In this section we want to provide an analogon of Theorem 4.12 for the scale of Triebel-Lizorkin spaces.
In Section 5.3 we have derived all necessary results for the scales(Fpq, Hq)and(Hp, Bq)to formulate this analog result. To clarify the situation we state this explicitly in the next theorem.
5.5: Mixed order systems on Triebel-Lizorkin spaces 121 Theorem 5.40 (Main Theorem on N-parabolic mixed order systems on Triebel-Lizorkin spaces). Let Ω := Sθ ×Σnδ, θ > π/2, and let L ∈ [HP[K](Ω)]m×m be an N-parabolic mixed order
Using the same notation as in Definitions 4.10 and 2.25 (respectively, Definition 5.27) 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. Using Theorem 5.35 instead of Theorem 4.6 and Theorem 4.7 the proof of Theorem 4.12 can be carried over literally.
Similarly to Definition 4.20, we introduce an admissible Triebel-Lizorkin scale.
Definition 5.41 (Admissible Triebel-Lizorkin scale). Let O1 and O2 be positive order functions such that O1− O2 is a monotone order function and let 1< p <∞,2p/(1 +p)< q <2p. Let the scale
(F`,K`)∈ {(Bp, Hq),(Fpq, Bq)}, `= 0, . . . , M be given such that there exist τ∈ {0, . . . , M−1} with
(F`,K`) = (Hp, Bq), `∈ {0, . . . , τ}, (F`,K`) = (Fpq, Hq), `∈ {τ+ 1, . . . , M}.
The scale(F`,K`)`=0,...,M is then called(O1,O2)-admissible if we have
(bτ(O2), mτ(O2)) 6= (bτ+1(O2), mτ+1(O2)), ifO1− O2 is increasing, (bτ(O1), mτ(O1)) 6= (bτ+1(O1), mτ+1(O1)), ifO1− O2 is decreasing.
Note that this definition is also meaningful if O1− O2 has trivial index, i.e. there exists α, β ∈R such that (O1− O2)(γ) =αγ+β for allγ >0and thereforeO1− O2 is increasing as well as decreasing.
Proposition 5.42 (Embedding condition III). Let us consider the situation of Theorem 5.40 with 1< p <∞,2p/(1 +p)< q <2p, and
(F`,K`)∈ {(Fpq, Bq),(Bp, Hq)}, `= 0, . . . , M.
If the scale(F`,K`)`=0,...,M is(OHi,OFj)-admissible for alli, j= 0, . . . , M, then the embedding conditions (5.29) and (5.30) are fulfilled. In this sense the scales(Fpq, Hq)and(Hp, Bq)are compatible ifp∈(1,∞) andq∈(2p/(1 +p),2p).
Proof. The proof is exactly the same as the proof of Proposition 4.21. Instead of the embeddings in Lemma 4.5 and Propositions 4.17 and 4.18 we here have to apply Propositions 5.37, 5.38, and 5.39.
As in Chapter 4 we state a condensed version of Theorem 5.40.
Corollary 5.43. Let L ∈[HP[K](Sθ×Σnδ)]m×m, θ > π/2, be an N-parabolic mixed order system such that for each i, j= 1, . . . , m the order function sj+ti is positive or negative. Let%≥0,s0` ≥0,r0`∈R (`= 0, . . . , M) such that
OHi(γ) := max
` {[s0`+m`(ti)]γ+r0`+b`(ti)}, γ >0, OFj(γ) := max
` {[s0`−m`(sj)]γ+r0`−b`(sj)}, γ >0, i, j= 1, . . . m are positive order functions. Furthermore, let the scale
(F`,K`)∈ {(Fpq, Hq),(Hp, Bq)}, 1< p <∞, 2p/(1 +p)< q <2p, `= 0, . . . , M be(OHi,OFj)-admissible for alli, j= 1, . . . mand let
s0`>max{max{−m`(ti), m`(si)}:i= 1, . . . , m}
for all`∈ {0, . . . , τ}whereτ is taken from Definition 5.41. Using the same notation as in Definition 5.27 we define for i, j= 1, . . . mthe spaces
Hi:=
M
\
`=0
Ws
0
`+m`(ti), r`0+b`(ti)
F`,K`,% (Rn+1+ ), Fj:=
M
\
`=0
Ws
0
`−m`(sj), r0`−b`(sj) F`,K`,% (Rn+1+ ).
5.5: Mixed order systems on Triebel-Lizorkin spaces 123 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 (5.29) and (5.30) are fulfilled due to Proposition 5.42. Hence, the assertion follows from Theorem 5.40.
Chapter6
Singular integral operators on L p -L q
In most cases we can treat a boundary value problem on the half space by partial Fourier and Laplace transform followed by a reduction to the boundary. Roughly speaking, we often derive a formula for the solution of a boundary value problem in this situation which is formally given by
u(t, x0, xn) = [h(∇+, xn)Φ](t, x0), t >0, x= (x0, xn)∈Rn+ (6.1) where h is a suitable scalar function and Φ a function on the boundary, i.e. in the trace space of the canonical solution space for u. For the heat equation, for instance, the solution can be given by u(t, x0, xn) = (h(∇+, xn)g)(t, x0)where gis the trace ofuon the boundary and
h(λ, z, xn) := exp
− q
λ+|z|2−·xn
with |z|−:= −
n−1
X
k=1
zk2
!1/2
for(λ, z)∈Sθ×Σn−1δ .
For this reason we have to concentrate on parameter-dependentH∞-calculus and the regularities in these parameters. Due to the fact that the operatorh(∇+, xn)in (6.1) only makes sense for fixedxn >0 we have to rearrange the order of arguments. InLp-Lp-theory there is no trouble by this rearrangement but inLp-Lq-theory some problems arise from the fact that we cannot apply Fubini’s theorem for the time- andxn-variable. To handle this problem we have to interpret the operatorh(∇+, xn)by a two step functional calculus. We define such an operator by singular integral operators in Definition 6.16. This definition is related to the so-called ’Volevich trick’, cf. [Vol65].
Singular integral operators onLp-Lqcan be found in [Fer87], for example. There the author considers operator-valued kernels of product type but this is not sufficient for our applications. To derive the desired regularities, associated with an order function, and the connection to the jointH∞-calculus of
∇+ we have to develop own results on this topic.
6.1 Singular integral operators
In the sequel we define the class of integral kernels which are bounded from above by the kernel of the one-sided Hilbert transform and a weight function given by an order function. A special case of the kernels in the next definition can be found in [DHP03, Section 7.1], for instance.
Here and in the following we always assume1< p, q <∞.
Definition 6.1. Let Obe a strictly positive order function and M ∈N0. Then we define KM(Sθ×Σn−1δ ,O), θ > π/2, δ >0
as the set of all kernel functionsk:Sθ×Σn−1δ ×R+×R+→Cwith k(·,·, xn, yn)∈H∞(Sθ×Σn−1δ ), xn, yn>0
125
andk(λ, z,·, yn)∈CM(R+,C)for all(λ, z)∈Sθ×Σn−1δ ,yn>0 such that ∂xjnk(λ, z, xn, yn)
≤C·(ΞO(λ, z))j xn+yn for all(λ, z)∈Sθ×Σn−1δ ,xn, yn>0, andj≤M.
Remark 6.2(One-sided Hilbert transform). Foru∈Lp(R+)we define the one-sided Hilbert trans-form by
(H+u)(x) :=
Z ∞ 0
u(y)
x+ydy, x∈R+.
Then we have H+ ∈L(Lp(R+)). This can be easily seen by a reflection argument and the boundedness of the Hilbert transform on Lp(R).
Definition 6.3. Let X be a Banach space of class HT with property (α). For fixed λ∈Sθ and kernel k∈KM(Sθ×Σn−1δ ,O)we define the parameter-dependent singular integral operator
(GN[λ, k]g)(xn) :=
Z ∞ 0
[k(λ,∇N, xn, yn)g(yn)]dyn[Hr
q(Rn−1,X)], xn>0
forg∈Lq(R+, Hqr(Rn−1, X))andN :=Hqr(Rn−1, X),r∈R. For fixedxn andλ the integral exists due to Remark 6.2 andkk(λ,∇N, xn, yn)kL(Hr
q(Rn−1,X))≤C(xn+yn)−1 for allyn>0. This especially yields GN[λ, k]∈L(Lq(R+, Hqr(Rn−1, X))).
As an application of Proposition B.25 we derive the next result.
Proposition 6.4. Letk∈KM(Sθ×Σn−1δ ,O)andN :=Hqr(Rn−1, X),r∈R. Then following assertions hold:
(i) The set
{k(λ,∇N, xn, yn) :λ∈Sθ} ⊆L(Hqr(Rn−1, X)), xn, yn>0 isR-bounded with
Rp({k(λ,∇N, xn, yn) :λ∈Sθ})≤C· 1
xn+yn, xn, yn>0. (6.2) (ii) The set{GN[λ, k] :λ∈Sθ} ⊆L(Lq(R+, Hqr(Rn−1, X))) isR-bounded.
Proof. (i) Due to the definition ofKM(Sθ×Σn−1δ ,O)there existsC >0such that
(xn+yn)k(·,·, xn, yn)∈H∞(Sθ×Σn−1δ ), kk(·,·, xn, yn)k∞≤C(xn+yn)−1 for allxn, yn>0. Theorem 2.34 and Lemma 1.16 directly yield theR-boundedness of
{f(∇N) :f ∈H∞(Sθ×Σn−1δ ),kfk∞≤C} ⊆ N. Thus, we derive (i) from Remark B.14 (vi).
(ii) To apply Proposition B.25 we defineK:={k(λ,∇N, xn, yn) :λ∈Sθ}and k0(xn, yn) := (xn+yn)−1, xn, yn>0.
The assumptions of Proposition B.25 are fulfilled due to (6.2) and Remark 6.2. So we deduce the R-boundedness of{GN[λ, k] :λ∈Sθ} ⊆L(Lq(R+, Hqr(Rn−1, X))).
6.1: Singular integral operators 127 Definition 6.5. Let X be Banach space of classHT. We define the rearranging operator by
U:Lq(Rn−1, Lq(R+, X))→Lq(R+, Lq(Rn−1, X))
where[(U f)(xn)](x0) := [f(x0)](xn)forf ∈Lq(Rn−1, Lq(R+, X)),x0 ∈Rn−1,xn>0.
Lemma 6.6. We get
U ∈LIsom Hqr(Rn−1, Hq`(R+, X)), Hq`(R+, Hqr(Rn−1, X))
(6.3) for all`, r≥0.
Proof. The case`, r= 0is based on Fubini’s theorem and can be found in [AE09, III X 6.22], for example.
The rest can be shown by easy calculations.
In the next proposition we introduce singular integral operators onLp-Lq given by kernels in the class KM(Sθ×Σnδ,O).
Definition and Proposition 6.7(Singular integral operator onLp-Lq). LetX be a Banach space of classHT with property (α). Letk∈KM(Sθ×Σn−1δ ,O)andW :=0Hp,%s (R+, Lq(R+, Hqr(Rn−1, X))), s, r≥0, %≥0,1< p, q <∞. Then we define the singular integral operator realized onW by
GW[k] :=T(DMt %)∈L(0Hp,%s (R+, Lq(R+, Hqr(Rn−1, X)))) (6.4) where
T(λ) := (GN[λ, k])+∈0Hp,%s (R+, Lq(R+, Hqr(Rn−1, X))), λ∈Sθ withN :=Hqr(Rn−1, X)andM%:=0Hp,%s (R+, Lq(R+, Hqr(Rn−1, X))).
Proof. Due to Proposition 6.4 (ii) and Lemma 2.42 (ii) we have λ∈Sθ7→(GN[λ, k])+
∈HR∞(Sθ,B(DtM%))
whereB(DMt %)is the commutator ofDMt %. An application of Theorem 1.21 and Theorem 2.39 then show the asserted boundedness ofGW[k].
To derive more information on the regularity in thexn-variable of GW[k]g we have to consider the behavior of parameter-dependentH∞-calculus under derivatives with respect to this parameter.
Lemma 6.8. Lets≥0,m∈N0, and letX be a Banach space of classHT. Then
0Hp,%s (R(t)+ , Hqm(R(x)+ , X)) is the set of all functionsf ∈0Hp,%s (R(t)+ , Lq(R(x)+ , X))such that
(∂x+)kf ∈0Hp,%s (R(t)+, Lq(R(x)+ , X)) (6.5) for allk∈ {0, . . . , m}.
Proof. Let f ∈ 0Hp,%s (R(t)+ , Hqm(R(x)+ , X)). Then it is obvious that (6.5) holds for all k ≤ m by the definition of the canonical extension.
Letf ∈0Hp,%s (R(t)+, Lq(R(x)+ , X)) be such that (6.5) holds. We then have∂xk(f(t))∈Lq(R(x)+ , X)for almost allt∈R+ andk≤m. This obviously yieldsf(t)∈Hqm(R(x)+ , X)for almost allt∈R+. Using the definition of the norm in0Hp,%s (R(t)+, Hqm(R(x)+ , X))we obtain f ∈0Hp,%s (R(t)+ , Hqm(R(x)+ , X)).
In the next result we state the behavior of a parameter-dependentH∞-calculus and derivatives with respect to this parameter.
Lemma 6.9. Let X be a Banach space of class HT with property(α). LetI⊆Rbe an interval and let f: Σn−1δ ×I→Cbe a function such that
f(z,·) ∈ CM(I,C), M ∈N0, z∈Σn−1δ , (∂xjf)(·, x) ∈ H∞(Σn−1δ ), x∈I, j≤M.
WithN :=Hqr(Rn−1, X),r∈R, we then derive
∂xj(T(x)g) = (∂xjf)(∇N, x)g, T(x) :=f(∇N, x), g∈Hqr(Rn−1, X), x∈I, j≤M andT(·)g∈CM(I,N).
Proof. It is obvious that it is sufficient to consider the caseM = 1. Letg∈Hqr(Rn−1, X)and (gj)j∈N⊆S(Rn, X)with gj→g inHqr(Rn−1, X). Then we have
∂x(T(x)gj) =∂x(F−1f(iξ, x)Fgj) =F−1(∂xf)(iξ, x)Fgj= (∂xf)(∇N, x)gj. Now it is easy to show∂x(T(x)g) = (∂xf)(∇N, x)g for allx∈I.
For the treatment of singular integral operators on Bessel-potential spaces with mixed regularity we introduce an abbreviation for those spaces. For any Banach spaceX of classHT we define
Xj(O, κ) :=
M(O)
\
`=0
0Hp,%j·m`(O)(R+, Hqκ(R+, Hqj·b`(O)(Rn−1, X))), %≥0, j∈N0, κ≥0.
Here M(O) ∈ N is given such that O(γ) = max{m`(O)γ +b`(O) :` = 0, . . . , M(O)}, γ > 0. With the help of these spaces we can describe the gained regularity in thexn-variable by a singular integral operator defined by (6.4).
Lemma 6.10. LetXbe a Banach space of classHT with property(α). Moreover letf ∈H∞(Sθ×Σn−1δ ), θ > π/2. Then we have
(f(λ,∇N))+n =U f(λ,∇M)U−1
where N := Lq(Rn−1, X) and M := Lq(Rn−1, Lq(R+, X)). For the sake of clearness +n denotes the natural extension toLq(R+, Lq(Rn−1, X)).
Proof. We have (f(λ,∇N))+n = f(λ,(∇N)+n) due to Lemma 2.45 (iv). With Proposition 1.26 and U−1(∇N)+nU =∇Mwe then getf(λ,(∇N)+n) =U f(λ,∇M)U−1, which yields the assertion.
Lemma 6.11. Let X be a Banach space of class HT with property(α). Let k∈KM(Sθ×Σn−1δ ,O), W:=Lp,%(R+, Lq(R+, Lq(Rn−1, X))),
j≤M, andf ∈Xj(O,0). Then we have
(∂njk)Φ−jN(O) ∈ K0(Sθ×Σn−1δ ,O),
∂nj(GW[k]f) = GW[(∂njk)Φ−jN(O)](UΦjN(O)(∇W+0)U−1f) whereW0 :=Lp,%(R+, Lq(Rn−1, Lq(R+, X))).
Proof. Forg:=UΦjN(O)(∇W+0)U−1f ∈ W we get GW[k]f =GW[k]
UΦ−jN(O)(∇W+0)U−1g . DefiningT(λ) := GN[λ, k]+
,S(λ) :=
UΦ−jN(O)(λ,∇N)U−1+
, andN :=Lq(Rn−1, Lq(R+, X))we get GW[k]f = T(DWt )S(DWt )g= (T S)(DWt )g
6.2: Extension symbols 129