on q by an approach which follows the intrinsic structure of the Bessel-valued Triebel-Lizorkin spaces. The results of Chapters 2 and 4 mainly depend on Fourier multiplier theorems. Versions of Fourier multiplier theorems on Triebel-Lizorkin spaces can be found in [BK05] and [BK09], for example.

5.3.1 The joint time-space H

^∞

-calculus on Bessel-valued Triebel-Lizorkin spaces

As in Section 2.4 we define the time derivative operator. Let 1 < p < ∞, 2p/(1 +p) < q < 2p,

% ≥ 0, s >0, and r ∈ R. Then we define the Bessel-valued time-derivative operator on the space M%:=0F_pq,%^s (R+, H_q^r(Rⁿ))by

D^M_t ^%:D(D_t^M%)⊆0F_pq,%^s (R+, H_q^r(Rⁿ)) → 0F_pq,%^s (R+, H_q^r(Rⁿ)), u 7→ ∂tu

where D(D_t^M%) :=₀F_pq,%^s+1(R+, H_q^r(Rⁿ)). This operator inherits all properties from the time derivative operators considered in Section 2.4.

As in Section 2.6 we define the natural extension of∇^N (N :=H_q^r(Rⁿ)) to the Bessel-valued Triebel-Lizorkin space₀F_pq,%^s (R+, H_q^r(Rⁿ)).

Definition and Lemma 5.29(Natural extension on Triebel-Lizorkin spaces).LetX, Y be Banach spaces of classHT. For a densely defined closed linear operatorA:D(A)⊆X→Y the natural extension of Ato₀F_pq,%^s (R+, X)given by

A⁺:D(A⁺)⊆0F_pq,%^s (R+, X) → 0F_pq,%^s (R+, Y), u 7→ A◦u

is a well-defined operator with dense domainD(A⁺) :=0F_pq,%^s (R+, D(A)).

Proof. We can directly show

A⁺∈L(₀F_pq,%^s (R+, D(A)),₀F_pq,%^s (R+, Y))

by using the definition of the norms. Hence A⁺ is well defined. As in the proof of Definition and Lemma 2.40 we have

D(R+,)⊗D(A),→^d D(R+, X),→^d 0F_pq,%^s (R+, X)

according to Remark 5.9. Due to D(R⁺,)⊗D(A) ⊆ 0F_pq,%^s (R⁺, D(A)) the operator A⁺ is densely defined.

The result of Lemma 2.42 also carries over to the setting of Triebel-Lizorkin spaces.

Lemma 5.30. In the same situation as in Definition and Lemma 5.29 we have A⁺∈L(₀F_pq,%^s (R+, X),₀F_pq,%^s (R+, Y))

if A∈L(X, Y). In particular, we have

A⁺∈LIsom(0F_pq,%^s (R+, X),0F_pq,%^s (R+, Y)) if and only if A∈LIsom(X, Y). In both cases we have

kA⁺k_L(0Fs

pq,%(R+,X),0F_pq,%^s (R+,Y))≤ kAk_L(X,Y).

To obtain the bounded joint time-spaceH^∞-calculus on Triebel-Lizorkin spaces there are two strate-gies. The first one is to adapt the proof in Section 2.6, where the result is derived by an iterative approach.

The second one is a direct interpolation argument by Theorem 1.27, which we will use here. So the analog to Theorem 2.47 reads as follows:

5.3: Auxiliary results on Bessel-valued Triebel-Lizorkin spaces 115 Theorem 5.31 (Joint time-space H^∞-calculus on Triebel-Lizorkin spaces). Let 1 < p < ∞, 2p/(1 +p)< q <2p,r∈R,s >0,%≥0, and

W:=W_F,K,%^s,r (Rⁿ⁺¹+ ) =0F_pq,%^s (R+, H_q^r(Rⁿ)), (F,K) := (Fpq, Hq).

Denoting the translatedW-realization of∇+ by

∇^W_+,µ:= (µ+D_t^M%,(∇^N)⁺), M%:=W, N :=H_q^r(Rⁿ), µ≥0 andΩ :=S_θ×Σⁿ_δ,θ > π/2, we get the following results.

(i) The tuple∇^W_+,µ has a bounded jointH^∞(Ω)-calculus on W.

(ii) TheH^∞-calculus inW is compatible with theH^∞-calculus inW⁰ :=Lp,%(R⁺, H_q^r(Rⁿ)), i.e.

f(∇^W_+,µ)⊆f(∇^W_+,µ⁰) holds for allf ∈H_P(Ω).

(iii) We have f(∇^W_+,µ) =fµ(∇^W₊)for allf ∈HP(Ω).

(iv) If we defineV :=W_F,K,%+µ^s,r (Rⁿ⁺¹+ ), then we have

f(∇^V₊)u=Mµ⁻¹f(∇^W_+,µ)Mµu for allf ∈HP(Ω) andu∈ V.

Proof. (i) Let p0, p1, q0, q1 ∈ (1,∞), and θ ∈ (0,1) be given as in Corollary 5.11. Considering the operator tuples

∇^W_+,µ⁰ := (µ+D_t^M%,0,(∇^N0)⁺), W0:=M%,0:=0H_p^s_0,%(R+, H₂^r(Rⁿ)), N0:=H₂^r(Rⁿ),

∇^W_+,µ¹ := (µ+D_t^M%,1,(∇^N1)⁺), W1:=M%,1:=0B^s_p1,%(R+, H_p^r₁(Rⁿ)), N1:=H_p^r₁(Rⁿ) we see that∇^W_+,µ⁰ and ∇^W_+,µ⁰ fulfill the assumptions of Theorem 1.27. The compatibility condition for the resolvents (1.4) can be obtained by Lemma 2.28 (ii) and the representation by Fourier multipliers in Theorem 2.34 and Theorem 2.39. According to Corollary 5.11 we have

[W0,W1]θ = W, [D(D^M_t ^%,0), D(D_t^M%,1)]θ=D(D^M_t ^%).

As in the proof of Theorem 2.34 (i) we have to show an interpolation result for the domainsD(D_j^N0) and D(D^N_j¹)but the situation here is somewhat different because the spaces N0 and N1 are not contained in each other. Due to Theorem 2.34 (i) we have 1 ∈ ρ(D^N_j⁰)∩ρ(D_j^N1). With this we define the operator

A:N0+N1 → D(D^N_j⁰) +D(D^N_j¹),

u=u0+u1 7→ (1− D^N_j⁰)⁻¹u0+ (1− D^N_j^k)⁻¹u1

and obtainA_|N ∈L_Isom(N, X)withX := [D(D_j^N0), D(D_j^N1)]_θ. The representation of the resolvents by Fourier multipliers and Lemma 2.28 (ii) then yields the representationA_|N = [Tm^(q)]_|N by the Lq-Fourier multiplierm(ξ) := (1−iξj)⁻¹. Note that we already have 1∈ρ(D^N_j )and

[T_m^(q)]_|N = (1− D_j^N)⁻¹∈L_Isom(N, D(D^N_j )) according to Theorem 2.34. Thus, we obtain

hD(D_j^N0), D(D_j^N1)i

θ

=D(D^N_j ). (5.20)

It is obvious that D(D^N_j⁰)is a Hilbert space with the graph normkuk:= (kuk²_N0+kD_j^N0uk²_N0)^1/2, u∈D(D_j^N0). Proposition 5.10 and (5.20) then yield

hD((D_j^N0)⁺), D((D_j^N1)⁺)i

θ

= h

0H_p^s

0,%(R+, D(D^N_j⁰)),₀B_p^s

1,%(R+, D(D^N_j¹))i

θ

= 0F_pq,%^s R⁺,h

D(D^N_j⁰), D(D^N_j¹)i

θ

= 0F_pq,%^s R+, D(D^N_j )

= D((D^N_j )⁺).

Altogether, we obtain that ∇^W_+,µ is the interpolated tuple of ∇^W_+,µ⁰ and ∇^W_+,µ¹ in Theorem 1.27.

Theorem 1.27 then yields the sectoriality and bisectoriality ofµ+D^M_t ^% and(D^N_j )⁺,(j= 1, . . . , n), respectively.

(ii) Due to Proposition 5.7 the representation off(∇^W_+,µ)can be derived by the compatibility result in Proposition 1.31.

(iii) This follows from Lemma 1.30.

(iv) Using Proposition 1.26 this similarly follows as in Theorem 2.47.

Remark 5.32(Properties of op₊[Ψ_s]and op[Λ_r]on Triebel-Lizorkin spaces). Fors≥0,r, r⁰∈R,

%≥0,1< p <∞, andq∈(2p/(1 +p),2p). We have h

op[Λ_r]

|Hq^r0+r(Rⁿ)

i+

∈ L_Isom(₀F_p,q,%^s0 (R+, H_q^r0+r(Rⁿ)),₀F_p,q,%^s0 (R+, H_q^r0(Rⁿ))), s⁰≥0, op₊[Ψs]_|

0F_p,q,%^s0+s(R+,H_q^r0(Rⁿ)) ∈ LIsom(0F_p,q,%^s0+s(R+, H_q^r0(Rⁿ)),0F_p,q,%^s0 (R+, H_q^r0(Rⁿ))), s⁰ >0

where op₊[Ψs] := M%⁻¹r⁺₀ op[Ψs(·+%)]e⁺₀M%. This result follows directly from (5.12), Lemma 5.29, Remark 2.18, and Proposition 2.33.

Proposition 5.33(Shifts on Triebel-Lizorkin spaces). Let 1< p <∞,2p/(1 +p)< q <2p,r⁰∈R, s⁰>0,s, r≥0,%≥0,(F,K) := (Fpq, H_q), andW :=W_F,K,%^s0,r0 (Rⁿ⁺¹+ ). Then the symbolsΛ_r andΨ_s give rise to the isomorphisms

Λ_r(∇^W₊) =h

op[Λ_r]_|Kr0+r(Rⁿ)

i⁺

∈ L_Isom

W_F,K,%^s0,r0+r(Rⁿ⁺¹+ ), W_F,K,%^s0,r0 (Rⁿ⁺¹+ ) , Ψ_s(∇^W₊) =op₊[Ψ_s]

|W_F,K,%^s0+s,r0(Rⁿ⁺¹+ ) ∈ L_Isom

W_F,K,%^s0+s,r0(Rⁿ⁺¹+ ), W_F,K,%^s0,r0 (Rⁿ⁺¹+ ) and we additionally haveD(Λ_r(∇^W₊)) =W_F,K,%^s0,r0+r(Rⁿ⁺¹+ )andD(Ψ_s(∇^W₊)) =W_F,K,%^s0+s,r0(Rⁿ⁺¹+ ).

Proof. (i) It is obvious that Λ⁻¹_r ∈ H^∞(Ω) and Λr ∈ HP(Ω) with Ω := Sθ×Σⁿ_δ, θ > π/2. From Theorem 5.31 (ii), Proposition 2.49, and Remark 5.32 we infer

(Λ⁻¹_r )(∇^W₊) = (Λ⁻¹_r )(∇^W₊⁰)_|W, W⁰ :=L_p,%(R+, H_q^r0(Rⁿ)),

= h

op[Λ_−r]_|Hr0 q (Rⁿ)

i+

∈L_Isom(W, W_F,K,%^s0,r0+r(Rⁿ⁺¹+ )).

The rest follows as in the proof of Proposition 2.49 by Theorem 1.19.

(ii) This can be proved in the same way as (i).

5.3: Auxiliary results on Bessel-valued Triebel-Lizorkin spaces 117

5.3.2 H

^∞

-calculus of N-parabolic symbols on Bessel-valued Triebel-Lizorkin spaces

The shift operators in Proposition 4.3 were an essential ingredient for the proof of the main result on mixed order systems in Chapter 4. Thus, we now have to provide a version for Triebel-Lizorkin spaces to derive an analogon.

Proposition 5.34. Let J ∈N⁰,NV := (ri, si)i=0,...J+1⊆[0,∞)² be the vertices (starting at the origin and indexed in the counter-clockwise direction) of a Newton polygonN,1< p <∞,2p/(1 +p)< q <2p, (F,K) := (Fpq, Hq),s⁰>0, and r⁰∈R. If ΦN is defined as in Proposition 4.3 andW:=W_F,K,%^s0,r0 (Rⁿ⁺¹+ ) (%≥0), then for all µ≥0the operator ΦN(∇^W_+,µ)is invertible with

D(ΦN(∇^W_+,µ)) = \

(r,s)∈NV

W_F,K,%^s0+s,r0+r(Rⁿ⁺¹+ ),

ΦN(∇^W_+,µ)∈LIsom





\

(r,s)∈NV

W_F,K,%^s0+s,r0+r(Rⁿ⁺¹+ ),W



.

Proof. The proof of Proposition 4.3 can be transferred literally by using Theorem 1.19, Proposition 5.33, and (4.4).

The next theorem is the analogon of Theorem 4.6 and Theorem 4.7 for Triebel-Lizorkin spaces.

Theorem 5.35(H^∞-calculus for symbols inHP on Triebel-Lizorkin spaces). LetQ∈HP[K](Ω), Ω :=Sθ×Σⁿ_δ, θ > π/2, 1 < p < ∞, 2p/(1 +p) < q < 2p, (F,K) := (Fpq, Hq), s⁰ >0, r⁰ ∈ R, and W:=W_F,K,%^s0,r0 (Rⁿ⁺¹+ ). Then we deduce the following assertions.

(i) Let O be an upper increasing order function of Q with αin(O) < s⁰ and define the strictly posi-tive order function O+(γ) := O(γ) +αin(O)γ+βin(O). If we define W− := W_F,K,%^s00,r00(Rⁿ⁺¹+ ) with s⁰⁰:=s⁰−αin(O)>0 andr⁰⁰:=r⁰−βin(O)∈R, then we get for allµ≥λ0(Q,O)and all ℘∈K

D(Q_µ[℘](∇^W₊⁻)) ⊇

M

\

`=0

W_F,K,%^{s0+m`(O),r0+b`(O)}(Rⁿ⁺¹+ )

= \

(r,s)∈NV(O+)

W_F,K,%^s00+s,r00+r(Rⁿ⁺¹+ ) =:V and the restriction of the maximal realization toV yields the bounded map

K→L(V,W), ℘7→h

Qµ[℘](∇^(%)₊ )i

|V.

(ii) Let O be an upper decreasing order function of Qwith αde(O)< s⁰ and define the strictly positive order function O+(γ) := −O(γ) +αde(O)γ+βde(O). If we define W− := W_F,K,%^s00,r00(Rⁿ⁺¹+ ) with s⁰⁰:=s⁰−αde(O)>0 andr⁰⁰:=r⁰−βde(O)∈R, then we get for allµ≥λ0(Q,O)and all ℘∈K

D(Qµ[℘](∇^W₊⁻)) ⊇ W (5.21)

and the restriction of the maximal realization toW yields the bounded map K→L(W,V), ℘7→h

Qµ[℘](∇^(%)₊ )i

|W

where

V :=

M

\

`=0

W_F,K,%^{s0−m`(O),r0−b`(O)}(Rⁿ⁺¹+ ) = \

(r,s)∈N_V(O₊)

W_F,K,%^s00+s,r00+r(Rⁿ⁺¹+ ).

Proof. Using Proposition 5.34 instead of Proposition 4.3 the proofs are literally the same as in Theorem 4.6 and Theorem 4.7.

Corollary 5.36. Let 1< p <∞, 2p/(1 +p)< q <2p,(F,K) := (F_pq, H_q),s⁰ >0,r⁰ ∈R,%≥0, and W :=W_F,K,%^s0,r0 (Rⁿ⁺¹+ ). Further let Ω :=Sθ×Σⁿ_δ, θ > π/2. If P ∈SN[K](Ω) is an N-parabolic symbol, then there existsµ >0 such that

P_µ[℘](∇^W₊)∈L_Isom





\

(r,s)∈N_V(P)

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



, ℘∈K

and the norm of P_µ[℘](∇^W₊) can be estimated from above by a constant independent of the parameter

℘∈K.

Proof. This proof can also be carried over literally from Corollary 4.9 by using Theorem 5.35 instead of Theorems 4.6 and 4.7.

Im Dokument General parabolic mixed order systems in Lp and applications (Seite 123-128)