Elliptic and Parabolic Boundary Value Problems in Weighted Function Spaces

In this paper we study elliptic and parabolic boundary value problems with inhomogeneous boundary conditions in weighted function spaces of Sobolev, Bessel potential, Besov and Triebel-Lizorkin type. As one of the main results, we solve the problem of weighted Lq-maximal regularity in weighted Besov and Triebel-Lizorkin spaces for the parabolic case, where the spatial weight is a power weight in the Muckenhoupt A∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$A_{\infty }$\end{document}-class. In the Besov space case we have the restriction that the microscopic parameter equals to q. Going beyond the Ap-range, where p is the integrability parameter of the Besov or Triebel-Lizorkin space under consideration, yields extra flexibility in the sharp regularity of the boundary inhomogeneities. This extra flexibility allows us to treat rougher boundary data and provides a quantitative smoothing effect on the interior of the domain. The main ingredient is an analysis of anisotropic Poisson operators.

An important feature of the power weights (1-1) is that they allow to treat "rougher" behaviour in the initial time and on the boundary by increasing the parameters µ and γ, respectively. In [57,59,63,66,73] this is for instance reflected in the lower regularity of the initial/initial-boundary data that can be dealt with. In the L p -approach to parabolic problems with Dirichlet boundary noise, where the noise is a source of roughness on the boundary, weights are even necessary to obtain function space-valued solution processes [1,30,62].
As in [59], in this paper we exploit this feature of the power weights  in the study of vector-valued parabolic initial-boundary value problems of the form (1)(2) ∂ t u(x, t) + A(x, D, t)u(x, t) = f (x, t), x ∈ O, t ∈ J, B j (x ′ , D, t)u(x ′ , t) = g j (x ′ , t), x ′ ∈ ∂O, t ∈ J, j = 1, . . . , m, u(x, 0) = u 0 (x), Here, J is a finite time interval, O ⊂ R n is a C ∞ -domain with a compact boundary ∂O and the coefficients of the differential operator A and the boundary operators B 1 , . . . , B n are B(X)-valued, where X is a UMD Banach space. One could for instance take X = C N , describing a system of N initial-boundary value problems. Our structural assumptions on A, B 1 , . . . , B m are an ellipticity condition and a condition of Lopatinskii-Shapiro type. For homogeneous boundary data (i.e. g j = 0, j = 1, . . . , m) these problems include linearizations of reaction-diffusion systems and of phase field models with Dirichlet, Neumann and Robin conditions. However, if one wants to use linearization techniques to treat such problems with non-linear boundary conditions, it is crucial to have a sharp theory for the fully inhomogeneous problem.
Maximal regularity provides sharp/optimal estimates for PDEs. Indeed, maximal regularity means that there is an isomorphism between the data and the solution of the problem in suitable function spaces. It is an important tool in the theory of nonlinear PDEs: having established maximal regularity for the linearized problem, the nonlinear problem can be treated with tools as the contraction principle and the implicit function theorem (see [74]).
The main result of this paper is concerned with weighted L q -maximal regularity in weighted Triebel-Lizorkin spaces for (1)(2), where we use the weights . In order to elaborate on this, let us for reasons of exposition consider as a specific easy example of (1-2) the heat equation with the Dirichlet boundary condition where J = (0, T ) with T ∈ (0, ∞) and where O is a smooth domain in R n with a compact boundary ∂O.
In order to introduce the weighted L q -maximal regularity problem for (1)(2)(3) in an abstract setting, let q ∈ (1, ∞), µ ∈ (−1, q − 1) and E ⊂ D ′ (O) a Banach space of distributions on O such that there exists a notion of trace on the associated second order space E 2 = {u ∈ D ′ (O) : D α u ∈ E, |α| ≤ 2} that is described by a bounded linear operator Tr ∂O : E 2 −→ F for some suitable Banach space.
In the L q,µ -E-maximal regularity approach to (1-3) one is looking for solutions u in the maximal regularity space where the boundary condition u |∂O = g has to be interpreted as Tr ∂O u = g. The problem (1-3) is said to enjoy the property of maximal L q,µ -E-regularity if there exists a (necessarily unique) space of initial-boundary data D i.b. ⊂ L q (J, v µ ; F) × E such that for every f ∈ L q (J, v µ ; E) it holds that (1-3) has a unique solution u in (1)(2)(3)(4) if and only if (g, u 0 ) ∈ D i.b. . In this situation there exists a Banach norm on D i.b. , unique up to equivalence, with which makes the associated solution operator a topological linear isomorphism between the data space L q (J, v µ ; E) ⊕ D i.b. and the solution space W 1 q (J, v µ ; E) ∩ L q (J, v µ ; E 2 ). The maximal L q,µ -E-regularity problem for (1)(2)(3) consists of establishing maximal L q,µ -E-regularity for (1)(2)(3) and explicitly determining the space D i.b. .
Establishing L q,µ -L p,γ -maximal regularity with p q allows one to treat more nonlinearities than in the case p = q, as it provides more flexibility for scaling or criticality arguments (see e.g. [33], [51], [75], [76], [77]). Such arguments have turned out to be crucial in applications to the Navier-Stokes equations, convection-diffusion equations, the Nerst-Planck-Poisson equations in electro-chemistry, chemotaxis equations and the MHD equation (see [75], [77]).
One way to avoid these difficulties is to work in weighted Besov and Triebel-Lizorkin spaces instead of E = L p (O, w ∂O γ ). The advantage of the scales of weighted Besov and Triebel-Lizorkin spaces is the strong harmonic analytic nature of these function spaces, leading the availability of many powerful tools (see e.g. [9,10,11,40,41,42,43,60,67,68,69,84]). In particular, there is a Mikhlin-Hörmander Fourier multiplier theorem.
Here δ is again given by δ = δ p,γ,s = s 2 + 1 − 1+γ p . Note however, that we assume that the time integrability parameter and the microscopic parameter in space coincide.
The main technical ingredient in this paper is an analysis of anisotropic Poisson operators and their mapping properties on weighted mixed-norm anisotropic function spaces. The Poisson operators under consideration naturally occur as (or in) solution operators to the model problems where A(D) and B j (D) are homogeneous with constant coefficients. Moreover, they are operators K of the form (1-10) for some anisotropic Poisson symbol-kernel k.
The anisotropic Poisson operator (1-10) is an anisotropic (x ′ , t)-independent version of the classical Poisson operator from the Boutet the Monvel calculus. The Boutet the Monvel calculus is a pseudo-differential calculus that in some sense can be considered as a relatively small "algebra", containing the elliptic boundary value problems as well as their solution operators (or parametrices). The calculus was introduced by, as the name already suggests, Boutet de Monvel [5,6], having its origin in the works of Vishik and Eskin [86], and was further developed in e.g. [35,36,37,78,46]; for an introduction to or an overview of the subject we refer the reader to [37,38,82].
A parameter-dependent version of the Boutet de Monvel calculus has been introduced and worked out by Grubb and collaborators (see [37] and the references given therein). This calculus contains the parameter-elliptic boundary value problems as well as their solution operators (or parametrices). In particular, resolvent analysis can be carried out in this calculus.
In the present paper we also consider a variant of the parameter-dependent Poisson operators from [37] in the x ′ -independent setting. Besides that this is one of the key ingredients in our treatment of the parabolic problems (1-2) through the anisotropic Poisson operators (1)(2)(3)(4)(5)(6)(7)(8)(9)(10), it also forms the basis for our parameter-dependent estimates in weighted Besov, Triebel-Lizorkin and Bessel potential spaces for the elliptic boundary value problems These parameter-dependent estimates are an extension of [60] on second order elliptic boundary value problems subject to the Dirichlet boundary condition, which was in turn in the spirit of [22,39] (see Remark 6.3).
In the latter the scales of weighted B-and F -spaces, the dual scales to the scales of weighted B-and F-spaces, are also included. These scales naturally appear in duality theory and can for instance be used in the study of parabolic boundary value problems with multiplicative noise at the boundary in a setting of weighted L p -spaces, see Remark 6.9.
Outline. The outline of the paper is as follows.
• Section 2: Preliminaries from weighted (mixed-norm anisotropic) function spaces, distribution theory, UMD Banach spaces and L q -maximal regularity, differential boundary value systems. • Section 3: Sobolev embedding and trace results for mixed-norm anisotropic function spaces. • Section 4: Introduction and basic properties of Poisson operators, solution operators to model problems and mapping properties. • Section 5: L q,µ -maximal regularity for parabolic boundary value problems (1-2).
In the whole paper let Let X be a Banach space and (S , A , µ) a measure space. Throughout the paper we write L 0 (S ; X) for the space of all equivalence classes of strongly measurable functions f : S → X, where as usual the equivalence relation is the one of functions that coincide almost everywhere. A weight on a measure space (S , A , µ) is a measurable function w : S −→ [0, ∞] that takes it values almost everywhere in (0, ∞). We denote by W(S ) the sets of all weights on (S , A , µ). For w ∈ W(S ) and p ∈ [1, ∞) we denote by L p (S , w) the space of all f ∈ L 0 (S ; C) with We equip L p (S , w) with the norm · L p (S ,w) , which turns it into a Banach space. As an extension (and in fact consequence) of (2-1), for p ∈ (1, ∞) we have [L p (S , w)] * = L p ′ (S , w ′ p ) isometrically with respect to the pairing . Given a Banach space X, we denote by L p (S , w; X) the associated Bochner space For p ∈ (1, ∞) we denote by A p = A p (R n ) the class of all Muckenhoupt A p -weights, which are all the locally integrable weights for which the A p -characteristic . For the important model problem case O = R n + we simply write w γ := w R n + γ = dist( · , ∂R n + ) γ . Furthermore, in connection with the pairing (2-1), for p ∈ (1, ∞) we have in view of (2-4).
2.2. UMD Spaces and L q maximal Regularity. The general references for this subsection are [44,45,54]. The UMD property of Banach spaces is defined through the unconditionality of martingale differences, which is a primarily probabilistic notion. A deep result due to Bourgain and Burkholder gives a pure analytic characterization in terms of the Hilbert transform: a Banach space X has the UMD property if and only if it is of class HT, i.e. the Hilbert transform H has a bounded extension H X to L p (R; X) for any/some p ∈ (1, ∞). A Banach space with the UMD property is called a UMD Banach space. Some facts: • Every Hilbert space is a UMD space; • If X is a UMD space, (S , Σ, µ) is σ-finite and p ∈ (1, ∞), then L p (S ; X) is a UMD space. • UMD spaces are reflexive.
• Closed subspaces and quotients of UMD spaces are again UMD spaces. In particular, weighted Besov and Triebel-Lizorkin spaces (see Section 2.4) are UMD spaces in the reflexive range.
Let A be a closed linear operator on a Banach space X. For q ∈ (1, ∞) and v ∈ A q (R) we say that A enjoys the property of In the specific case of the power weight v = v µ with q ∈ (−1, q − 1), we speak of L q,µ (R)maximal regularity and L q,µ (R + )-maximal regularity.
Note that L q (v, R)-maximal regularity and L q (v, R + )-maximal regularity can also be formulated in terms of evolution equations. For instance, A enjoys the property of L q (v, R + )maximal regularity if and only if, for each f ∈ L q (v, R + ; X), there exists a unique solution References for L q (R)-maximal regularity and L q (R + )-maximal regularity include [4,71] and [27,54]. Works on L q (R + , v)-maximal regularity include [12,13,31].
Finally, as −A is the generator of an exponentially stable analytic semigroup on X, the variation of constants formula yields L q (R + , v)-maximal regularity. Indeed, viewing ) and L q (R + , v; X) as closed subspaces of W 1 q (R, v; X) ∩ L q (R, v; D(A)) and L q (R, v; X), respectively, through extension by zero, the formula shows that (∂ t + A) −1 maps L q (R + , v; X) to 0 W 1 q (R + , v; X) ∩ L q (R + , v; D(A)). As an application of its operator-valued Fourier multiplier theorem, Weis [87] characterized L q (R + )-maximal regularity in terms of R-sectoriality in the setting of UMD Banach spaces. The corresponding result for L q (R)-maximal regularity involves R-bisectoriality, see [4]. Using [31,Theorem 3.5] and Theorem A.1, these results carry over to the weighted setting.
Let us introduce the notion of R-boundedness. Let X be a Banach space. Let (ε k ) k∈N be a Rademacher sequence on some probability space (Ω, F , P), i.e. a sequence of independent random variables with P(ε k = 1) = P(ε k = −1) = 1 2 . A collection of operators T ⊂ B(X) is called R-bounded if there exists a finite constant C ≥ 0 such that, for all K ∈ N, T 0 , . . . , T K ∈ T and x 0 , . . . , . The least such constant C is called the R-bound of T and is denoted by R(T ).
The space Rad p (N; X), where p ∈ [1, ∞), is defined as the Banach space of sequence (x k ) k∈N for which there is convergence of ∞ k=0 ε k x k in L p (Ω; X), endowed with the norm .
As a consequence of the Kahane-Khintchine inequalities, Rad p (N; X) = Rad q (N; X) with an equivalence of norms. We put Rad(N; X) = Rad 2 (N; X). Note that a collection of operators T ⊂ B(X) is called R-bounded if and only if {diag(T 0 , . . . , T K ) : T 0 , . . . , T K ∈ T } ⊂ B(Rad(N; X)) is a uniformly bounded family of operators, in which case the R-bound coincides with that uniform bound; here Furthermore, note that, as a consequence of the Kahane-Khintchine inequalities and Fubini, given p ∈ [1, ∞) and a σ-finite measure space (S , A , µ), there is a natural isomorphism of Banach spaces Rad(N; L p (S ; X)) ≃ L p (S ; Rad(N; X)).
Having introduced the notion of R-boundedness, we can now give the definition of Rsectoriality, which is an R-boundedness version of sectoriality.
Recall that an unbounded operator A on a Banach space X is a sectorial operator if A is injective, closed, has dense range and there exists a φ ∈ (0, π) such that Σ π−φ ⊂ ρ(−A) and The infimum over all possible φ is called the angle of sectoriality and is denoted by ω(A).
In this case we also say that A is sectorial of angle ω(A). The condition that A has dense range is automatically fulfilled if X is reflexive (see [45,Proposition 10.1.9]). We say that an unbounded operator A on a Banach space X is an R-sectorial operator if A is injective, closed, has dense range and there exists a φ ∈ (0, π) such that Σ π−φ ⊂ ρ(−A) and The infimum over all possible φ is called the angle of R-sectoriality and is denoted by ω R (A). In this case we also say that A is R-sectorial of angle ω R (A).
A way to approach L q -maximal regularity is through operator sum methods, as initiated by Dore & Venni [28]. Using the Kalton-Weis operator sum theorem [49,Theorem 6.3] in combination with [63, Proposition 2.7], we obtain the following result: Proposition 2.3. Let X be a UMD space, q ∈ (1, ∞) and v ∈ A q (R). If A is a closed linear operator on a Banach space X with 0 ∈ ρ(A) that is R-sectorial of angle ω R (A) < π 2 , then A enjoys the properties of L q (v, R)-maximal regularity and L q (v, R + )-maximal regularity.

Decomposition and Anisotropy
is called the d -decomposition of R n . For x ∈ R n we accordingly write x = (x 1 , . . . , x l ) and x j = (x j,1 , . . . , x j,d j ), where x j ∈ R d j and x j,i ∈ R ( j = 1, . . . , l; i = 1, . . . , d j ). We also say that we view R n as being d -decomposed. Furthermore, for each k ∈ {1, . . . , l} we define the inclusion map . . , 0, x k , 0, . . . , 0), and the projection map For x ∈ R l and y ∈ R n we define x j y j,i .
Given a ∈ (0, ∞) l , we define the (d , a)-anisotropic dilation δ (d ,a) λ on R n by λ > 0 to be the mapping δ (d ,a) λ on R n given by the formula x ∈ R n .
A (d , a)-anisotropic distance function on R n is a function u : for all x ∈ R n and λ > 0. (iii) There exists a c > 0 such that u(x + y) ≤ c(u(x) + u(y)) for all x, y ∈ R n .
All (d , a)-anisotropic distance functions on R n are equivalent: Given two (d , a)-anisotropic distance functions u and v on R n , there exist constants m, In this paper we will use the (d , a)-anisotropic distance function | · | d ,a : R n −→ [0, ∞) given by the formula 2.4. Distribution Theory and Function Spaces. As general references for this subsection we would like to mention [3,9,56].
2.4.1. Distribution Theory and Some Generic Function Space Theory. Let X be a Banach space. The spaces of X-valued distributions and X-valued tempered distributions on R n are defined as D ′ (R n ; X) := L(D(R n ), X) and S ′ (R n ; X) := L(S(R n ), X), respectively; for the theory of vector-valued distributions we refer to [3] . More generally, given Banach spaces E ֒→ D ′ (U 1 ; X 1 ) and F ֒→ D ′ (U 2 ; X 2 ), T ∈ B(E, F) and open subsets Given a Banach space Z, O M (R n ; Z) denotes the space of slowly increasing Z-valued smooth functions on R n . Pointwise multiplication ( f, g) → f g yields separately continuous bilinear mappings (2)(3)(4)(5)(6) O M (R n ; B(X)) × S(R n ; X) −→ S(R n ; X), O M (R n ; B(X)) × S ′ (R n ; X) −→ S ′ (R n ; X).
As a consequence, (m, f ) → F −1 [mf ] yields separately continuous bilinear mappings (2)(3)(4)(5)(6). We use the following notation: Let E ֒→ D ′ (U; X) be a Banach space of distributions on an open subset U ⊂ R n . For a finite set of multi-indices J ⊂ N d we define the Sobolev space W J [E] as the space of all f ∈ E with D α f ∈ E for every α ∈ J, equipped with the norm Suppose R n is d -decomposed as in Section 2.3. For a Banach space Z, a ∈ (0, ∞) l and N ∈ N we define M (d In case a = 1 we simply speak of admissible.
To each σ ∈ R we associate the operators is the standard i-th basis vector in R d j . If E ֒→ S ′ (R n ; X) is a (d , a)-admissible Banach space for a given a ∈ (0, ∞) l , then Furthermore, Let E, F ֒→ S ′ (R n ; X) be (d , a)-admissible Banach spaces for a given a ∈ (0, ∞) l . If ( · , · ) is an interpolation functor (e.g. a)-admissible as well. Moreover, it holds that (which can be seen through the pairing (2-2)). So we can define the associated Sobolev space of order k ∈ N l Using the pairing (2-2), we find that L p (R n , w; X) ֒→ S ′ (R n ; X) in the natural way. For a ∈ (0, ∞) l , s ∈ R and ς ∈ (0, ∞) l we can thus define the Bessel potential spaces If X is a UMD space and w ∈ l j=1 A rec p j (R d j ) 1 , then L p (R n , w; X) is (a, d )-admissible (see [31]). In particular, if X is a UMD space, then (2-8), (2-9) and (2-10) hold true with as the set of all sequences ϕ = (ϕ n ) n∈N ⊂ S(R n ) which are constructed in the following way: given a ϕ 0 ∈ S(R n ) satisfying should already work, but this is not available in the literature and not needed in this paper anyway.
It holds that f = ∞ n=0 S n f in S ′ (R n ; X) respectively in S(R n ; X) whenever f ∈ S ′ (R n ; X) respectively f ∈ S(R n ; X). Given Up to an equivalence of extended norms on S ′ (R n ; X), · B s,a p,q,d (R n ,w;X) and · F s,a p,q,d (R n ,w;X) do not depend on the particular choice of ϕ ∈ Φ d ,a (R n ). Let us note some basic relations between these spaces. Monotonicity of ℓ q -spaces yields that, . Furthermore, Minkowksi's inequality gives (2-15) B s,a p,min{p 1 ,...,p l ,q},d (R n , w; X) ֒→ F s,a p,q,d (R n , w; X) ֒→ B s,a p,max{p 1 ,...,p l ,q},d (R n , w; X). The real interpolation of weighted isotropic scalar-valued Triebel-Lizorkin spaces from [9, Theorem 3.5] (see [67,Proposition 6.1] in the Banach space-valued A p -setting), can be extended to the mixed-norm anisotropic Banach space-valued setting. This yields the following interpolation identity: , A ∈ {B, F} and s ∈ R. There exists N ∈ N, only depending on a, p, q, n, w, Proof. For simplicity of notation we only treat the case A = F. Let N be as in (2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17) for ) as a consequence of the Kahane-Khintchine inequalities and Fubini. Finally, the observa- The following result is a representation for anisotropic mixed-norm Triebel-Lizorkin spaces in terms of classical isotropic Triebel-Lizorkin spaces (see Paragraph 2.4.2.b).
This intersection representation is actually a corollary of a more general intersection representation in [58], see [58,Example 5.5]. In the above form it can also be found in [56,Theorem 5.2.35]. For the case X = C, d 1 = 1, w = 1 we refer to [21,Proposition 3.23].
There is the following analogue of Theorem 2.5 in the Besov space case: In the parameter range that we have defined the spaces H ς p,d (R n , w; X), H s,a p,d (R n , w; X), B s,a p,q,d (R n , w; X) and F s,a p,q,d (R n , w; X) above, the corresponding versions on open subsets O ⊂ R n are defined by restriction: Parameter-independent spaces. In the special case l = 1 and a = 1, the anisotropic mixednorm spaces introduced in Paragraph 2.4.2.a reduce to classical isotropic Sobolev, Bessel potential, Besov and Triebel-Lizorkin spaces W k respectively. In the case that O is a C ∞ -domain and w = w ∂O γ , we use the notation: is an admissible Banach space of tempered distributions. By lifting, H s p (R n , w; X) is admissible as well. In fact, there is an operator-valued Mikhlin theorem for H s p (R n , w; X) (obtained by lifting from L p (R n , w; X)): , extends to a bounded linear operator on H s p (R n , w; X) with T m B(H s p (R n ,w;X)) X,p,w,n m RM n+2 Proof. The case s = 0 can be obtained as in [70,Proposition 3.1], from which the case of general s ∈ R subsequently follows by lifting.
In the scalar-valued case X = C, we have In the Banach space-valued case, this identity is valid if and only if X is isomorphic to a Hilbert space. For general Banach spaces X we still have (see [67,Proposition 3.12]) where O ⊂ R n is R n + or a C ∞ -domain with compact boundary. For UMD spaces X there is a suitable randomized substitute for (2-21) (see [70, [67,68], if γ 0 > γ 1 and Proposition 2.8. Let X be a UMD space and p ∈ (1, ∞). Let w ∈ A p be such that . Theorem 2.9 (Rychkov's extension operator [81]). Let O be a special Lipschitz domain in R n or a Lipschitz domain in R n with a compact boundary and let X be a Banach space. Then there exists a linear operator with the properties that • Proof. The existence of such an operator for the unweighted scalar-valued variant was obtained in [81,Theorem 4.1]. However, the proof given there extends to the weighted Banach space-valued setting.
Tr ∂O : A s p,q,γ (O; X) −→ ∂A s p,q,γ (∂O; X) that are related by tr ∂O = Tr ∂O • E , where E is any choice of Rychkov's extension operator (from Theorem 2.9). There is compatibility for both of the trace operators tr ∂O and Tr ∂O on the different function spaces that are allowed above.
Let us now introduce reflexive Banach space-valued versions of the B-and F -scales, the scales dual to the B-and F-scales, respectively, as considered in [60]. Let X be a reflexive Banach space, . Notationally it will be convenient to define Parameter-dependent spaces. We now present an extension to the reflexive Banach spacevalued setting of the parameter-dependent function spaces discussed in [60, Section 6], which was in turn partly based on [39]. As the theory presented in [60, Section 6] carries over verbatim to this setting, we only state results without proofs. The reflexivity condition comes from duality arguments involving the dual scales that are needed outside the A p -range. Although for the B-and F-scales duality is only used in Corollary 2.11, for simplicity we restrict ourselves to the setting of reflexive Banach spaces from the start. For σ ∈ R and µ ∈ [0, ∞) we define Ξ σ µ ∈ L(S(R n ; X)) ∩ L(S ′ (R n ; X)) by and denote by A (R n ,w;X) < ∞} equipped with this norm. For the Bessel-potential scale we proceed in a similar way. Suppose that X is a UMD Banach space and let p ∈ (1, ∞) and w ∈ A p (R n ). We define and write H s,µ,s 0 p (R n , w; X) for the space { f ∈ S ′ (R n ; X) : f H s,µ,s 0 p (R n ,w;X) < ∞} endowed with this norm.
It trivially holds that For an open subset U ⊂ R n we put . Next we consider a vector-valued version of the parameter-dependent Besov spaces as introduced in [39], but in the notation of [60,Section 6]. Let X be a reflexive Banach space, where M µ ∈ L(S(R n ; X)) ∩ L(S ′ (R n ; X)) denotes the operator of dilation by µ −1 . We furthermore write B s,µ p,q (R n ; X) for the space { f ∈ S ′ (R n ; X) : f B s,µ p,q (R n ;X) < ∞} equipped with this norm. Then For a compact smooth manifold M we define B s,µ p,q (M) in terms of B s,µ p,q (R n ) in the standard way. Then the analogues of (2-30) and (2-31) for B s,µ p,q (M) are valid. It will be convenient to write and let s ∈ ( 1+γ p , ∞) and s 0 ∈ (−∞, 1+γ p ). Then that is, The respective assertion also holds for the Bessel potential scale if X is a UMD Banach space and if γ ∈ (−1, p − 1). Then 2.5. Differential Boundary Value Systems.

The Equations.
Here we introduce some of the notation and terminology that will be used in Sections 5 and 6 on parabolic and elliptic boundary value problems. Let X be a Banach space, O ⊂ R n a C ∞ -domain with a compact boundary ∂O and J ⊂ R an interval. Let m ∈ N 1 and let m 1 , . . . , m m ∈ N satisfy m i ≤ 2m − 1 for each i ∈ {1, . . . , m}.
Systems on O: Consider we are going to specify later in Section 6. For the moment we just assume the top order coefficients a α , |α| = 2m, and b i,β , |β| = m i , to be bounded and uniformly continuous. The lower order terms are allowed to be distributions. We call (A(D), which we are going to specify later in Section 5. We call (A(D), B 1 (D), . . . , B m (D)) a B(X)-valued boundary value system (of order 2m) on O × J.

Ellipticity and Lopatinskii-Shapiro Conditions. Let us now turn to the two structural assumptions on
The condition (E) φ is parameter ellipticity. In order to state it, we denote by the subscript # the principal part of a differential operator: given a differential operator then it in addition holds that σ(A # (∞, ξ, t)) ⊂ C + for all t ∈ J and |ξ| = 1. By A # (∞, ξ, t) we mean that the limit lim |x|→∞ A # (x, ξ, t) exists for all t ∈ J and all |ξ| = 1 and that A # (∞, ξ, t) is defined as this limit.
The condition (LS) φ is a condition of Lopatinskii-Shapiro type. Before we can state it, we need to introduce some notation. For each x ∈ ∂O we fix an orthogonal matrix O ν(x) that rotates the outer unit normal ν(x) of ∂O at x to (0, . . . , 0, −1) ∈ R n , and define the rotated operators (A ν , B ν ) by In the scalar-valued case, there are several equivalent characterizations for the Lopatinskii-Shapiro condition. It is a common approach to consider the polynomial , then we can formulate the following result: . This condition is sometimes called covering condition. A proof for this statement can for example be found in Chapter 3.2 of [79]. A similar condition can be formulated using the so-called Lopatinskii matrix. Proposition 3.1. Let X be a Banach space, p, p ∈ (1, ∞) l , q, q ∈ [1, ∞], s, s ∈ R, a ∈ (0, ∞) l , and w, w ∈ l j=1 A ∞ (R d j ). Suppose that • p 1 ≤ p 1 , p j = p j and w j = w j for j ∈ {2, . . . , l}; • w 1 (x 1 ) = |x 1 | γ 1 and w 1 (  p,q,d (R n , w; X) ֒→ F s,a p, q,d (R n , w; X). One of the nice things about this embedding, which has already turned out to be a powerful technical tool in the isotropic case (see e.g. [60,57,62,69]), is the (inner) trace space invariance in the sharp case s = s + a 1 γ 1 − γ 1 p 1 , see Proposition 3.7 below. In the two other embedding results in this section, Lemmas 3.4 and 3.5 below, there also is such an invariance.

(3-3)
Proof. This can be shown as the scalar-valued case in [57, (27)]. Note that the duality arguments therein remain valid as X is a UMD space and therefore reflexive.
Proof. The proof given in [ [59,Theorem 4.6], which is the only case that is used in this paper.
For the statement of Proposition 3.7 we need some notation and terminology that we first introduce.
3.2.1. Some notation. We slightly modify the notation from [59, Sections 4.3.1 & 4.3.2] to our setting. 3.2.1.a. The working definition of the trace. Let ϕ ∈ Φ d ,a (R n ) with associated family of convolution operators (S k ) k∈N ⊂ L(S ′ (R n ; X)) be fixed. In order to motivate the definition to be given in a moment, let us first recall that f = ∞ k=0 S k f in S(R n ; X) (respectively in S ′ (R n ; X)) whenever f ∈ S(R n ; X) (respectively f ∈ S ′ (R n ; X)), from which it is easy to see that f ∈ S(R n ; X).
Furthermore, given a general tempered distribution f ∈ S ′ (R n ; X), recall that S k f ∈ O M (R n ; X). In particular, each S k f has a well-defined classical trace with respect to This suggests to define the trace operator τ = τ ϕ : on the domain D(τ ϕ ) consisting of all f ∈ S ′ (R n ; X) for which this defining series converges in S ′ (R n−d 1 ; X). Note that F −1 E ′ (R n ; X) is a subspace of D(τ ϕ ) on which τ ϕ coincides with the classical trace of continuous functions with respect to {0 d 1 } × R n−d 1 ; of course, for an f belonging to F −1 E ′ (R n ; X) there are only finitely many S k f non-zero.
3.2.1.b. The distributional trace operator. Let us now introduce the concept of distributional trace operator. The motivation for introducing it comes from Lemma 3.6. The distributional trace operator r (with respect to the plane {0 d 1 } × R n−d 1 ) is defined as follows. Viewing C(R d 1 ; D ′ (R n−d 1 ; X)) as subspace of D ′ (R n ; X) = D ′ (R d 1 × R n−d 1 ; X) via the canonical identification D ′ (R d 1 ; D ′ (R n−d 1 ; X)) = D ′ (R d 1 × R n−d 1 ; X) (arising from the Schwartz kernel theorem), we define r ∈ L(C(R d 1 ; D ′ (R n−d 1 ; X)), D ′ (R n−d 1 ; X)) as the 'evaluation in 0 map'

Then, in view of
we have that the distributional trace operator r coincides on C(R n ; X) with the classical trace operator with respect to the plane {0 d 1 } × R n−d 1 , i.e., The following lemma can be established as in [47,Section 4.2.1].
defines a convergent series in S ′ (R n ; X) with for some constant c > 0 independent of g. Moreover, the operator ext defined via this formula is a linear operator ext : which acts as a right inverse of r :
Then the trace operator τ = τ ϕ (3-5) is well-defined on F s,a p,q,d (R n , w; X), where it is independent of ϕ, and restricts to a retraction for which the extension operator ext from Lemma 3.6 (with d = d ′ and a = a ′ ) restricts to a corresponding coretraction.
Proof. Using the Sobolev embedding from Proposition 3.1 with p = p (see Remark 3.3) in combination with the invariance of the space on the right-hand side of (3)(4)(5)(6)(7)(8) under this embedding, we may without loss of generality assume that p 1 = q. So ]. Now the proof goes analogously to the proof of [56, Theorem 5.2.52].

Poisson Operators
4.1. Symbol Classes. In this subsection we give the definition and derive some properties of the symbol classes we want to work with. We will restrict ourselves to symbols with constant coefficients and infinite regularity in the parameter-dependent case. For the main results of this paper, treating the general symbol classes which are usually considered in the framework of the Boutet de Monvel calculus is not necessary. Nonetheless we will treat them in a forthcoming paper for the discussion of pseudo-differential boundary value problems. Our symbol classes are variants of the classical symbol classes considered in [37] and the other references we gave in the introduction.
In this section, our parameter-dependent symbols usually depend on a complex variable. If we say that the symbol is differentiable with respect to that variable, we interpret this complex variable as an element of R 2 and mean that the symbol is differentiable in the real sense. Likewise, if there is a complex variable appearing in the Bessel potential, we treat it as a variable in R 2 .
In the whole section, we use the notation and conventions of Section 2.3. So let l ∈ N and d ∈ N l 1 such that |d | = n.
In the special case l = 1 we also omit d in the notation and write S d,∞ a (R n × Σ; Z) and p (d,∞) a,k instead.
for all α ′ ∈ N n−1 and all m, m ′ ∈ N. The elements of S d (R n−1 ; S L p (R + ; Z)) will be called parameter-independent Poisson symbol-kernels of order d + 1 or degree d. (b) We denote by S d d ,a (R n−1 ; S L p (R + ; Z)) the space of all smooth functions for every α ′ ∈ N n−1 an all m, m ′ ∈ N. The elements of S d d ,a (R n−1 ; S L p (R + ; Z)) will be called anisotropic parameter-independent Poisson symbol-kernels of order d + a 1 or degree d. In the special case a 1 = . . . = a l we omit d in the notation and write  1 and a = (a 1 ,â, a l+1 ) ∈ (0, ∞) × (0, ∞) l−1 × (0, ∞). (a) By S d,∞ (R n−1 × Σ; S L p (R + ; Z)) we denote the space of all smooth functions for all α ′ ∈ N n−1 , γ ∈ N 2 and all m, m ′ ∈ N. The elements of S d,∞ (R n−1 ×Σ; S L p (R + ; Z)) will be called parameter-dependent Poisson symbol-kernels of order d + 1 or degree d and regularity ∞.
for every α ′ ∈ N n−1 , m, m ′ ∈ N and γ ∈ N 2 . The elements of S d,∞ d ,a (R n−1 ×Σ; S L p (R + ; Z)) will be called anisotropic parameter-dependent Poisson symbol-kernels of order d + a 1 or degree d and regularity ∞. In the special case a 1 = . . . = a l we omit d in the notation and write S d,∞ a (R n−1 × Σ; S L p (R + ; Z)) and k S d,∞ a (S Lp (R + ;Z)),α ′ ,m,m ′ ,γ instead. d ,a (R n−1 ×Σ; S L p (R + ; Z)) is independent of p. The respective assertion also holds in the isotropic or parameter-independent case.
(b): We will derive this from (a) by a scaling argument. For simplicity of notation we restrict ourselves to the isotropic parameter-dependent case. Consider a smooth function Applying the seminorm estimates associated with (a) to h α ′ ,γ ( · , ξ ′ , µ) the desired result follows.
Remark 4.6. (a) Occasionally, we will need the estimates in the definitions of the Poisson symbol-kernel classes with m being a non-negative real number instead of a natural number. But the respective estimates follow by using Young's inequality. Indeed, for example in the anisotropic parameter-dependent case we have for all θ ∈ [0, 1] that Using the triangle inequality for the L p (R + ; Z)-norm yields the desired estimate. (b) Let q > 0. Then we have that S d,∞ d ,a = S qd,∞ d ,qa for all the anisotropic symbol classes, since The respective assertion also holds for the other symbol-kernel classes as well as for the Hörmander symbols. (d) Let d 1 , d 2 ∈ R and suppose that we have a continuous bilinear mapping Z 1 × Z 2 → Z for the Banach spaces Z 1 , Z 2 and Z. Then, the bilinear mapping is continuous. The respective assertions also hold for the other classes of Hörmander symbols.
Definition 4.9. (a) Given a Hörmander symbol with constant coefficients p or p µ := p(·, µ) in the parameter-dependent case, we define the associated operator or respectively. (b) Given a Poisson symbol-kernel k or k µ := k(·, ·, µ) in the parameter-dependent case, we define the associated operator respectively.
Lemma 4.12. Consider the situation of Definition 4.11. We have the continuous embedding . The respective assertion holds within the isotropic or parameter-independent classes.
Proof. We only prove the result for the anisotropic and parameter-dependent case, as the other cases can be proven in the exact same way. For given α ∈ N n and γ ∈ N 2 we obtain Here we used the first part of Remark 4.6.
Note that Lemma 4.12 shows us that we can define an operator to a symbol in S d,∞ d ,a (R n−1 × Σ; S L ∞ (R; Z)) by the means of Definition 4.9.
Proof. Before we give the proof we should note that our approach here is strongly influenced by [46,Proposition 4.1] where the Poisson symbol-kernel was also mapped to the corresponding Hörmander symbol (cf. [36,37]). We only prove the result for the anisotropic and parameter-dependent case, as the other cases can be proven in the exact same way. The proof consists of three steps: (i) We show that the Seeley extension is bounded from S d,∞ d ,a (R n−1 × Σ; S L 1 (R + ; B(X, Y))) to S d,∞ d ,a (R n−1 × Σ; S L 1 (R; B(X, Y))).
So let us prove the three steps one by one: (i) For the Seeley extension we fix two sequences (a k ) k∈N , It was proven in [83] that such sequences indeed exist. Moreover, we take a function φ ∈ C ∞ (R + ) with φ(t) = 1 for 0 ≤ t ≤ 1 and φ(t) = 0 for t ≥ 2. Then, the Seeley extension for a function f ∈ S d,∞ d ,a (R n−1 × Σ; S L p (R + ; Z)) is defined by The assertion regarding the smoothness has already been proved by Seeley in [83]. Hence, we only have to show that the estimates of the symbol classed are preserved under the Seeley extension. But they indeed hold as (ii) This follows directly from the above computation together with the definition of the symbol classes and the fact that F x 1 →ξ 1 maps L 1 (R; B(X, Y)) continuously into L ∞ (R; B(X, Y)). (iii) For all g ∈ S(R n−1 ) and all x ∈ R n + we have that This finishes the proof.
Proof. In order to keep notations shorter, we first show the assertion for constant p 3 . Hence, we omit it in the notation and estimate the term By the product rule and the triangle inequality, it suffices to estimate expressions of the form But for such an expression, we obtain A similar computation shows the respective assertion for the case that p 1 is constant and p 3 is arbitrary. The formula for the operators is trivial.

Solution Operators for Model Problems.
In this subsection we consider the boundary value model problems ∂ t u + (1 + η + A(D))u = 0 on R n + × R, B j (D)u = g j , on R n−1 × R, j = 1, . . . , m, for η ≥ 0. Here, A(D), B 1 (D), . . . , B m (D) is a constant coefficient homogeneous B(X)valued differential boundary value system on R n + as considered in Section 2.5. In this subsection we restrict ourselves to g 1 , . . . , g m ∈ S(R n−1 ; X) so that we can later extend the solution by density to the desired spaces.
Proof. Under Fourier transformation in time, (4-2) turns into . . , m. The result thus follows from Proposition 4.16 through a substitution as in Remark 4.7.
In order to prove Proposition 4.16, we use a certain solution formula to (4-1). Following the considerations in [19, Proposition 6.2] we can represent the solution in the Fourier image as • A 0 is some smooth function with values in B(X 2m , X 2m ) that one obtains from λ − A(D x 1 , ξ ′ ) after some reduction to a first-order system, • M is some smooth function with values in B(X m , X 2m ) which maps the values of the boundary operator applied to the stable solution to the vector containing all normal derivatives of this solution up to order 2m − 1, • ρ is a positive parameter that can be chosen in different ways and in dependence of ξ ′ and λ, Another operator that we will use later is the spectral projection P − of the matrix A 0 to the part of the spectrum that lies above the real line. This spectral projection hast the property that P − (b, σ)M(b, σ) = M(b, σ). For our purposes, we will rewrite the above representation in the following way: For j = 1, . . . , m we write The functions (ξ ′ , λ) → e iρA 0 (b,σ)x 1 M ρ, j (b, σ) (note that ρ, b and σ depend on (ξ, λ) where we oppress the dependence in the notation for the sake of readability) are exactly the Poisson symbol-kernels k j in Proposition 4.16. In the following, we will show that they satisfy the symbol-kernel estimates in order to prove Proposition 4.16.
Proof. For the sake of simpler notation, we only consider the case Σ 1 = . . . = Σ N = R.
For the rest of this section, in (4-3) we fix In particular, if we choose a = (a 1 , a 2 ) = ( 1 2m , 1) then we obtain so that (b, σ) coincides with the function in Proposition 4.20.
Proposition 4.21. Let again a 1 , a 2 > 0 such that 1 a 1 , 1 a 2 ∈ N and let A be smooth with values in some Banach space Z. We further assume that A and all its derivatives are bounded on the range of (b, σ). Then, we have that A • (b, σ) ∈ S 0,∞ a (R n−1 × Σ, Z). Proof. We show by induction on |α ′ | + |γ| that D α ′ ξ ′ D γ λ (A • (b, σ)) is a linear combination of terms of the form (D α ′ ξ ′ D γ λ A) • (b, σ) · f with f ∈ S −a 1 |α ′ |−a 2 |γ|,∞ a (R n−1 × Σ), α ′ ∈ N n−1 and γ ∈ N 2 . Obviously, this is true for |α ′ | + |γ| = 0. So let j ∈ {1, . . . , n − 1}. By induction hypothesis, we have that it suffices to treat the summands separately, i.e. we consider . By the product rule and the chain rule, we have σ)] By the induction hypothesis and Remark 4.6 (c) and (d) we have that The same computation for ∂ λ 1 and ∂ λ 2 instead of ∂ j also shows the desired behavior and hence, the induction is finished. Finally, the assertion follows now from Proposition 4.20 and Remark 4.6 (c) and (d).
Corollary 4.24. We have that Proof. This is obtained by computing the L 1 -norms in Proposition 4.23.
Putting together the above gives Proposition 4.16: Proof , s ∈ R and a ∈ (0, ∞) l . Then ( k, g) → OPK( k)g defines continuous bilinear operators . Note that Theorem 4.25 is an extension of the isotropic unweighted scalar-valued setting in [46,Theorem 4.3] in case of constant coefficients. But in contrast to [46,Theorem 4.3] we take p ∈ (1, ∞) l . Since we use Proposition 3.1 in the proof we do not allow p ∈ [1, ∞) l in this formulation. However, it should be possible to remove this restriction, see Remark 3.2. The proof of Theorem 4.25, which adjusts the line of arguments in [46,Theorem 4.3] to our situation, will be given at the end of this section.
Proof. Let s, γ, σ and η be as in Lemma 3.4. Then note that we have the embedding (3-3) while Observing that the result thus follows from Theorem 4.25.

Parabolic Problems
In this section we consider the linear vector-valued parabolic initial-boundary value problem (1-2). As the main result of the paper, we solve the L q,µ -H s p,γ -maximal regularity problem, the L q,µ -F s p,r,γ -maximal regularity problem and the L q,µ -B s p,q,γ -maximal regularity problem for (1-2) in Theorem 5.3. This simultaneously generalizes [59,Theorem 3.4] and [57,Theorem 4.2].
Before we can state Theorem 5.3, we first need to introduce some notation.
5.1. Some notation and assumptions. Let O be either R n + or a C ∞ -domain in R n with a compact boundary ∂O and J = (0, T ) with T ∈ (0, ∞). Let X be a Banach space and let A(D), B 1 (D), . . . , B m (D) be a B(X)-valued differential boundary value system on O × J as considered in Section 2.5 where the coefficients satisfy certain smoothness conditions which we are going to introduce later. Put m * := max{m 1 , . . . , m m } and m * := min{m 1 , . . . , m m }.
In the L q,µ -E-maximal regularity approach in Theorem 5.3 we look for solutions and characterize the data f , g = (g 1 , . . . , g m ) and u 0 for which this actually can be solved. Let us now introduce some notation for the function spaces appearing in this problem. For an open interval I ⊂ R and v ∈ A q (R), we put  In Theorem 5.3 we will in particularly see that which basically just is a trace theory part of the problem. In view of the commutativity of taking traces, tr ∂O • tr t=0 = tr t=0 = tr ∂O , when well-defined, we also have to impose a compatibility condition on g and u 0 in . In order to formulate this precisely, let us define 1 + µ q in the definition of IB q,µ (I; E), let us remark the following. For simplicity of notation we restrict ourselves to cases (a) and (b), case (c) being analogous. Suppose κ j,E > 1+µ q . Then (g j , u 0 ) → tr t=0 g j − B t=0 j (D)u 0 is a well-defined bounded linear operator B q,µ (I; E) ⊕ I q,µ (I; E) → L p (∂O; X) as B q,µ, j (I; E) ֒→ F κ j,E q,p (I, v µ ; L p (∂O; X)) and Suppose that X is a UMD space, that A(D), B 1 (D), . . . , B m (D) satisfies the smoothness conditions (SDP), (SDL), (SBP) and (SBL) as well as the conditions (E) φ , (LS) φ for some φ ∈ (0, π 2 ), and that κ j,E 1+µ q for all j ∈ {1, . . . , m}. Then the problem (1-2) enjoys the property of maximal L q,µ -E-regularity with IB q,µ (J; E) as the optimal space of initialboundary data, i.e.
Remark 5.4. In the L q,µ -L p,γ -case the proof simplifies a bit on the function space theoretic side of the problem, yielding a simpler proof than the previous ones ( [20] (µ = 0, γ = 0), [66] (q = p, µ ∈ [0, p − 1), γ = 0) and [59]). where M q,µ (I; E), D q,µ (I; E), IB q,µ (I; E) are as in the beginning of this section. Lemma 5.9. Let X be a UMD Banach space and assume that (A, B 1 , . . . , B m ) is homogeneous with constant-coefficients on O = R n + and satisfies (E) φ and (LS) φ for some φ ∈ (0, π 2 ). Let q ∈ (1, ∞) and v ∈ A q (R). Let E and E 2m be given as in either (a), is a well-defined bounded linear operator and the differential parabolic boundary value problem admits a bounded linear solution operator Finally, let us prove the uniqueness of solutions. For this it suffices to show that S(R n + × R; X) is dense in M. Indeed, using this density, the uniqueness statement follows from a combination of (5-3), the uniqueness statement in Corollary 4.18 and the continuity of our solution operator S (η) : B −→ M.
For this density, note that W 1 q (R, v; E 2m ) is dense in M by a standard convolution argument (in the time variable). So yielding the required density.
by the compactness of D 1 and D 2 .
Proof of Proposition 5.7. We first show that the differential parabolic boundary value problem E 2m has 0 in its resolvent and is R-sectorial with angle < π 2 , which in the cases (b) and (c) is contained in Lemma 5.11 and which in case (a) can be derived as in [19,Corollary 5.6] using the operator-valued Mikhlin theorem for H s p (R n , w γ ; X) (see Proposition 2.7). As a consequence (see Section 2.2), the parabolic problem Choosing an extension operator E ∈ B L q (R, v; E), L q (R, v; E) , recalling (5-3), denoting by r + ∈ B(M, M) the operator of restriction from R n × R to R n + × R and denoting by S the solution operator from Lemma 5.9, we find that T ( f, g 1 , . . . , g m ) := r + RE f − S B(D)r + RE f + S (g 1 , . . . , g m ) defines a solution operator as desired.
Finally, the uniqueness follows from the uniqueness obtained in Lemma 5.9.
Lemma 5.12. Let X be a UMD Banach space and assume that (A, B 1 , . . . , B m ) is homogeneous with constant-coefficients on O = R n + and satisfies (E) φ and (LS) φ for some φ ∈ (0, π 2 ). Let q ∈ (1, ∞) and v ∈ A q (R). Let E and E 2m be given as in either Proof. As a consequence of Proposition 5.7, 1+η+A B satisfies the conditions of Lemma 2.1 with ||| · ||| = · E 2m . Therefore, there is an equivalence of norms in D(1 + η + A B ) = D(A B ) = {u ∈ E 2m : B(D)u = 0} and 1 + η + A B is a closed linear operator on E enjoying the property of L q (R, v)-maximal regularity. Moreover, it follows from the polynomial bounds in Proposition 5.7 and Lemma 2.1 that C + ⊂ ̺(1 + A B ) and that η → (η + 1 + A B ) −1 is polynomially bounded. Thus, 1 + A B satisfies the conditions of Lemma 2.2 and the desired result follows. Lemma 5.13. Let the notations be as in Subsection 5.1 with v = v µ , µ ∈ (−1, q − 1), and suppose that X is a UMD space. Then Tr t=0 : u → u(0) is a retraction Tr t=0 : M q,µ (J; E) −→ I q,µ (J; E).
Proof. This can be derived from [ Proof of Proposition 5.8. That u → (u ′ + (1 + A(D))u, B(D)u, u(0)) is a bounded operator M q,µ (R + ; E) −→ D q,µ (R + ; E) ⊕ B q,µ (R + ; E) ⊕ I q,µ (E) follows from a combination of Proposition 5.7 (choosing an extension operator M q,µ (R + ; E) → M q,µ (R; E)) and Lemma 5.13. That it maps to D q,µ (R + ; E) ⊕ IB q,µ (R + ; E) can be seen as follows: we only need to show that Tr t=0 B j (D)u = B j (D)Tr t=0 u, u ∈ M q,µ (R + ; E), (5)(6)(7)(8) when κ j,E > 1+µ q (also see Remark 5.2), which simply follows from Here this density follows from a standard convolution argument (in the time variable) in combination with an extension/restriction argument. In the following, we use the notation 6.3. Operator Theoretic Results. The L q -maximal regularity established in Theorem 5.3 for the special case of homogeneous initial-boundary data gives L q -maximal regularity and thus R-sectoriality for the realizations of the corresponding elliptic differential operators: Corollary 6.6. Let O be either R n + or a C ∞ -domain in R n with a compact boundary ∂O. Let X be a UMD Banach space and let (A(D), B 1 (D), . . . , B m (D)) be a B(X)-valued differential boundary value system on O as considered in Section 2.5 and put m * := max{m 1 , . . . , m m }. Let E and E 2m be given as in Case (a), Case (b) or Case (c) as in Section 5.1. Let (A(D), B 1 (D), . . . , B m (D)) be a B(X)-valued differential boundary value system of order 2m on O that satisfies (E) φ and (LS) φ for some φ ∈ (0, π 2 ). Moreover, we assume that the coefficients satisfy the conditions (SDP) s , (SDL) s , (SBP) s and (SBL) s from Section 6.1. For every θ ∈ (φ, π 2 ) there exists µ θ > 0 such that µ θ + A is R-sectorial with angle ω R (µ θ + A B ) ≤ θ.
The following result is an immediate corollary to Theorem 6.2. For every θ ∈ (φ, π) there exists µ θ > 0 such that µ θ + A B is sectorial with angle φ µ θ +A ≤ θ. Remark 6.8. From the R-sectoriality and sectoriality in Corollary 6.6 and Corollary 6.7, respectively, one could derive boundedness of the H ∞ -functional calculus using interpolation techniques from [26,48]: [48,Corollary 7.8] and [26,Theorem 3.1] give a bounded H ∞ -calculus of the part of A B in the Rademacher interpolation space E, D(A) θ and in the real interpolation space (E, D(A)) θ,q , respectively. In this way one could improve the R-sectoriality to a bounded H ∞ -functional calculus in Corollary 6.6 and the sectoriality to a bounded H ∞ -functional calculus in the B-and B-cases Corollary 6.7. However, the required knowledge of interpolation with boundary conditions does not seem to be available in the literature at the moment. Remark 6.9. The scales of weighted B-and F -spaces, the dual scales to the scales of weighted B-and F-spaces, naturally appear in duality theory. In [60] they were used to describe the adjoint operators for realizations of second order elliptic operators subject to the Dirichlet boundary condition in weighted B-and F-spaces (see [60,Remark 9.13]), which was an important ingredient in the application to the heat equation with multiplicative noise of Dirichlet type at the boundary in weighted L p -spaces in [62] through the so-called Dirichlet map (see [60,Theorem 1.2]). The incorporation of the scales of weighted Band F -spaces in Theorem 6.2 and Corollary 6.7 would allow us similarly to describe the adjoint of the operator A B from Corollary 6.6, which could then be used to extend [62] to more general parabolic boundary value problems with multiplicative noise at the boundary.