Instationary Generalized Stokes Equations in Partially Periodic Domains

We consider an instationary generalized Stokes system with nonhomogeneous divergence data under a periodic condition in only some directions. The problem is set in the whole space, the half space or in (after an identification of the periodic directions with a torus) bounded domains with sufficiently regular boundary. We show unique solvability for all times in Muckenhoupt weighted Lebesgue spaces. The divergence condition is dealt with by analyzing the associated reduced Stokes system and in particular by showing maximal regularity of the partially periodic reduced Stokes operator.


Introduction
Consider the partially periodic instationary generalized Stokes problem and Ω is a domain in G := R n1 × T n2 L with T L := R/LZ, L > 0 and n := n 1 + n 2 ≥ 2. The topology and differentiable structure on G are the canonical ones inherited from R n , so that (1.1) governs a flow which is periodic of length L in the direction of the variables y := (x n1+1 , . . . , x n ). Such partially periodic models are relevant in mathematical fluid mechanics, for example in the analysis of flows in spiraling tubes or layer-like domains with periodic boundary conditions. Assumption 1.1. We want to consider problem (1.1) in • the partially periodic whole space G := {(x , y) | x := (x 1 , . . . , x n1 ) ∈ R n1 , y := (x n1+1 , . . . , x n ) ∈ T n2 L }; • the partially periodic half space G + := {x ∈ G | x 1 > 0}; • bounded partially periodic C 1,1 -domains, that is bounded, open and connected Ω ⊂ G, where the boundary can be described locally (after a possible rotation of the coordinate system) as the graph of a C 1,1 -function.
The nonperiodic case n 2 = 0 has been extensively investigated in the literature in a variety of domains. Bothe and Prüss [3] considered general instationary Stokes systems in bounded and exterior domains for Dirichlet, Neumann and Navier boundary conditions. Unique solvability of (1.1) in Sobolev spaces for Jonas Sauer was partly supported by DFG and JSPS via the International Research Training Group 1529. 290 J. Sauer JMFM a large class of domains including bounded and exterior domains, asymptotically flat layers, infinite cylinders, perturbed half spaces and aperture domains was obtained by Abels [2], where also variable viscosity and mixed boundary conditions are admitted. The main idea of Abels [2] is to use maximal L p regularity of some associated Stokes operator. We want to follow this train of thought and establish a theory which enables us to show a corresponding regularity result for the partially periodic reduced Stokes operator, i.e, for all n 2 ∈ {0, . . . , n − 1}. For n 2 > 0, there are only very few results in the literature. In the case of a homogeneous divergence condition, i.e, g = 0, there are early results by Iooss [13] in the L 2 framework. In L p Sobolev spaces, problem (1.1) has been treated by Denk and Nau [8,17] in the case of a straight cylinder and by the author [20,22,23] in the whole space case Ω = G. In particular, Theorem 3.5 in [23] shows that the partially periodic Stokes operator admits maximal L p regularity in L q ω,σ (G) for all q ∈ (1, ∞) and all ω ∈ A q (G) for n ≥ 3 with an A q -consistent estimate. Here, A q (G) is the class of Muckenhoupt weights ω ∈ A q (R n ) which are periodic of length L with respect to the variables y = (x n1+1 , . . . , x n ), cf. [22,Proposition 2]. Recall that a nonnegative ω ∈ L 1 loc (R n ) is in the Muckenhoupt class A q (R n ), if where ω := ω −q /q and the supremum runs over all balls B ⊂ R n . The reason to include weighted spaces lies in an extrapolation theorem in the spirit of García-Cuervo and Rubio de Francia [12], which roughly states that uniform bounds in weighted spaces immediately extend to R-bounds. More precisely, the following proposition can be found in [20,Theorem 2]. Here, we call a constant c = c(ω) that depends on Muckenhoupt weights A q -consistent, if for each d > 0 we have sup{c(ω) : ω is an A q (G)-weight with A q (ω) < d} < ∞. Proposition 1.2. Suppose that r, q ∈ (1, ∞), ω ∈ A q (G) and that Ω ⊂ G is measurable. Moreover, assume that T is a family of linear operators such that for all ν ∈ A r (G) there is an A r -consistent constant c r = c r (ν) > 0 with for all f ∈ L r ν (Ω) and all T ∈ T . Then every T ∈ T can be extended to L q ω (Ω) and T is R-bounded with an A q -consistent R-bound c q .
Since R-boundedness of solutions to the corresponding resolvent equations is connected to maximal L p regularity via the Theorem of Weis (see Proposition 2.2 below), Proposition 1.2 suggests that key in understanding problem (1.1) is a thorough investigation in weighted spaces of the resolvent problem where λ ∈ Σ ϑ := {λ ∈ C : | arg λ| < ϑ, λ = 0}, ϑ ∈ (0, π). In particular, we aim at obtaining a priori estimates which are A q -consistent. The purpose of the present paper is threefold: • extend the results in [2] to the partially periodic case and to weighted spaces, • extend the results on the corresponding resolvent equations in [22] to all dimensions n ≥ 2 and to domains with boundary, and • extend the results on the maximal L p regularity of the partially periodic Stokes operator in [23] to non-homogeneous divergence data g and to domains with boundary. Our main results are stated in the following two theorems. Here, the space of initial values is defined as the real interpolation space B 2−2/p q,p,ω (Ω) := L q ω (Ω), W 2,q ω (Ω) ∩ W 1,q 0,ω (Ω) 1−1/p,p , which can be regarded as Vol. 20 (2018) Instationary Generalized Stokes Equations 291 a partially periodic, weighted space of Besov type. The precise definition of the respective function spaces can be found in Sect. 3. Theorem 1.3. Let n ≥ 2 and let Ω be as in Assumption 1.1. Assume T ∈ (0, ∞), p, q ∈ (1, ∞) and ω ∈ A q (G). Then there is an A q -consistent constant c = c(n, p, q, ω, Ω, T) > 0 such that for all T ∈ (0, T], all f ∈ L p (0, T ; L q ω (Ω) n ), all g ∈ L p (0, T ; W 1,q ω (Ω)) with ∂ t g ∈ L p (0, T ; W −1,q 0,ω (Ω)) and all u 0 ∈ B 2−2/p p,q,ω (Ω) n satisfying the compatibility condition there is a unique (u, p) ∈ L p (0, T ; W 2,q ω (Ω) n ) ∩ W 1,p (0, T ; L q ω (Ω) n ) × L p (0, T ; W 1,q ω (Ω)) solving (1.1) and it holds the estimate (

1.3)
If Ω is bounded, the assertion remains true for T = ∞.
As explained above, the key ingredient in the proof of Theorem 1.3 is the following result.

4)
where c = c(n, q, ω, ϑ, Ω) > 0 is A q -consistent. In the case of a bounded domain, the term on the left-hand side may be replaced by u W 2,q ω (Ω) .
Remark 1.5. Note that for bounded partially periodic C 1,1 -domains, a homogeneous flux condition is built in into our functional analytic setting: Consider for example a periodic cylinder Ω := D × R/LZ, where D ⊂ R n−1 is the unit disc. Then for the corresponding pressure p ∈ W 1,q ω (Ω) from Theorem 1.4 it does not only hold that ∇p ∈ L q ω (Ω) n , but also that p itself is periodic in the sense p| xn↓0 = p| xn↑L . Therefore, the pressure drop within one periodic cell is zero, which results in a homogeneous flux condition.
The paper is structured as follows: Firstly, in Sect. 2 we prove that Theorem 1.3 can be deduced from Theorem 1.4 by arguments similar to the ones in [1,2]. The notation and basic results on weighted Lebesgue and Sobolev spaces defined over domains in the group G are provided in Sect. 3. The main part of this paper are Sects. 4-6, which are devoted to establishing Theorem 1.4. In Sect. 4, the case Ω = G is treated. Sections 5 and 6 are concerned with Theorem 1.4 in the cases of the half space and bounded periodic C 1,1 -domains, respectively. It should be pointed out that the treatment of bounded domains in Sect. 6 is very different in style compared to the Sects. 4 and 5. In fact, since bounded domains have a finite measure and are relatively compact, standard localization techniques can be applied to show the corresponding regularity estimates, which reduces a large part of the problem to the nonperiodic case. Observe that in doing so, it is also necessary to use non-periodic results, as one is rotating the coordinate system during the process of localization, which is not compatible with having distinguished directions of periodicity.
Finally, in "Appendix", we give a construction of the Helmholtz decomposition in weighted partially periodic spaces.

Proof of Theorem 1.3
Let us show that Theorem 1.4 indeed implies Theorem 1.3. Therefore, consider the reduced partially periodic Stokes equations . Observe that the unique solvability follows from the Helmholtz projection, more precisely from Lemma 7.1.
If Ω is a bounded domain, also λ = 0 is permitted.
, which exists due to Lemma 7.1, and where also λ = 0 is allowed in the case of a bounded domain. Then it is immediate that (∇p, ∇ϕ) = (Δu, ϕ) − (∇div u, ϕ), ϕ ∈ W 1,q ω (Ω) and hence p = P u. Thus u solves (2.1) and For q ∈ (1, ∞) and ω ∈ A q (G), we define the partially periodic reduced Stokes operator A red q,ω on L q ω (Ω) via We want to show that A red q,ω admits maximal L p regularity. Here, a generator −A of a bounded analytic semi-group on a Banach space X is said to admit maximal L p -regularity, if for all f ∈ L p (0, ∞; X) and u 0 ∈ (X, D(A)) 1−1/p,p , the mild solution to is a.e. D(A)-valued, a.e. differentiable with values in X and such that both u t and Au belong to L p (0, ∞; X). Recall the Theorem of Weis [25].

Proposition 2.2.
Let p ∈ (1, ∞) and assume that −A is the generator of a bounded analytic semi-group in an UMD space X. Then A admits maximal L p -regularity if and only if the operator family Vol. 20 (2018)
If Ω is bounded, also T = ∞ is permitted.
Proof. Lemma 2.1 shows that the family of operators . Proposition 1.2 shows that it is even R-bounded. Thus, the Theorem of Weis applies. Note that in [23,Theorem 2.11], an A q -consistent version of Weis' Theorem has been given, which justifies the claimed A q -consistency of the bound c. Since A red q,ω is invertible on bounded domains by Lemma 2.1, the additional remark also follows from the Theorem of Weis.
We can now give the proof of Theorem 1.3. Uniqueness of solutions to (1.1) follows directly from Theorem 2.3, since for f = 0 and g = 0 we have A red q,ω u = −Δu + ∇p. Hence, we can concentrate on the existence part. Let f , g, and u 0 be given as in the theorem and define for almost all t ∈ (0, T ) the pressure p r (t) via . By the assumptions on f and g, we see ∇p r ∈ L p (0, T ; L q ω (Ω)). Then (u, p) is the desired solution to (1.1), where u is obtained by Theorem 2.3 with f r := f − ∇p r , and where p := P u + p r . Indeed, it remains only to verify div u = g.

Preliminaries
If equipped with addition as group operation and the canonical quotient topology inherited from R n , G := R n1 × T n2 L is turned into a locally compact abelian group. Thus, under the canonical identification of G with R n1 × [0, L) n2 the Haar measure μ on G is given up to a normalization factor by the product of the Lebesgue measure on R n1 and the Lebesgue measure on [0, L) n2 , that is Let Ω ⊂ G be a domain, i.e., an open connected subset of G. For q ∈ [1, ∞] and a partially periodic Muckenhoupt weight ω ∈ A q (G), the weighted Lebesgue space L q ω (Ω) is the space of all q-integrable functions with respect to the measure ω dμ. Note here, that the classes A 1 (R n ) and A ∞ (R n ) can be defined in a similar manner as for q ∈ (1, ∞), see e.g. [24] for details on Muckenhoupt weights. The dual space of L q ω (Ω) can be identified with L q ω (Ω) via the duality pairing (u, v) := Ω u v dμ. Since the topology of G is inherited by R n , we can talk in virtue of the canonical quotient mapping about the space of smooth functions C ∞ (G) and the Schwartz-Bruhat space S(G) [4,19]. It is well-known that the Pontryagin dual of G isĜ = R n1 × Λ n2 L , where Λ L := 2π L Z. The differentiable structure onĜ and in particular the Schwartz-Bruhat space S(Ĝ) is introduced in a similar way as for G. We refer to [15,22] for details. We remark S(G) → L q ω (G) → S (G), see [22,Lemma 2] (and [21,Lemma 3.6] in the case q = 1).
We define weighted Sobolev spaces and homogeneous Sobolev spaces in terms of weak derivatives, that is where the equivalence relation ∼ identifies two functions u 1  Remark 3.1. It should be noted that for all q ∈ [1, ∞] and ω ∈ A q (G), W 0,q ω (Ω) = L q ω (Ω), and that for all m ∈ N 0 the spaces W m,q ω (Ω) and W m,q ω (Ω) equipped with their respective norms yield Banach spaces. Moreover, Lemma 3 in [22] shows that C ∞ 0 (G) is dense in W m,q ω (G) as long as q ∈ (1, ∞). Since the approximating sequence constructed there depends neither on the exponent of integrability nor on the Muckenhoupt weight, we see that Muckenhoupt weights behave well under mirroring: For a generic function ϕ on G, let us define Lemma 3.2. Let q ∈ (1, ∞) and ω ∈ A q (R n ) and define Then ω ∈ A q (R n ) and we have the estimate A q ( ω) ≤ 2 q A q (ω). Moreover ω = ω * . By the canonical identification of G and R n1 × [0, L) n2 , we can associate to any domain Ω ⊂ G a domainΩ ⊂ R n . It is instructive to think ofΩ as one periodic cell of the domain Ω. We call a subset Ω ⊂ G a (bounded) Lipschitz domain, if the correspondingΩ ⊂ R n is a (bounded) Lipschitz domain.
Moreover, we divide the boundary ofΩ into the two parts Σ and ∂Ω G , where Σ are the faces at the end of the cell (if there are such) and ∂Ω G coincides with ∂Ω under the canonical identification of G and For bounded domains, weighted spaces can be embedded into non-weighted ones by the open-ended property of Muckenhoupt weights. Let Ω ⊂ G be a (possibly unbounded) Lipschitz domain. We use Lemma 3.3 to introduce the function spaces W 1,q 0,ω (Ω) and W 1,q 0,ω (Ω) in the canonical way, namely as the subspaces of W 1,q ω (Ω) (resp. W 1,q ω (Ω)) of functions whose trace vanishes locally. Again, we introduce the corresponding dual spaces for all u ∈ W 1,q ω (Ω) with h(u) = 0. Proof. See [11, Corollary 2.1] for the corresponding non-periodic result.

Corollary 3.5.
Let Ω ⊂ G be a bounded Lipschitz domain, q ∈ (1, ∞) and ω ∈ A q (G). Then there is an The same assertion has been proven in [11,Corollary 2.2] in the non-periodic setting for u ∈ W 2,q ω (Ω) with u| ∂Ω = 0, i.e., u vanishes on the whole of ∂Ω. Revising the proof, we see that it suffices that u vanishes on ∂Ω G .
Multipliers that are smooth only outside the origin play an important rôle in the field of partial differential equations. Therefore, we state the following theorem on 0-homogeneous multipliers.

Theorem 4.3.
Let M ∈ C n (R n \ {0}) be homogeneous of degree 0. Then the origin 0 ∈ R n is contained in the Lebesgue set of M . In particular, for q ∈ (1, ∞) and ω ∈ A q (G) the partially periodic Riesz Proof. Due to the homogeneity of M it holds which shows that 0 is in the Lebesgue set of M .
For the assertion about the Riesz transformation, define We note that m j = M j |Ĝ, where for η = 0 this is to be understood in the sense that lim r→0 , the assertion follows from Theorem 4.3.
the open-ended property of Muckenhoupt weights, we obtain with P (x) := (L + |x |) n1 as in the proof of Lemma 2 in [22] the estimate Next, for k ∈ N, the volume of a cuboid U k with edges of length 2 k in the direction of the variables x and length L in the direction of y, can be computed as μ(U k ) = 2 kn1 . Also, the function P is bounded on U k by P L ∞ (U k ) 2 kn1 , where means that it can be estimated modulo a constant c = c(n 1 , L). Since u ∈ L q ω (G) is harmonic, we obtain by the mean value formula , then we use the same computation as above for Then ψ can be extended to a periodic function, and hence Bρ uΔψ dx = 0. Therefore, by Weyl's Lemma, u is harmonic in B ρ . Since the origin of the ball was arbitrary, u is harmonic everywhere.
In order to show the density of Then v is harmonic and by part (i) it follows v = 0. Hahn-Banach's theorem yields the assertion.

Weak Solutions to the Laplace Equation
Consider the weak Laplace operator for j ∈ {1, . . . , n}, where we have used Corollary 4.4, which also gives A q (G)-consistency and hence A q (G)-consistency of the constant c. This shows . Therefore, Δ q,ω is injective and has closed range. By reasons of symmetry it holds Δ q,ω = Δ q ,ω and thus also the adjoint operator is injective and has closed range. Consequently, Δ q,ω is an isomorphism by the closed range theorem.
(G) and so F = 0. An application of Hahn-Banach's theorem yields the assertion.
Let us now turn to the resolvent problem of the Laplace equation. Assume λ ∈ Σ ϑ for some ϑ ∈ (0, π) and consider the operator Vol. 20 (2018) Instationary Generalized Stokes Equations 299 , then an application of the Fourier transform gives u = 0, which shows the injectivity of (λ − Δ) q,ω .
Concerning the surjectivity, we find for By density, this extends to all ϕ ∈ W 1,q ω (G). Furthermore, estimates (4.3) give An application of the Fourier transform shows that v = 0 and hence u 1 = u 2 . (iii) The proof follows analogously as in part (i), only without f 0 and u 0 . Then |λ| u L q ω (G) and Let us conclude this section with a regularity result. For a functional F ∈ W −1,q which is well-defined by Proposition 4.9 (with λ = 1) and due to the facts (ii) By truncation, a function in L q (G) can be approximated by functions in L q (G) with compact support. Hence, let f ∈ L q (G) have compact support and set (f ) := G f dμ. For R > 0, let Q R ⊂ G be a cuboid with length R 1/n1 in direction of the variables x and length L in direction of the variables y and consider the characteristic function χ QR . If we write Note that for all Then u and consequently also ∇ 2 u ∈ L q ω (G) are harmonic. Lemma 4.5 shows ∇ 2 u = 0. For existence, we note that we may assume f ∈ C ∞ 0 (G) by density. Moreover, by Corollary 4.10 and Remark 4.11 the assertion is true for all f ∈ L q ω (G) ∩ W −1,q ω (G). Hence, an approximation procedure using part (ii) yields a solution u ∈ W 2,q (G). Furthermore, for each j ∈ {1, . . . , n} we , whence we obtain u ∈ W 2,q ω (G). (iv) This is just another application of Proposition 4.7(ii).
Proof. (i) To prove uniqueness, let (u, p) ∈ W 2,q ω (G) n × W 1,q ω (G) be a solution to (4.8) with data (f, g) = (0, 0). Then ∇div u = 0, which shows that div u is constant. Therefore Δp = 0, and so p and ∇p ∈ L q ω (G) are harmonic. In view of Lemma 4.5 we receive ∇p = 0. It follows Δu = 0 and Corollary 4.12 gives ∇ 2 u = 0. For existence, note that by Proposition 4.7 there is a unique pressure q ∈ W 1,q ω (G) satisfying Δ q,ω q = div f ∈ W −1,q ω (G). Moreover, there is an A q -consistent constant c = c(n, q, ω) > 0 such that , where c = c(n, q, ω) > 0 is an A q -consistent. It remains to verify that ∇div u = ∇g. Since v := ∇div u − ∇g ∈ L q ω (G) n is harmonic, this is ensured by Lemma 4.5. (ii) In the proof of part (i), the regularity of q stems from Proposition 4.7 (i). Consequently, by Propo-

Proof of Theorem 1.4 in the Whole Space
In view of Proposition 4.7 we may define the pressure p ∈ W 1,q ω (G) as the solution to the weak Laplace equation with right-hand side div f + (λ − Δ)g ∈ W −1,q ω (G). Moreover, let us define v g := ∇W ∈ L q ω (G) n , where W := Δ −1 q,ω g. Note that Corollary 4.10(i) implies v g ∈ W 2,q ω (G) n . Therefore, we can apply [22, Theorem 1] to solve Note that there is an A q -consistent c = c(n, q, ω, ϑ) > 0 such that Setting u := v + v g , we obtain a solution (u, p) ∈ W 2,q ω (G) n × W 1,q ω (G) to (1.2) with a corresponding A q -consistent a priori estimate. This proves the existence part of the theorem.

Trace Spaces
It will be convenient to introduce the group H := R n1−1 × T n2 L , such that G + = R + × H. We usually use the symbol x to refer to an element in H. Note that doing so, we have several notations for a point x ∈ G + corresponding to the different splittings x). We introduce trace spaces in the weighted set-up as quotient spaces, identifying the boundary of G + with H. Note that we can introduce a differentiable structure on H similar to G, and consequently the spaces C ∞ 0 (H) and S(H) are well-defined. Definition 5.1. Let q ∈ (1, ∞), ω ∈ A q (G) and m ∈ N. Then we define the weighted trace spaces T m,q ω (H) := W m,q ω (G + )/ ∼, T m,q ω (H) := W m,q ω (G + )/ ∼, where the equivalence relation identifies two functions whose difference has locally a vanishing trace. The topologies of T m,q ω (H) and T m,q ω (H) are given by the quotient topology. In particular, T m,q ω (H) and T m,q ω (H) are Banach spaces.
vanishes locally}, and this norm is independent of the choice of the respective representative u ∈ W m,q ω (G + ). We will write γ(u) := [u] in the following. With this notation it is obvious that the trace operator γ : W m,q ω (G + ) → T m,q ω (H) is bounded, linear and surjective. An analogous statement can be made in the case of homogeneous spaces, i.e., about the trace operator γ : W m,q ω (G + ) → T m,q ω (H). Remark 5.3. There are certain cases, in which the trace spaces can be identified with fractional order Sobolev spaces. For example, in the nonperiodic case G + = R n + , it is well known that weights of the form ω α (x) := dist(x, ∂R n + ) α are in the class A q (R n ) for α ∈ (−1, q − 1) and that T 1,q ωα (R n−1 ) = W 1− 1+α q ,q (R n−1 ), see [18].

Instationary Generalized Stokes Equations 305
Proof. Follows from the corresponding results in G and the fact that for N ∈ N, i ∈ {1, . . . , N}, m i ∈ N 0 , 1 ≤ q i < ∞ and ω i ∈ A qi (R n ) there is an extension operator Λ : (G), see [6]. In fact, in [6] the assertion is proved only for n 2 = 0, but revising the proof, it is readily seen that also the general case is admissible. Lemma 5.6. Let q ∈ (1, ∞), ω ∈ A q (G) and k ∈ N. Then C ∞ 0 (H) is dense in T 1,q ω (H), T 2,q ω (H) and T k,q ω (H).

J. Sauer JMFM
Proof. It suffices to show the assertion for u ∈ W 1,q 0,ω (G + ), since trivially Eu ∈ L q ω (G) for u ∈ L q ω (G + ). Moreover, it suffices to prove that ∂ i Eu coincides almost everywhere on G with the zero extension E(∂ i u) for all i ∈ {1, . . . , n}, since then 0,ω (G + ) and let B ρ be a ball with radius ρ L. Furthermore, let ψ ∈ C ∞ 0 (G) be such that ψ = 1 on B ρ/2 and supp ψ ⊂ B ρ . Then ψu ∈ W 1,r 0 (Q) for some r > 1, where Q := B ρ ∩ G + . Take a sequence {u m } m∈N ⊂ C ∞ 0 (Q) approximating ψu in the space W 1,r 0 (Q) and compute for every Thus, ∂ i Eu = E(∂ i u) as an identity in S (G) and the assertion is proven.

Proof. By Lemma 3.2 we can assume
. It follows γ(ϕ) = 0 and supp ϕ ⊂ Q, where Q = G + ∩ U for some smooth and compact U ⊂ G. By Lemma 3.3, we know that there is s > 1 such that u| Q ∈ W 1,s (Q). Moreover, ϕ ∈ W 1,s 0 (Q) and we thus find a sequence {ϕ k } ⊂ C ∞ 0 (Q) converging to ϕ in W 1,s 0 (Q). Let us denote by v the odd extension of u to the whole group. Then in the situation of part (i) it holds since u is harmonic on G + . Therefore, G v Δϕ dμ = 0 for all ϕ ∈ C ∞ 0 (G) and Lemma 4.5 (ii) shows that v is harmonic. In particular, also ∇v is harmonic and so ∇v ∈ C ∞ (G). Moreover, we have Since v is smooth across the interface of G + and G − , this implies the regularity ∇v ∈ L q1 ω1 (G) + L q2 ω2 (G). Lemma 4.5 (i) gives now ∇v = 0, whence we conclude u = 0 by the boundary condition γ(u) = 0.
The second assertion follows analogously.
(i) There is a linear, A q -consistently bounded extension operator . Proof. Follows directly from Theorem 5.12.
We can even improve the regularity result about the extension operator R λ . To do so, we need the following lemma.
In fact, the extension operators R and R λ have a very familiar representation in terms of Poisson operators. To see this, we will first prove a preliminary lemma. In this lemma, we use Plancherel's theorem on the locally compact abelian group H. Note that the Fourier transform yields an isometry from L 2 (H) to L 2 (Ĥ) only if the corresponding Haar measures on H andĤ are normalized accordingly. In our case, this gives an additional c n > 0 depending on the dimension n such that f L 2 (H) = c n F H f L 2 (Ĥ) . However, as it turns out, we are only interested in finiteness of the L 2 -norms, and we can therefore suppress the dimensional constant c n in the following.
Proof. Once we have shown ∇F ∈ W m,2 (G + ) for all m ∈ N 0 , Sobolev's embedding theorem immediately gives F ∈ C ∞ (G + ), and similarly for f and f λ . Hence, we focus on the assertion about the L 2 (G + ) regularity. We want to employ Plancherel's theorem. Note that we are in the unweighted case, and hence we can write L 2 (G + ) = L 2 (R + ; L 2 (H)). Observe ψ := F H ψ ∈ S(Ĥ). Thus, by elementary computation, we obtain Moreover, for k ∈ {2, . . . , n}, it holds Finally, since λ = R − , there is δ > 0 such that Re( √ λ + s 2 ) ≥ δ(1 + s) for all s > 0, see for example the proof of [9, Lemma 2.5]. Therefore |e −x1 √ λ+s 2 | ≤ e −x1δ(1+s) , and it follows Sauer JMFM Therefore f λ , ∇f, ∇F ∈ L 2 (G + ). To take care of the higher derivatives, note that for k ∈ {2, . . . , n} it holds Observe that m, m λ , m F and M/s are bounded near the origin. Hence, in any case we are in one of the situations discussed above and it follows f λ , ∇f, ∇F ∈ W 1,2 (G + ). Iterating this process, we can estimate every order of differentiability.
With these preparations in mind, we are ready to identify the Poisson operator with the extension operator R λ . Let us denote by T λ the linear operator defined on S(H) via where x = (x 1 , x) ∈ G + . Note, that the case λ = 0 is included here. In accordance with our notation for R and R λ , we will drop the index λ = 0 if no confusion can arise.
is the unique extension of the operator T to a bounded linear operator on T 1,q ω (H) with the property γ • R = id T 1,q ω (H) . Moreover, it holds for almost all There is an A q -consistent constant c = c(n, q, ω, ϑ) > 0 such that is the unique extension of the operator T λ to a bounded linear operator on T 1,q ω (H) with the property γ • R λ = id T 1,q ω (H) . Furthermore, it holds for almost all shows that T φ is harmonic. Hence, for each φ ∈ S(H), T φ ∈ W 1,2 (G + ) is a solution to (5.2) with F = 0. Since S(H) ⊂ T 1,2 (G) ∩ T 1,q ω (G) for all ω ∈ A q (G) by Corollary 5.6, the regularity assertion in Theorem 5.12 gives T φ = Rφ ∈ W 1,q ω (G) and an A q -consistent constant c = c(n, q, ω) such that ∇T φ L q ω (G) = ∇Rφ L q ω (G) ≤ c|φ| T 1,q ω (G) . Moreover, for j ∈ {2, . . . , n} we have ∂ j φ ∈ S(H), and hence by the same arguments it follows where we have used Lemma 5.8 in the last estimate. Since T φ is harmonic, we obtain ∂ 2 1 T φ = − n j=2 ∂ 2 j T φ and so ∂ 2 1 T φ L q ω (G) ≤ (n − 1)c|φ| T 2,q ω (G) . Summarizing, we have proved the claimed a priori estimates. Corollary 5.6 shows that S(H) is dense in T 1,q ω (G) and therefore part (i) is proven.

Weak Solutions to the Stokes Equations
In this section, we investigate weak solutions to the Stokes equations in the periodic half space, i.e., we consider the problem where c = c(n, q, ω) > 0 is A q -consistent. Moreover, for φ ∈ T 2,2 (H) n , this weak solution solves (5.5) even in a strong sense, i.e., (w, q) ∈ W 2,2 (G + ) n × W 1,2 (G + ) and there is a positive constant c = c(n) > 0 such that Proof. By Lemma 5.6, S(H) is dense in both T 1,q ω (H) and T 2,2 (H), and so it suffices to construct a solution with the correct regularity and a priori estimate for φ ∈ S(H) n . Hence, let φ ∈ S(H) n be fixed.
We define the pressure where R is the extension operator defined in Corollary 5.14. Then by Theorem 5.20 it follows that there is an A q -consistent c = c(n, q, ω) > 0 such that Note that also q L 2 (G+) ≤ c |φ| T 1,2 (G+) and hence ∇q ∈ W −1,2 0 (G + ) n ∩ W −1,q 0,ω (G + ) n . Moreover, we define for every j ∈ {1, . . . , n} the component w j via  Lemma 5.19. Thus, w is smooth and it holds D α ∇w ∈ L 2 (G + ) for all α ∈ N n 0 , in particular for |α| = 0. Let us verify that γ(w) = φ, div w = 0 and Δw = ∇q. Since φ and w are smooth, we can evaluate point-wise and obtain easily γ(w) = φ by considering x 1 0. Concerning the divergence, we compute Thus, it suffices to show that the inner sum vanishes for each i ∈ {1, . . . , n} separately. Let us apply F H to (5.10) for notational convenience. Then, for i = 1 we obtain withφ 1 := F H φ 1 and using (5.9) Similarly, for i ∈ {2, . . . , n}, Thus, we have proven div w = 0. It remains to show Δw = ∇q, that is Δw j = ∂ j q for all j ∈ {1, . . . , n}. Again, we start with j = 1 and compute F H ∂ 1 q, where we apply the Fourier transform merely for the sake of readability. By definition of q and by the representation of R in Theorem 5.20, On the other hand, whence we see Comparing (5.13) to (5.11), the relation Δw 1 = ∂ 1 q follows. To show Δw j = ∂ j q for j ∈ {2, . . . , n}, we proceed analogously. It holds Vol. 20 (2018)
It is left to show the a priori estimate (5.7). For j ∈ {2, . . . , n} we know that ∂ j w ∈ W 1,2 (G + ) is a weak solution to the Laplace equation (5.2) with right hand side ∇∂ j q ∈ W −1,2 0 (G + ) and with boundary data ∂ j φ ∈ T 1,2 (H). Theorem 5.12 and Theorem 5.20 applied to q = 2 and ω = 1 show with a constant c > 0. For the estimates of the derivatives with respect to the first variable we use the Stokes equations in order to obtain whence the estimate (5.7) follows.
Let q ∈ (1, ∞) and ω ∈ A q (G). Similarly as in the case of the periodic whole space G, we introduce the Banach spaces and , furnished with the respective product space norms. With this notation, we have the following theorem.

Strong Solutions to the Stokes Equations
Let us consider strong solutions to the Stokes equations in the periodic half space. More precisely, we investigate the problem We have the following regularity result. solution to (5.18), then (u, p) ∈ W 2,q2 ω2 (G + ) × W 1,q2 ω2 (G + ) and there is an A q2 (G)-consistent constant c = c(n, q 2 , ω 2 ) > 0 such that Proof. Recall the definition of the spaces X q ω (G + ) and Y q ω (G + ) in (5.16). Let j ∈ {2, . . . , n} and observe that ( where the regularity assertion for ∂ j φ stems from Lemma 5.8. Moreover, (∂ j u, ∂ j p) ∈ X q1 ω1 (G + ) is a weak solution to the Stokes equations (5.5) with data (∂ j f, ∂ j g, ∂ j φ). The regularity assertion in Theorem 5.22 gives (∂ j u, ∂ j p) ∈ X q2 ω2 (G + ) and an A q2 (G)-consistent c = c(n, q 2 , ω 2 ) > 0 such that . For the derivatives with respect to the first variable, we use the Stokes equations (5.18) and observe for k ∈ {2, . . . , n} which implies u ∈ W 2,q2 ω2 (G + ), ∇p = f + Δu ∈ L q2 ω2 (G + ) and the full a priori estimate.

Estimates on the Boundary
Define for i ∈ {2, . . . , n} and φ ∈ S(H) the Riesz transformation on H by

J. Sauer JMFM
Moreover, define the operator for λ ∈ C \ R − . As usual, we will drop the index λ in the case λ = 0.

Proof of Theorem 1.4 in the Half Space
We divide the proof into five steps.
Step 1: Scaling Argument We claim that it is sufficient to prove the theorem for λ ∈ Σ ϑ with |λ| = 1. For ε > 0, we write ψ ε (x) := ψ(x/ε) for a generic function ψ. Observe that ω ∈ A q (G) if and only if ω ε ∈ A q (G ε ) with A q (ω ε ) = A q (ω), where the locally compact abelian group G ε := R n1 × T)εL n2 is equipped with the Haar measure μ ε defined via Note that the length of periodicity L > 0 does not enter in the constant of the transference principle of Theorem 4.1, and hence all results obtained so far hold true also in G ε with estimates independent of ε > 0.