Maximal Regularity with Weights for Parabolic Problems with Inhomogeneous Boundary Conditions

In this paper we establish weighted $L^{q}$-$L^{p}$-maximal regularity for linear vector-valued parabolic initial-boundary value problems with inhomogeneous boundary conditions of static type. The weights we consider are power weights in time and in space, and yield flexibility in the optimal regularity of the initial-boundary data and allow to avoid compatibility conditions at the boundary. The novelty of the followed approach is the use of weighted anisotropic mixed-norm Banach space-valued function spaces of Sobolev, Bessel potential, Triebel-Lizorkin and Besov type, whose trace theory is also subject of study.


Introduction
This paper is concerned with weighted maximal L q -L p -regularity for vector-valued parabolic initial-boundary value problems of the form (1) x ∈ O, t ∈ J, B j (x ′ , D, t)u(x ′ , t) = g j (x ′ , t), x ′ ∈ ∂O, t ∈ J, j = 1, . . . , n, u(x, 0) = u 0 (x), Here, J is a finite time interval, O ⊂ d 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 = ¼ N , describing a system of N initial-boundary value problems. Our structural assumptions on A, B 1 , . . . , B n are an ellipticity condition and a condition of Lopatinskii-Shapiro type. For homogeneous boundary data (i.e. g j = 0, j = 1, . . . , n) 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. During the last 25 years, the theory of maximal regularity turned out to be an important tool in the theory of nonlinear PDEs. Maximal regularity means that there is an isomorphism between the data and the solution of the problem in suitable function spaces. 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. Let us mention [7,15] for approaches in spaces of continuous functions, [1,44] for approaches in Hölder spaces and [3,5,13,14,24,52,54] for approaches in L p -spaces (with p ∈ (1, ∞)).
As an application of his operator-valued Fourier multiplier theorem, Weis [64] characterized maximal L p -regularity for abstract Cauchy problems in UMD Banach spaces in terms of an R-boundedness condition on the operator under consideration. A second approach to the maximal L p -regularity problem is via the operator sum method, as initiated by Da Prato & Grisvard [17] and extended by Dore & Venni [23] and Kalton & Weis [36]. For more details on these approaches and for more information on (the history of) the maximal L p -regularity problem in general, we refer to [18,38].
In the maximal L q -L p -regularity approach to (1) one is looking for solutions u in the "maximal regularity space" (2) W 1 q (J; L p (O; X)) ∩ L q (J; W 2n p (O; X)). To be more precise, the problem (1) is said to enjoy the property of maximal L q -L pregularity if there exists a (necessarily unique) space of initial-boundary data D i.b. ⊂ L q (J; L p (∂O; X)) n × L p (O; X) such that for every f ∈ L q (J; L p (O; X)) it holds that (1) has a unique solution u in (2) if and only if (g = (g 1 , . . . , g n ), u 0 ) ∈ D i.b. . In this situation there exists a Banach norm on D i.b. , unique up to equivalence, with D i.b. ֒→ L q (J; L p (∂O; X)) n ⊕ L p (O; X), which makes the associated solution operator a topological linear isomorphism between the data space L q (J; L p (O; X))⊕D i.b. and the solution space W 1 q (J; L p (O; X))∩L q (J; W 2n p (O; X)). The maximal L q -L p -regularity problem for (1) consists of establishing maximal L q -L pregularity for (1) and explicitly determining the space D i.b. .
The maximal L q -L p -regularity problem for (1) was solved by Denk, Hieber & Prüss [19], who used operator sum methods in combination with tools from vector-valued harmonic analysis. Earlier works on this problem are [39] (q = p) and [63] (p ≤ q) for scalarvalued 2nd order problems with Dirichlet and Neumann boundary conditions. Later, the results of [19] for the case that q = p have been extended by Meyries & Schnaubelt [47] to the setting of temporal power weights v µ (t) = t µ , µ ∈ [0, q − 1); also see [46]. Works in which maximal L q -L p -regularity of other problems with inhomogeneous boundary conditions are studied, include [20,21,22,24,47] (the case q = p) and [49,60] (the case q p).
It is desirable to have maximal L q -L p -regularity for the full range q, p ∈ (1, ∞), as this enables one to treat more nonlinearities. For instance, one often requires large q and p due to better Sobolev embeddings, and q p due to scaling invariance of PDEs (see e.g. [29]). However, for (1) the case q p is more involved than the case q = p due to the inhomogeneous boundary conditions. This is not only reflected in the proof, but also in the space of initial-boundary data ( [19,Theorem 2.3] versus [19,Theorem 2.2]). Already for the heat equation with Dirichlet boundary conditions, the boundary data g has to be in the intersection space denotes a Triebel-Lizorkin space and W s p = B s p,p a non-integer order Sobolev-Slobodeckii space.
In this paper we will extend the results of [19,47], concerning the maximal L q -L pregularity problem for (1), to the setting of power weights in time and in space for the full range q, p ∈ (1, ∞). In contrast to [19,47], we will not only view the spaces (2) and (3) as intersection spaces, but also as anisotropic mixed-norm function spaces on J × O and J × ∂O, respectively. Identifications of intersection spaces of the type (3) with anisotropic mixed-norm Triebel-Lizorkin spaces have been considered in a previous paper [42], all in a generality including the weighted vector-valued setting. The advantage of these identifications is that they allows us to use weighted vector-valued versions of trace results of Johnsen & Sickel [35]. These trace results will be studied in their own right in the present paper.
The weights we consider are the power weights , where µ ∈ (−1, q − 1) and γ ∈ (−1, p − 1). These weights yield flexibility in the optimal regularity of the initial-boundary data and allow to avoid compatibility conditions at the boundary, which is nicely illustrated by the result (see Example 3.6) that the corresponding version of (3) becomes Note that one requires less regularity of g by increasing γ.
The idea to work in weighted spaces equipped with weights like (4) has already proven to be very useful in several situations. In an abstract semigroup setting temporal weights were introduced by Clément & Simonett [16] and Prüss & Simonett [53], in the context of maximal continous regularity and maximal L p -regularity, respectively. Other works on maximal temporally weighted L p -regularity are [37,40] for quasilinear parabolic evolution equations and [47] for parabolic problems with inhomogeneous boundary conditions. Concerning the use of spatial weights, we would like to mention [9,45,51] for boundary value problems and [2,10,25,55,61] for problems with boundary noise.
The paper is organized as follows. In Section 2 we discuss the necessary preliminaries, in Section 3 we state the main result of this paper, Theorem 3.4, in Section 4 we establish the necessary trace theory, in Section 5 we consider a Sobolev embedding theorem, and in Section 6 we finally prove Theorem 3.4.
Acknowledgements. The author would like to thank Mark Veraar for the supervision of his master thesis [41], which led to the present paper.  [30].
Important for this paper are the power weights of the form w = dist( · , ∂O) γ , where O is a C ∞ -domain in d and where γ ∈ (−1, ∞). If γ ∈ (−1, ∞) and p ∈ (1, ∞), then (see [26,Lemma 2.3] is called the d -decomposition of d . For x ∈ d we accordingly write x = (x 1 , . . . , x l ) and x j = (x j,1 , . . . , x j,d j ), where x j ∈ d j and x j,i ∈ ( j = 1, . . . , l; i = 1, . . . , d j ). We also say that we view d as being d -decomposed. Furthermore, for each k ∈ {1, . . . , l} we define the inclusion map and the projection map We equip L p,d ( d , w) with the norm || · || L p,d ( d ,w) , which turns it into a Banach space. Given a Banach space X, we denote by L p,d ( d , w; X) the associated Bochner space

Anisotropy.
Suppose that d is d -decomposed as in Section 2.1. Given a ∈ (0, ∞) l , we define the (d , a)-anisotropic dilation δ (d ,a) λ on d by λ > 0 to be the mapping δ (d ,a) λ on d given by the formula for all x ∈ d and λ > 0. (iii) There exists a c > 0 such that u(x + y) ≤ c(u(x) + u(y)) for all x, y ∈ d . All (d , a)-anisotropic distance functions on d are equivalent: Given two (d , a)-anisotropic distance functions u and v on d , there exist constants m, In this paper we will use the (d , a)-anisotropic distance function | · | d ,a : d −→ [0, ∞) given by the formula 2.3. Fourier Multipliers. Let X be a Banach space. The space of X-valued tempered distributions on d is defined as S ′ ( d ; X) := L(S( d ; X)); for the theory of vector-valued distributions we refer to [4] (and [3, Section III.4]). We write L 1 ( d ; . To a symbol m ∈ L ∞ ( d ; B(X)) we associate the Fourier multiplier operator In this case T m has a unique extension to a bounded linear operator on L p,d ( d , w; X) due to denseness of S( d ; X) in L p,d ( d , w; X), which we still denote by T m . We denote by M p,d ,w (X) the set of all Fourier multipliers m ∈ L ∞ ( d ; B(X)) on L p,d ( d , w; X). Equipped with the norm ||m|| M p,d ,w (X) := ||T m || B(L p,d ( d ,w;X) , M p,d ,w (X) becomes a Banach algebra (under the natural pointwise operations) for which the natural inclusion M p,d ,w (X) ֒→ B(L p,d ( d , w; X)) is an isometric Banach algebra homomorphism; see [38] for the unweighted non-mixed-norm setting.
For each a ∈ (0, ∞) l and N ∈ we define We furthermore define RM (X) as the space of all operator-valued symbols m ∈ C 1 ( \ {0}; B(X)) for which we have the R-bound see e.g. [18,32] for the notion of R-boundedness.
and a ∈ (0, ∞) l , then there exists an N ∈ for which If X is a UMD space, p ∈ (1, ∞) and w ∈ A p ( ), then For these results we refer to [41] and the references given there.

Function Spaces.
For the theory of vector-valued distributions we refer to [4] (and [3, Section III.4]). For vector-valued function spaces we refer to [50] (weighted setting) and the references given therein. Anisotropic spaces can be found in [6,35,41]; for the statements below on weighted anisotropic vector-valued function space we refer to [41]. Suppose that d is d -decomposed as in Section 2.1. Let X be a Banach space and let a ∈ (0, ∞) l . For 0 A,B ( d ) as the set of all sequences ϕ = (ϕ n ) n∈ ⊂ S( d ) which are constructed in the following way: given a ϕ 0 ∈ S( d ) satisfying To ϕ ∈ Φ d ,a ( d ) we associate the family of convolution operators ( Here O M ( d ; X) denotes the space of slowly increasing X-valued smooth functions on d . It Up to an equivalence of extended norms on S ′ ( d ; X), || · || B s,a p,q,d ( d ,w;X) and || · || F s,a do not depend on the particular choice of ϕ ∈ Φ d ,a ( d ).

The Main Result
In order to give a precise description of the maximal weighted L q -L p -regularity approach for (1), let O be either d + or a smooth domain in d with a compact boundary ∂O. Furthermore, let X be a Banach space, let let v µ and w ∂O γ be as in (4), put and let n, n 1 , . . . , n n ∈ be natural numbers with n j ≤ 2n − 1 for each j ∈ {1, . . . , n}. Suppose that for each α ∈ d , |α| ≤ 2n, γ,µ , ¿ p,q γ,µ ) and that for each j ∈ {1, . . . , n} and β ∈ d , |β| ≤ n j , µ ) have to be interpreted in the sense of bounded extension from the space of X-valued compactly supported smooth functions. Define A(D) ∈ B( p,q γ,µ , ¿ p,q γ,µ ) and In the above notation, given f ∈ ¿ p,q γ,µ and g = (g 1 , . . . , g n ) ∈ [» p,q µ ] n , one can ask the question whether the initial-boundary value problem (20) has a unique solution u ∈ p,q γ,µ . Definition 3.1. We say that the problem (20) enjoys the property of maximal L q µ -L p γregularity if there exists a (necessarily unique) linear space . In this situation we call D i.b. the optimal space of initial-boundary data and D the optimal space of data.
has been equipped with a Banach norm generating such a topology, then the solution operator is an isomorphism of Banach spaces, or equivalently, The maximal L q µ -L p γ -regularity problem for (20) consists of establishing maximal L q µ -L p γ -regularity for (20) and explicitly determining the space D i.b. together with a norm as in Remark 3.2. As the main result of this paper, Theorem 3.4, we will solve the maximal L q µ -L p γ -regularity problem for (20) under the assumption that X is a UMD space and under suitable assumptions on the operators A(D), B 1 (D), . . . , B n (D). (A, B 1 , . . . , B n ). As in [19,47], we will pose two type of conditions on the operators A, B 1 , . . . , B n for which we can solve the maximal L q µ -L p γ -regularity problem for (20): smoothness assumptions on the coefficients and structural assumptions.

Theorem 3.4. Let the notations be as above. Suppose that X is a UMD space, that
Then the problem (20) enjoys the property of maximal L q µ -L p γ -regularity with ½ p,q γ,µ as the optimal space of initial-boundary data, i.e., the problem (20) admits a unique solution u ∈ Moreover, the corresponding solution operator S : ¿ p,q γ,µ ⊕½ p,q γ,µ −→ p,q γ,µ is an isomorphism of Banach spaces.
Remark 3.5. The compatibility condition tr t=0 g j − B t=0 j (D)u 0 = 0 in the definition of ½ p,q γ,µ is basically imposed when (g j , u 0 ) → tr t=0 g j − B t=0 j (D)u 0 makes sense as a continuous linear operator from p,q γ,µ, j ⊕ p,q γ,µ to some topological vector space V. That it is indeed a well-defined continuous linear operator from p,q γ,µ, j ⊕ p,q γ,µ to L 0 (∂O; X) when κ j,γ > 1+µ q can be seen by combining the following two points: Furthermore, it holds that 2n(1 − 1+µ q ) > n j + 1+γ q , so each tr ∂O D β , |β| ≤ n j , is a continuous linear operator from by the trace theory from Section 4.1. Therefore, B t=0 j (D) = |β|≤n j b j,β (0, · )tr ∂O D β makes sense as a continuous linear operator from in combination with the trace theory from Section 4.1 yields that tr t=0 is a welldefined continuous linear operator from Notice the dependence of the space of initial-boundary data on the weight parameters µ and γ. For fixed q, p ∈ (1, ∞) we can roughly speaking decrease the required smoothness (or regularity) of g and u 0 by increasing γ and µ, respectively. Furthermore, compatibility conditions can be avoided by choosing µ and γ big enough. So the weights make it possible to solve (20) for more initial-boundary data (compared to the unweighed setting). On the other hand, by choosing µ and γ closer to −1 (depending on the initial-boundary data) we can find more information about the behavior of u near the initial-time and near the boundary, respectively.
The dependence on the weight parameters µ and γ is illustrated in the following example of the heat equation with Dirichlet and Neumann boundary conditions: Example 3.6. Let N ∈ and let p, q, γ, µ be as above.

(i) The heat equation with Dirichlet boundary condition:
The heat equation with Neumann boundary condition:

Trace Theory
In this section we establish the necessary trace theory for the maximal L q µ -L p γ -regularity problem for (20).

Traces of Isotropic Spaces.
In this subsection we state trace results for the isotropic spaces, for which we refer to [43] (also see the references there). Note that these are of course special cases of the more general anisotropic mixed-norm spaces, for which trace theory (for the model problem case of a half-space) can be found in the next subsections and in [41].
The following notation will be convenient: There is a universal coretraction in the sense that there exists an operator ext ∂O ∈ L(S ′ (∂O; X), S ′ ( d ; X)) (independent of A , p, q, γ, s) which restricts to a coretraction for the operator

Traces of Intersection Spaces.
For the maximal L q µ -L p γ -regularity problem for (20) we need to determine the temporal and spatial trace spaces of Sobolev and Bessel potential spaces of intersection type. As the temporal trace spaces can be obtained from the trace results in [49], we will focus on the spatial traces.
By the trace theory of the previous subsection, the trace operator tr ∂O can be defined pointwise in time on the intersection spaces in the following theorem. It will be convenient to use the notation tr ∂O [¾] = ¿ to say that tr ∂O is a retraction from ¾ onto ¿. and The main idea behind the proof of Theorem 4.4 is, as in [59], to exploit the independence of the trace space of a Triebel-Lizorkin space on its microscopic parameter. As in [59], our approach does not require any restrictions on the Banach space X.
The UMD restriction on Y comes from the localization procedure for Bessel potential spaces used in the proof, which can be omitted in the case O = d + . This localization procedure for Bessel potential spaces could be replaced by a localization procedure for weighted anisotropic mixed-norm Triebel-Lizorkin spaces, which would not require any restrictions on the Banach space Y. However, we have chosen to avoid this as localization of such Triebel-Lizorkin spaces spaces has not been considered in the literature before while we do not need that generality anyway. For localization in the scalar-valued isotropic non-mixed norm case we refer to [43].
From the natural identifications and Corollary 4.9 it follows that In contrast to Theorem 4.6, the trace result [35, Theorem 2.2] is formulated for the distributional trace operator; see Remark 4.8 for more information. However, all estimates in the proof of that result are carried out for the 'working definition of the trace'. The proof of Theorem 4.6 presented below basically consists of modifications of these estimates to our setting. As this can get quite technical at some points, we have decided to give the proof in full detail.
with associated family of convolution operators (S n ) n∈ ⊂ L(S ′ ( d ; X)) be fixed. In order to motivate the definition to be given in a moment, let us first recall that Furthermore, given a general tempered distribution f ∈ S ′ ( d ; X), recall that S n f ∈ O M ( d ; X); in particular, each S n f has a well defined classical trace with respect to {0} × d−1 . This suggests to define the trace operator τ = τ ϕ :

Then, in view of
we have that the distributional trace operator r coincides on C( d ; X) with the classical trace operator with respect to the hyperplane {0} × d−1 , i.e., The following lemma can be established as in [35,Section 4.2.1].
for some constant c > 0 independent of g. Moreover, the operator ext defined via this formula is a linear operator which acts as a right inverse of r :

Trace spaces of Triebel-Lizorkin, Sobolev and Bessel potential spaces.
Theorem 4.6. Let X be a Banach space, where it is independent of ϕ, and restricts to a retraction for which the extension operator ext from Lemma 4.5 (withd = d ′′ andã = a ′′ ) restricts to a corresponding coretraction.
Remark 4.7. In the situation of Theorem 4.6, suppose that q < ∞. Then S( d ; X) is a dense linear subspace of F s,a p,q,d ( d , (w γ , w ′′ ); X) and τ is just the unique extension of the classical trace operator to a bounded linear operator (28).
Remark 4.8. In contrary to the unweighted case considered in [35], one cannot use translation arguments to show that , the inclusion F s,a p,q,d ( d , (w γ , w ′′ ); X) ֒→ C( ; S ′ ( d−1 ; X)) can be obtained as follows: pickings with s >s > a 1 p 1 (1 + γ + ), there holds the chain of inclusions F s,a p,q,d ( d , (w γ , w ′′ ); X) ֒→ B˜s ,a p,1,d ( d , (w γ , w ′′ ); X) (30) ֒→ Here the restriction s > a 1 p 1 (1 + γ + ) when γ < 0 is natural in view of the necessity of s > Note that the trace space of the weighted anisotropic Triebel-Lizorkin space is independent of the microscopic parameter q ∈ [1, ∞]. As a consequence, if ¾ is a normed space , then the trace result of Theorem 4.6 also holds for ¾ in place of F s,a p,q,d ( d , (w γ , w ′′ ); X). In particular, we have: , a ∈ (0, 1) l , a = sa −1 . Then the trace operator τ = τ ϕ (25) is well-defined on ¾, where it is independent of ϕ, and restricts to a retraction for which the extension operator ext from Lemma 4.5 (withd = d ′′ andã = a ′′ ) restricts to a corresponding coretraction.

Traces by duality for Besov spaces.
extends to a retraction (29) tr  γ 1 ; X) . . .). Proof. Thanks to the Sobolev embedding of Proposition 5.1 it is enough to treat the case w ∈ l j=1 A p j ( d j ), which can be obtained by l iterations of Proposition 4.10.
Remark 4.12. The above proposition and its corollary remain valid for q = ∞. In this case the norm estimate corresponding to (29) can be obtained in a similar way, from which the unique extendability to a bounded linear operator (29) can be derived via the Fatou property, (10) and the case q = 1. The remaining statements can be established in the same way as for the case q < ∞.  X) in B s,a p,q,d ( d , w; X). This could also be established in the standard way by the Sobolev embedding Proposition 5.1, see for instance [48,Proposition 7.4].
Let X be a Banach space. Then which can be seen by testing on the dense subspace Estimates in the classical Besov and Triebel-Lizorkin spaces for the tensor product with the one-dimensional delta-distribution δ 0 can be found in [33,Proposition 2.6], where a different proof is followed than the one below.
In order to perform all the estimates in Lemma 4.14 we need the following two lemmas.
Applying Lemma 4.15 we obtain the estimate (ii) Observing that the desired estimate can be derived in the same way as in (i).
Proof of Proposition 4.10. Let us first establish (29) and (30). Thanks to the Sobolev embedding Proposition 5.1 we may restrict ourselves to the case γ ∈ (−1, under the natural identifications; also see the discussion preceding Lemma 4.14. In this way we explicitly have These two inequalities imply (29) and (30) In view of (the modified version of) (27) and Lemma A.3, it suffices to estimate .
A simple computation even shows that

The proof of Theorem 4.6.
For the proof of Theorem 4.6 we need three lemmas. Two lemmas concern estimates in Triebel-Lizorkin spaces for series satisfying certain Fourier support conditions, which can be found in Appendix A. The other lemma is Lemma 4.16.
Proof of Theorem 4.6. Let the notations be as in Proposition 4.5. We will show that, for an arbitrary ϕ ∈ Φ d ,a ( d ), (I) τ ϕ exists on F s,a p,q,d ( d , (w γ , w ′′ ); X) and defines a continuous operator (II) The extension operator ext from Proposition 4.5 (withd = d ′′ andã = a ′′ ) restricts to a continuous operator , the right inverse part in the first assertion follows from (I) and (II). The independence of ϕ in the first assertion follows from denseness of S( d ; X) in F s,a p,q,d ( d , (w γ , w ′′ ); X) in case q < ∞, from which the case q = ∞ can be deduced via a combination of (10) and (11).

It now follows that
from which we in turn obtain .
j ] d j for each k ∈ and some c > 0, the desired estimate (33) is now a consequence of Proposition A.6.
(II): We may with out loss of generality assume that q = 1.
and write g n = T n g for each n. By construction of ext we have ext g = ∞ n=0 ρ(2 na 1 · ) ⊗ g n in S ′ ( d ; X) with each ρ(2 na 1 · ) ⊗ g n satisfying (27) for a c > 1 independent of g. In view of Lemma A.2, it is thus enough to show that .
In order to establish the estimate (34), we define, for each x ′′ ∈ d−1 , We furthermore first choose a natural number N > 1 p 1 (1 + γ) and subsequently pick a constant C 1 > 0 for which the Schwartz function ρ ∈ S( ) satisfies the inequality |ρ(2 na 1 x 1 )| ≤ C 1 |2 na 1 x 1 | −N for every n ∈ and all x 1 0.

Sobolev embedding for Besov spaces
The result below is a direct extension of part of [48,Proposition 1.1]. We refer to [34] for embedding results for unweighted anisotropic mixed-norm Besov space and we refer to [31] for embedding results of weighted Besov spaces. Proposition 5.1. Let X be a Banach space, p,p ∈ (1, ∞) l , q,q ∈ [1, ∞], s,s ∈ , a ∈ (0, ∞) l , and w,w ∈ l j=1 A ∞ ( d j ). Suppose that J ⊂ {1, . . . , l} is such that • p j =p j and w j =w j for j J; • w j (x j ) = |x j | γ j andw j (x j ) = |x j |γ j for j ∈ J for some γ j ,γ j > −d j satisfying Furthermore, assume that q ≤q and that s − i∈I Proof. This is an immediate consequence of inequality of Plancherel-Pólya-Nikol'skii type given in Lemma 5.2.
Lemma 5.2. Let X be a Banach space, p,p ∈ (1, ∞) l , and w,w ∈ l j=1 W( d j ). Suppose that J ⊂ {1, . . . , l} is such that • p j =p j and w j =w j for j J; • w j (x j ) = |x j | γ j andw j (x j ) = |x j |γ j for j ∈ J for some γ j ,γ j > −d j satisfying Then there exists a constant C > 0 such that, for all f ∈ S ′ ( d ; Proof.
Step II. The case J = {l}: ..+d l−1 ; X)) (Schwartz kernel theorem), f corresponds to an element of S ′ ( d l ; C( d 1 +...+d l−1 ; X)) having compact Fourier support contained in [−R l , R l ] d l . Given a compact subset K ⊂ d 1 +...+d l−1 we have the continuous linear operator with compact Fourier support contained in [−R l , R l ] d l , so that we may apply Step I to obtain that , the desired result follows by taking K = K n = [−n, n] d l and letting n → ∞.
Step IV. The general case: Just apply Step III repeatedly (#J times).

Proof of the Main Result
In this section we prove the main result of this paper, Theorem 3.4.

Necessary Conditions on the Initial-Boundary Data.
Let the notations and assumptions be as in Theorem 3.4. Suppose that g = (B 1 (D)u, . . . , B n (D)u) and u 0 = tr t=0 u for some u ∈ p,q γ,µ . We show that (g, u 0 ) ∈ ½ p,q γ,µ . It follows from [49, Theorem 1.1] (also see [54,Theorem 3.4.8]) that Using standard techniques one can derive the same result with replaced by J and d replaced by O: (38) tr t=0 [ p,q γ,µ ] = p,q γ,µ . In particular, we must have u 0 ∈ p,q γ,µ .
In this subsection we study the elliptic boundary value problem By the trace result of Corollary 4.3, in order to get a solution v ∈ W 2n p ( d + , w γ ; X) we need g = (g 1 , . . . , g n ) ∈ n j=1 F 2nκ j,γ p,p ( d−1 ; X). In Proposition 6.2 we will see that there is existence and uniqueness plus a certain representation for the solution (which we will use to solve (49)). In this representation we have the operator from the following lemma.
Then L σ λ restricts to a topological linear isomorphism from H s+2nσ w; E)) defines an analytic mapping for every σ ∈ and s ∈ .
Before we can state Proposition 6.2, we first need to introduce some notation. Given a UMD Banach space X and a natural number k ∈ , we have, for the UMD space E = L p ( + , | · | γ ; X), the natural inclusion ; E) and the natural identification . By Lemma 6.1 we accordingly have that, for λ ∈ ¼ \ (−∞, 0], that the partial Fourier multiplier operator

restricts to a bounded linear operator
Moreover, we even get an analytic operator-valued mapping In particular, we have with analytic dependence on the parameter λ ∈ ¼ \ (−∞, 0]. Proposition 6.2. Let X be a UMD Banach space, p ∈ (1, ∞), γ ∈ (−1, p − 1), and assume that (A, B 1 , . . . , B n ) satisfies (E) and (LS) for some φ ∈ (0, π). Then, for each λ ∈ Σ π−φ , there exists an operator Moreover, for each j ∈ {1, . . . , n} we have that defines an analytic mapping, for which the operators D αS , |α| ≤ 2n, can be represented as for analytic operator-valued mappings satisfying the R-bounds Comments on the proof of Proposition 6.2. This proposition can be proved in the same way as [19,Lemma 4.3&Lemma 4.4]. In fact, in the unweighted case this is just a modification of [19,Lemma 4.3&Lemma 4.4] (also see the formulation of [46, Lemma 2.2.6]).
In this formulation the proposition the weight w γ can actually be replaced by any weight w on d which is uniformly A p in the y-variable. Indeed, in the proof the weight only comes into play in [18,Lemma 7.1]. For weights w of the form w(x ′ , y) = v(x ′ )|y| γ with v ∈ A p ( d−1 ) we can then still define S(λ) as above thanks to the available trace theory from Section 4.1.
Remark 6.4. In [19] the specific extension operator E λ = e − · L 1/2n λ was used in the construction of the solution operator S(λ) = (S 1 (λ), . . . , S n (λ)), which has the advantageous property that D y E λ = ıL 1/2n λ E λ . Whereas the in this way obtained representation formulae for all j ∈ {1, . . . , n}. Defining we thus have (0, g, 0) ∈ ½ p,q γ,µ if and only if g ∈ 0 . So we need to solve (49) for g ∈ 0 . We will solve (49) by passing to the corresponding problem on (instead of + ). The advantage of this is that it allows us to use the Fourier transform in time. This will give F t u(θ) = S(1 + ıθ)(F t g 1 (θ), . . . , F t g n (θ)), where S(1 + ıθ) is the solution operator from Proposition 6.2.
Lemma 6.7. {g j ∈ S( d ; X) : tr t=0 g j = 0} is dense in 0 j Proof. As a consequence of Theorem 2.1, Let (S n ) n∈ be the family of convolution operator corresponding to some ϕ = (ϕ n ) n∈ ∈ For the pointwise induced operator family we thus have S n SOT −→ I in 0 j . Since by [43], the desired density follows.
Proof of Lemma 6.6.
is the unique solution of the problem Applying the inverse Fourier transform F −1 t with respect to θ, we find (III) We next derive a representation formula for S that is well suited for proving the boundedness of S . To this end, fix a g = (g 1 , . . . , g n ) ∈ V n . Then we have, for each multi-index α ∈ d , |α| ≤ 2n, (IV) We next show that ||S g|| ||g|| for g ∈ V n . Being a solution of (52), S g satisfies Hence, it suffices to establish the estimate ||D α S g|| ¿ ||g|| for all multi-indices α ∈ d , |α| ≤ 2n. So fix such an |α| ≤ 2n. Then, in view of the representation formula (54), it is enough to show that We only treat the estimate (56), the estimate (55) follows from (15) (and the fact that L (p,q),(d,1) ( d+1 , (w γ , v µ ); X) is an admissibile Banach space of X-valued tempered distributions on d+1 in view of (6)), from which the d + ×case follows by restriction. In combination with (53) and Lemma 6.5 this yields by the Kahane contraction principle and (48); in particular, T 2 j,α (1 + ı·) satisfies the Mikhlin condition corresponding to (7). As a consequence, T 2 j,α (1 + ı·) defines a bounded Fourier multiplier operator on L q ( , v; L p ( d + , w γ ; X)). In combination with (57), this gives the estimate (56).
Proof of Theorem 3.4. In view of Section 6.1, it remains to establish existence and uniqueness of a solution u ∈ p,q γ,µ of (20) for given ( f, g, u 0 ) ∈ p,q γ,µ ⊕ ½ p,q γ,µ . By a standard (but quite technical) perturbation and localization procedure, it is enough to consider the model problem ∂ t u + (1 + A(D))u = f, B j (D)u = g j , j = 1, . . . , n, u(0) = u 0 , on the half-space, where A and B 1 , . . . , B n are top-order constant coefficient operators as considered in Section 6.2. This procedure is worked out in full detail in [46]; for further comments we refer to Appendix B.
Let ( f, g, u 0 ) ∈ ¿ p,q γ,µ ⊕½ p,q γ,µ . In view of Theorem 4.4 and the fact that tr t=0 •B j (D) = B j (D) on p,q γ,µ • tr t=0 when κ j,γ < 1+µ q , we may without loss of generality assume that u 0 = 0. By Corollary 6.8 we may furthermore assume that g = 0. Defining A B as the operator on . . , n} and given by the rule A B v := A(D)v, we need to show that 1 + A B enjoys the property of maximal L q µ -regularity: for every In the same way as in [18,Theorem 7.4] it can be shown that As Y is a UMD space, 1 + A B enjoys maximal L q µ -regularity for µ = 0; see e.g. [65,Section 4.4] and the references therein. By [12,53] this extrapolates to all µ ∈ (−1, q − 1) (i.e. all µ for which v µ ∈ A q ).

Appendix B. Comments on the localization and perturbation procedure
As already mentioned in the proof of Theorem 3.4, the localization and perturbation procedure for reducing to the model problem case on d + is worked out in full detail in [46]. However, there only the case q = p with temporal weights having a positive power is considered. For some of the estimates used there (parts) of the proofs do not longer work in our setting, where the main difficulty comes from q p. It is the goal of this appendix to consider these estimates.
Top order coefficients having small oscillations. The most crucial part in the localization and perturbation procedure where we need to take care of the estimates is [46, Proposition 2.3.1] on top order coefficients having small oscillations. To be more specific, we only consider the estimates in Step (IV) of its proof.
Before we go to these estimates, let us start with the lemma that makes it possible to reduce to the situation of top order coefficients having small oscillations.   X) which, for every a ∈ (0, ∞) l , s ∈ , p ∈ [1, ∞) l , q ∈ [1, ∞] and w ∈ A ∞ ( d j ), restricts to a bounded linear operator from F s,a p,q,d ( d [d ;i],T , w; X) to F s,a p,q,d ( d , w; X) whose operator norm can be estimated by a constant independent of X and T .
Proof. Extending f from d−1 ×I to d−1 × by using an extension operator of Fichtenholz type and extending g from d−1 × I to d−1 × by using an extension operator as in Lemma B.2, we may restrict ourselves to the case I = .