The heat equation with rough boundary conditions and holomorphic functional calculus

In this paper we consider the Laplace operator with Dirichlet boundary conditions on a smooth domain. We prove that it has a bounded $H^\infty$-calculus on weighted $L^p$-spaces for power weights which fall outside the classical class of $A_p$-weights. Furthermore, we characterize the domain of the operator and derive several consequences on elliptic and parabolic regularity. In particular, we obtain a new maximal regularity result for the heat equation with rough inhomogeneous boundary data.


Introduction
Often solutions to PDEs can have blow-up behavior near the boundary of an underlying domain O ⊆ R d . Using weighted spaces with weights of the form w O γ (x) := dist(x, ∂O) γ for appropriate values of γ , allows for additional flexibility and even obtain well-posedness for problems which appear ill-posed at first sight. PDEs in weighted spaces have been considered by many authors (see e.g. [24,42,46,47]). Moreover, the H ∞ -functional calculus properties of differential operators on weighted space have been treated in several papers as well (see e.g. [7,11,12,52,62]. The development of the H ∞ -calculus was motivated by the Kato square root problem (see [63] for a survey) which was eventually solved in [10]. An H ∞ -calculus approach to the solution was obtained later in [14]. Since the work [39] it has turned out that the H ∞ -calculus is an extremely efficient tool in the L p -theory of partial differential equations (see the monographs [23,71] and references therein).
In this paper we study the boundedness of the H ∞ -calculus of the Laplace operator with Dirichlet boundary conditions Dir for bounded C 2 -domains O. This operator and its generalizations have been studied in many papers (see [20,21,51]. Our contribution is that we study Dir and its functional calculus on weighted spaces which do not fall into the classical setting, but which are useful for certain partial differential equations. In particular, we prove the following result. A similar result holds on the half space R d + or small deformations of the half space. The range γ ∈ (p − 1, 2p − 1) falls outside the classical A p -setting and Theorem 1.1 is new in this range. The range γ ∈ (−1, p − 1) can be treated by classical methods, and it can be derived from the general A p -case which will be considered in Section 4.
The boundedness of the H ∞ -calculus has many interesting consequences for the operator Dir on L p (O, w O γ ). Loosely speaking, the boundedness of the H ∞ -calculus can be used as a black box to ensure existence of certain singular integrals. In particular, the boundedness of the H ∞ -calculus implies: • Continuous and discrete square function estimates (see [ On bounded domains we analyse the spectrum of Dir and in particular we show that the analytic semigroup generated by Dir is exponentially stable. Additionally we use the functional calculus to characterize several of the fractional domain spaces. The main difficulty in the proof of Theorem 1.1 in the non-A p setting is that standard tools from harmonic analysis are not available. For instance, the boundedness of the Hilbert transform, the boundedness of the Hardy-Littlewood maximal function operator, and the Littlewood-Paley decomposition all hold on L p (R d , w O γ ) if and only if γ ∈ (−1, p − 1) (see [33,Chapter 9] and [77]). Here one also needs to use the fact that the A p -condition holds if and only if γ ∈ (−1, p − 1). As a consequence, we have to find a new approach to obtain the domain characterizations, sectoriality estimates and the boundedness of the functional calculus.
We have already mentioned that Theorem 1.1 implies maximal regularity results. As a further application we will derive a maximal regularity result for the heat equation on weighted spaces with rough inhomogeneous boundary conditions. The main reason we can allow much rougher boundary data than in previous works is that we allow γ ∈ (p − 1, 2p − 1). Maximal regularity results can be used to study nonlinear equations in an effective way (see e.g. [72] and references therein). The result below is a special case of Theorem 7. 16. In order to make the result transparent without losing the main innovative part of the result, we state the result in the special case u 0 = 0, f = 0 and p = q and without weights in time. Conversely, the conditions on g are necessary in order for u to be in the intersection space. Note that δ ∈ (0, 1) can be taken arbitrarily close to zero by taking γ arbitrarily close to 2p − 1. Moreover, if γ ∈ (2p − 3, 2p − 1) then the compatibility condition g(0, ·) = 0 also vanishes. Theorem 1.2 was proved in [22] and [82] for γ = 0, and in this case the smoothness parameter equals δ = 1 − 1 2p . In [22] actually the general setting of higher order operators A with boundary conditions of Lopatinskii-Shapiro was considered. In [57] the first author extended the latter result to the weighted situation with γ ∈ (−1, p − 1), in which case δ ∈ ( 1 2 , 1) can only be taken arbitrarily close to 1 2 by taking γ close to p − 1. It would be interesting to investigate if one can extend special cases of [57] to other values of γ . In ours proofs the main technical reason that we can extend the range of γ 's in the Dirichlet setting is that the heat kernel on a half space has a zero of order one at the boundary. The heat kernel in the case of Neumann boundary conditions does not have this property. Moreover, the Neumann trace operator is not well-defined for γ ∈ (p − 1, 2p − 1). It is a natural question to ask for which kernels associated to higher order elliptic operators with different boundary conditions one has similar behavior at the boundary. In such cases one might be able to allow for rougher boundary data as well.
There exist several theories of elliptic and parabolic boundary value problems on other classes of function spaces than the L q (L p )-framework of the above. The case that L p is replaced by a weighted Besov or Triebel-Lizorkin space is considered by the first named author in [54] in the elliptic setting and in [55] in the parabolic setting. The advantage in that setting is that one can use Fourier multiplier theorems for A ∞ -weights. The results in [54,55] are independent from the results presented here since in the non-A p setting Triebel-Lizorkin spaces do not coincide with Sobolev spaces. Results in the framework of tent spaces have been obtained in [5,8,13] for elliptic equations and in [9] for parabolic equations. Here in some cases the boundary data is allowed to be in L p or L 2 .
The paper is organized as follows. In Section 3 we present some results on traces, Hardy inequalities and interpolation inequalities which will be needed. In Section 4 we consider the half space case with A p -weights. In Section 5 we consider the half space case for non-A p -weights. We extend the results to bounded domains in Section 6, where Theorem 1.1 can be derived from Corollary 6.2. In Section 7 we consider the heat equation with inhomogeneous boundary conditions and, in particular, we will derive Theorem 1.2. In many of our considerations we consider the vector-valued situation. This is mainly because it can be convenient to write Sobolev spaces as the intersection of several simpler vector-valued Sobolev spaces.
Acknowledgment. The authors would like to thank Dorothee Frey and Bas Nieraeth for helpful discussions on Section 5.4.
For two topological vector spaces X and Y (usually Banach spaces), L(X, Y ) denotes the space of continuous linear operators. We write A p B whenever A ≤ C p B where C p is a constant which depends on the parameter p. Similarly, we write A p B if A p B and B p A. Unless stated otherwise in the rest of the paper X is assumed to be a Banach space. is the space of X-valued tempered distributions. We refer to [2][3][4] for introductions to the theory of vectorvalued distribution.

Function spaces and weights
A locally integrable function w : O → (0, ∞) is called a weight. A weight w on R d will be called even if w(−x 1 , x) = w(x 1 , x) for x 1 > 0 and x ∈ R d−1 .
Although we will be mainly interested in a special class of weights, it will be natural to formulate some of the result for the class of Muckenhoupt A p -weights. For p ∈ (1, ∞) and a weight w : R d → (0, ∞), we say that w ∈ A p if Here the supremum is taken over all cubes Q ⊆ R d with sides parallel to the coordinate axes. For p ∈ (1, ∞) and a weight w : R d → (0, ∞) one has w ∈ A p if and only if the Hardy-Littlewood maximal function is bounded on L p (R d , w). We refer the reader to [33,Chapter 9] for standard properties of A p -weights. For a fixed p and a weight w ∈ A p , the weight For a set ⊆ R d with nonempty interior and w : → (0, ∞) let L 1 loc ( ; X) denote the set of all functions such that for all bounded open sets 0 with 0 ⊆ , we have f | 0 ∈ L 1 ( 0 , w; X). In this case f is called locally integrable on . If the p-dual weight w = w −1/(p−1) (w = 1 when p = 1) is locally integrable on O, then L p (O, w; X) → D (O; X).
For p ∈ (1, ∞), an integer k ≥ 0 and a weight w with is a Banach space. We refer to [49,50] for a detailed study of weighted Sobolev spaces. Finally, for a set ⊆ R d with nonempty interior we let W k,1 loc ( , w; X) denote the space of functions such that D α f ∈ L 1 loc ( , w; X) for all |α| ≤ k. Let us mention that density of C ∞ c (O; X) in W 1,p (O, w; X) is not true in general, not even for w ∈ A ∞ . A sufficient condition class is w ∈ A p (see [80,Corollary 2.1.6]). Further examples and counterexamples can be found in [49,Chapter 7 & 11] and [83].
We further would like to point out that in general W k,p (O, w) does not coincide with a Triebel-Lizorkin space F k p,2 (O, w) if w / ∈ A p . Moreover, in the X-valued setting this is even wrong for w = 1 unless X is isomorphic to a Hilbert space (see [36]).

Localization and C k -domains
For later it will be convenient to define, given a special C k c -domain O with k ∈ N 0 , the numbers where the infimum is taken over all h ∈ C k c (R d−1 ; R) for which O, after rotation and translation, can be represented as (2.1).
Note that, in the above definition, V may be replaced by any smaller open neighborhood of x. Hence, we may without loss of generality assume that W is a C k c -domain. Moreover, if k ∈ N 0 then for any > 0 we can arrange that [W ] C k < .
If U, V ⊆ R d are open and : U → V is a C 1 -diffeomorphism, then we define * : where j = det(∇ ) denotes the Jacobian. In this way Obviously, det(∇ ) = 1. For a weight w : In the important case that w(x) = |x 1 | γ , we have In this way for k ∈ N 0 , the mapping * defines a bounded isomorphism * : In the paper we will often use a localization procedure. We will usually leave out the details as they are standard. In the localization argument for the functional calculus (see Theorem 6.1) we do give the full details as a precise reference with weighted spaces seems unavailable. Given . . , N } and N n=0 η 2 n = 1 (see [48,Ch.8,Section 4]). These functions can be used to decompose the space The mappings I : E k −→ F k and P : F k −→ E k given by and satisfy PI = I , thus P is a retraction with coretraction I.
The infimum over all possible ϕ is called the angle of sectoriality and 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 [38,Proposition 10.1.9]).
Let H ∞ ( ω ) denote the space of all bounded holomorphic functions f : ω → C and let The infimum over all possible ω > ω(A) is called the angle of the H ∞ -calculus and is denoted by ω H ∞ (A). In this case we also say that A has a bounded H ∞ -calculus of angle ω H ∞ (A). For details on the H ∞ -functional calculus we refer the reader to [34] and [38].
The following well-known result on the domains of fractional powers and complex interpolation will be used frequently. For the definitions of the powers A α with α ∈ C we refer to [34,Chapter 3]. For details on complex interpolation we refer to [15,37,79].
We say that A has BIP (bounded imaginary powers) if for every s ∈ R, A is extends to a bounded operator on X. In this case one can show that there exists M, σ ≥ 0 such that (see [34,Corollary 3.5.7]) where the constant in the norm equivalence depends α, β, θ , the sectoriality constants and on the constant M and σ in (2.6). Often it is difficult to determine whether A + B with the above domain is a closed operator. Sufficient conditions are given in the following theorem which will be used several times throughout this paper (see [26,73]). Theorem 2.4 (Dore-Venni). Let X be a UMD space. Assume A and B are sectorial operators on X with commuting resolvents and assume A and B both have BIP with ω BIP (A) + ω BIP (B) < π. Then the following assertions hold: The following can be used to obtain boundedness of the H ∞ -calculus for translated operators ((1) is straightforward and (2) follows from [34,Corollary 5.5.5]): Remark 2.5. Let σ ∈ (0, π) and assume A is a sectorial operator of angle ≤ σ .
(1) If A has a bounded H ∞ -calculus of angle ≤ σ , then for all λ ≥ 0, A + λ has a bounded H ∞ -calculus of angle ≤ σ . (2) If there exists a λ > 0 such that A + λ has a bounded H ∞ -calculus of angle ≤ σ , then for all λ > 0, A + λ has a bounded H ∞ -calculus of angle ≤ σ .

UMD spaces and Fourier multipliers
Below the geometric condition UMD will often be needed for X. UMD stands for unconditional martingale differences. One can show that a Banach space X is a UMD space if and only if the Hilbert transform is bounded if and only if the vector-valued analogue of the Mihlin multiplier theorem holds. For details we refer to [37,Chapter 5]. Here we recall the important examples for our considerations.
• Every Hilbert space is a UMD space; The following is a weighted version of Mihlin's type multiplier theorem and can be found in [67, Proposition 3.1] Then T m extends to a bounded operator on L p (R d , w; X), and its operator norm only depends on d, X, p, [w] A p and C m .
. Then the following assertions hold: Here W 1,q 0 (R + , v; X) denotes the closed subspace of W 1,q (R + , v; X) of functions which are zero at t = 0.  (2) can be derived as a consequence by repeating part of the proof of [58,Theorem 6.8] where the case v(t) = |t| γ was considered.
For p ∈ (1, ∞), w ∈ A p and s ∈ R, we define the Bessel potential space H s,p (R d , w; X) as the space of all f ∈ S (R d ; X) for which F −1 [(1 + | · | 2 ) s/2 f ∈ L p (R d , w; X). This is a Banach space when equipped with the norm

For an open subset
. This is a Banach space when equipped with the norm The next result can be found in [67,Propositions 3.2 & 3.5]. The duality pairing mentioned in the statement below is the natural extension of The UMD condition is also necessary in the above result (see [37,Theorem 5.6.12]).

Proposition 2.9 (Intersection representation). Let
In the above we use the convention that w is extended in a constant way in the remaining d 2 coordinates. In this way w ∈ A p (R d ) as well.
Proof. → is obvious. To prove the converse Let α be a multiindex with k := |α| ≤ n. It suffices to prove D α u L p (w;X) ≤ C( u L p (w;X) + d j =1 D k j u L p (w;X) ). This follows by using the Fourier multiplier m: Here

Hardy's inequality, traces, density and interpolation
In this section we will prove some elementary estimates of Hardy and Sobolev type and obtain some density and interpolation results. We will present the results in the X-valued setting, and later on apply this in the special case X = L p (R d−1 ) to obtain extensions to higher dimensions in Theorem 5.7.
Details on traces in weighted Sobolev spaces can be found in [40] and [57]. We will need some simple existence results in one dimension.
Note that the local L 1 -condition on w holds in particular for w ∈ A p .
Proof. Let u ∈ W 1,p (R + , w; X). By Hölder's inequality and the assumption on w we have L p ((0, t), w; X) → L 1 (0, t; X). In particular u and u are locally integrable on [0, ∞). Let Then v is continuous on [0, t] and moreover v = u on (0, t) (see [37,Lemma 2.5.8]). It follows that there is a z ∈ X such that u = z + v for all s ∈ (0, t). In particular, u has a continuous extension u to [0, t] given by u = z + v.
To prove the required estimates we just write u instead of u. Let x ∈ [0, ∞).
The estimate in (1) follows from Now by Hölder's inequality we have p and the latter tends to zero as t → 0.
Proof. First consider γ < p − 1. Writing u(t) = t 0 u (s)ds, it follows that Now the result follows from Hardy's inequality (see [32,Theorem 10.3.1]). The case γ > p − 1 follows similarly by writing u(t) = ∞ t u (s)ds. Here we use the fact that, by approximation, it suffices to consider the case where u = 0 on [n, ∞).
For other exponents γ than the ones considered in Lemma 3.1 another embedding result follows. Note that this falls outside the class of A p -weights.

Traces and Sobolev embedding
For u ∈ W 1,1 loc (R d + ; X) we say that Tr(u) = 0 if Tr(ϕu) = 0 for every ϕ ∈ C ∞ with bounded support in R d + . Note that ϕu ∈ W 1,1 ([0, ∞); L 1 (R d−1 ; X)) whenever u ∈ W 1,p (R d + , w; X) and w − 1 . Thus the existence of the trace of ϕu follows from Lemma 3.1. For integers k ∈ N 0 , p ∈ (1, ∞) and w ∈ A p , we let W k,p The traces in the above formulas exists since W k,p (R d + , w; X) → W k,1 loc (R d + ; X). We extend the definitions of the above spaces to the non-A p -setting.
Here the trace exists if j := k − |α| > γ +1 p since then j ≥ 2 and, by Lemmas 3.1 and 3.3, For γ ∈ (−∞, −1) and k ∈ N 0 we further let W k,p This notation is suitable since for k ∈ N 1 , by Lemma 3.1, Using  Proof. Since L p (O, w) ⊗ X is dense in L p (O, w; X) it suffices to consider the scalar setting. We claim that it furthermore suffices to approximate functions which are compactly supported in O.

Density results
To prove the claim, let f ∈ L p (O, w) and let (K n ) n∈N be an exhaustion by compact sets of O. Observe that f 1 K n → f by the dominated convergence theorem. Therefore, it suffices to consider functions f with compact support in O. Extending such functions f by zero to R d , the claim follows.
Let q ∈ (p, ∞) be such that w ∈ A q . Then for all functions f ∈ L p (R d , w) with compact support K ⊆ O, by Hölder's inequality one has Therefore, it suffices to approximate such functions f in the L q (R d , w) norm. To do so one can use a standard argument (see [58,Lemma 2.2]) by using a mollifier with compact support.  The density result [49,Theorem 7.2] can be extended to the vector-valued setting: Next we will prove a density result for power weights of arbitrary order using functions with compact support in .

Proof. By a standard localization argument it suffices to consider
. By a simple truncation argument we may assume that u is compactly supported on R d + . To prove the required result we will truncate u near the plane . This will be proved below. Using the claim the proof can be finished as follows. It remains to show that each u ∈ Therefore, it suffices to approximate u in the W k,p (R d + ; X)-norm. This can be done by extension by zero on R d − followed by a standard mollifier argument (see [58,Lemma 2.2]). To prove the claim for convenience we will only consider d = 1. Since φ n does not depend on x the general case is similar. Fix m ∈ {0, . . . , k}. By Leibniz formula one has (φ n u) ( The latter tends to zero as n → ∞ by the dominated convergence theorem.
In the next result we prove a density result in real and complex interpolation spaces. It will be used as a technical ingredient in the proofs of Lemma 3.14 and Proposition 3.17.
Proof. First consider the real interpolation space. In the case γ < −1 the result follows from , by Lemma's 3.6 and 3.7 it suffices to consider u ∈ C ∞ c (R d + ; X) and to approximate it by functions in Note that, for example in the case d = 1, for one of the terms Now we obtain that there is a constant C independent of n such that The latter tends to zero by the assumptions. The other terms can be treated with similar arguments. Finally one can approximate each u n by using (3.2) and the arguments given there. The density in the complex case follows from The next standard lemma gives a sufficient condition for a function to be in W 1,1 loc (R d ; X) when it consists of two W 1,1 loc -functions which are glued together. To prove the result one can reduce to the one-dimensional setting and use the formula u(t) − u(0) = t 0 u (s) ds. We leave the details to the reader.
Finally we will need the following simple density result in the A p -case.

Proof. By localization it suffices to consider
In particular, this shows that E 0 is bounded. For the final assertion let u ∈ W k,p . By a truncation we may assume u has bounded support.
, the result follows.

Interpolation
We continue with two interpolation inequalities. The first one is [58, Lemma 5.8].
The above result holds on smooth domains as well provided we replace the homogeneous norms [·] W k,p by · W k,p . In order to extend this interpolation inequality to a class of non-A pweights, we will use the following pointwise multiplication mappings M and M −1 . Let . By duality we obtain a mapping M : Proof. Since the derivatives with respect to x i with i = 1 commute with M, we only prove the result in the case d = 1. Observe that Mu L p (R + ,w γ −p ;X) = u L p (R d + ,w γ ;X) . Moreover, by the product rule, we have (Mu) (j ) = ju (j −1) + Mu (j ) for j ∈ {0, 1, 2}. Therefore, where we applied Lemma 3.2. This proves the required boundedness of M.
. By Proposition 3.8 and Lemma 3.11 it suffices to prove the required estimate for u ∈ C ∞ c (R + ; X). By the product rule, Now it remains to observe that by Lemma 3.2 (applied i times) .
Proof. By an iteration argument as in [48,Exercise 1.5.6], it suffices to consider k = 2 and j = 1. Moreover, by a scaling involving u(λ·) it suffices to show that The case γ ∈ (−1, p − 1) is contained in Lemma 3.12, where we actually do not need to proceed through (3.3). So it remains to treat the case γ ∈ (−p − 1, −1) ∪ (p − 1, 2p − 1). By standard arguments (see e.g. [79, Lemma 1.10.1]), it suffices to show that We first assume that γ ∈ (p − 1, 2p − 1). Using Lemma 3.13 and real interpolation of operators, we see that M is bounded as an operator By a combination of [79, Lemma 1.10.1] and (3.3) for the case γ ∈ (−1, p − 1), the space on the right hand side is continuously embedded into From Lemma 3.9 and the fact that Combining this with Lemma 3.13 we obtain (3.3).
Combining this with W n,p (R d and Lemma 3.13 we obtain (3.3).

Proof. By a localization argument it suffices to consider the case
is sectorial having bounded imaginary powers with angle 0. By a combination of Proposition 2.3 and [29, Lemma 9.5], Now the result follows from the following intersection representation for n ∈ N: Here → is clear. To prove the converse let u be in the intersection space. We first claim that u ∈ W n,p (R d + , w γ ; X). Using a suitable extension operator it suffices to show the result with R + and R d + replaced by R and R d respectively. Now the claim follows from Proposition 2.9. To prove u ∈ W n,p . It remains to show Tr(D α u) = 0. By assumption and the claim D α 1 Now we extend the last identity to the non-A p setting for j = 1 and k = 2.
Proof. The case γ ∈ (−1, p − 1) is contained in Proposition 3.15. For the case γ ∈ (jp − 1, (j + 1)p − 1) with j = 1 or j = −1 we reduce to the previous case. By a localization argument it suffices to consider O = R d + . By Lemma 3.13 and since the complex interpolation method is exact we deduce Next we prove a version of Proposition 3.16 without boundary conditions by reducing to the case with boundary conditions.

Proof. By a localization argument it suffices to consider
It remains to establish the case γ ∈ (p − 1, 2p − 1). The inclusion ← follows from Proposition 3.16 and W 1,p . To prove →, by Lemma 3.9 it suffices to show that , using Lemma 3.13 twice and the result for the A p -case already proved, we obtain Next we turn to a different type of interpolation result where we interpolate all the possible parameters including the target spaces. There are many existing results in this direction (see [67,Proposition 3.7] and [72, Proposition 7.1] and references therein). In the unweighted case it has recently also appeared in [4, Theorem 4.5.5] using a different argument. Since our (independent) proof is of interest we present the details.
Observe that w ∈ A p by [33, Exercise 9.1.5]. The proof of the theorem will be given below. As a corollary of Proposition 3.17 and Theorem 3.18 we obtain (using the identification from Proposition 2.8) the following mixed-derivative theorem: Proof. By Proposition 2.8, Theorem 3.18, and Proposition 3.17, For the proof of Theorem 3.18 we need two preliminary results. The first result follows as in [ For the next result we need to introduce some notation. Let (ε k ) k≥0 be a Rademacher sequence on a probability space . Let σ : N → (0, ∞) be a weight function, p ∈ (1, ∞) and let Rad σ,p (X) denote the space of all sequences (x k ) k≥0 in X for which The above space is p-independent and the norms for different values of p are equivalent (see [38,Proposition 6.3.1]). If σ ≡ 1, we write Rad p (X) := Rad σ,p (X). Clearly (x k ) k≥0 → (σ (k)x k ) k≥0 defines an isometric isomorphism from Rad σ,p (X) onto Rad p (X). By [38,Corollary 6.4.12], if X does not contain a copy isomorphic to c 0 (which is the case for UMD spaces), then (x k ) k≥0 in Rad σ,p (X) implies that k≥0 ε k σ (k)x k converges in L p ( ; X) and in this case .

Interpolation of the unweighted spaces
holds if X 0 and X 1 are K-convex spaces (see [38,Theorem 7.4.16] for details). In particular, UMD spaces are K-convex (see [37,Proposition 4.3.10]). We need the following weighted version of complex interpolation of Rad-spaces.
Proof. We use the same method as in [79, 1.18.5]. Let .
with implicit constants only depending on p j , X j , s j , [w j ] A p j . Now to reduce the statement to Proposition 3.21 we use a retraction-coretraction argument (see [ . Let ψ n = n+1 k=n−1 φ k for n ≥ 0, and let ψ −1 = 0. Then ψ k = 1 on supp ( φ k ) for all k ≥ 0, and supp ( ψ 0 ) ⊆ {ξ : |ξ | ≤ 2} and supp ( ψ k ) ⊆ {ξ : k≥0 . The boundedness of S follows from (3.6). We claim that R is bounded and this will be explained below. By the special choice of ψ k we have RS = I . Therefore, the retraction-coretraction argument applies and the interpolation result follows.
To prove claim let E j = L p j ( ; Y j )). Due to (3.6) and by density it suffices to show that, for all finitely-nonzero sequences (f ) ≥0 in Y j and all n ≥ 0, (3.7) Below, for convenience of notation, we view sequences on N as sequences on Z through extension by zero. Under this convention, by the Fourier support properties of (ϕ k ) k and the R-boundedness of {ϕ k * : k ≥ 0} (see [67,Lemma 4.1]) and the implied R-boundedness of where in the last step we used the contraction principle (see [37,Proposition 3.24]).

Dir on R d + in the A p -setting
Let p ∈ (1, ∞) and w ∈ A p (R d  ∂ 2 j u.
Let G z : R d → R denote the standard heat kernel on R d : It is well-known that |G z * f | ≤ cos −d/2 (arg(z))Mf , where M denotes the Hardy-Littlewood maximal function (see [38,Section 8.2]). Therefore, with f (y) = sign(y 1 )f (|y 1 |, ỹ) and By the properties of G z , the operator T (z) is bounded on L p (R d + , w; X) for any w ∈ A p with T (z) ≤ M B(L p (w)) cos −d/2 (arg(z)). In Theorem 4.1 we will show that T (z) is an analytic C 0 -semigroup with generator Dir . Moreover, in case X is a UMD space we characterize D( Dir ) and prove that Dir is a sectorial operator with a bounded H ∞ -calculus of angle zero.
Recall that a weight w is called The next result is the main result of this section on the functional calculus of − Dir on L pspaces with A p -weights. The result on the whole of R d is well-known to experts, but seems not to have appeared anywhere. By a standard reflection argument we deduce the result on R d + . It can be seen as a warm-up for Theorem 5.7 where weights outside the A p -class are considered. Theorem 4.1. Let X be a UMD space. Let p ∈ (1, ∞) and let w ∈ A p be even. Then the following assertions hold: with equivalent norms, the analytic C 0 -semigroup (e z Dir ) z∈C + is uniformly bounded on any sector ω with ω ∈ (0, π/2) and

Moreover, all the implicit constants only depend on X, p, d and [w] A p .
For the proof we use a simple lemma on odd extensions. For u ∈ L p (R d , w; X), the functions u and E odd u denote the odd extension of u: Proof. The case k = 0 is easy, so let us assume k ∈ {1, 2}. For u ∈ W k,p Indeed, this follows from Lemmas 3.10 and 3.1. From (4.3) we find that u ∈ W 2,p (R d , w; X) and that the stated estimates hold. If u ∈ W k,p (R d , w), then by Lemma 3.6 we can find u n ∈ C ∞ c (R d ) ⊗ X such that u n → u in W 2,p (R d , w; X). Then also u n (−·, ·) → u in W 2,p (R d , w; X). Now v n := (u n + u n (−·, ·))/2 satisfies v n ∈ C ∞ c (R d ; X) and v n (0, ·) = 0 and v n → u in W 2,p (R d + , w; X). Since Tr(v n ) = 0 the continuity of the trace implies Tr(u) = 0 as well. This part of the proof also implies the desired density result.
All the statements now follow. Dividing by r 2− d p and taking the limit r → ∞ gives (4.4). The existence and uniqueness in the second claim follow from the sectoriality in Theorem 4.1. Moreover, together with (4.4), the sectoriality yields the estimates for |α| = 0 and |α| = 2 in (4.5). The case |α| = 1 subsequently follows from Lemma 3.12. p, q ∈ (1, ∞), v ∈ A q (R), w ∈ A p (R d ) and assume w is even. Let J ∈ {R, R + }. Then the following assertions hold: For all λ > 0 and f ∈ L q (J, v; L p (R d + , w; X)) there exists a unique u ∈ W 1,q (J, v; L p (R d + , w; X)) ∩ L q (J, v; W 2,p Dir (R d + , w; X)) such that u + (λ − Dir )u = f , u(0) = 0 in case J = R + . Moreover, the following estimate holds

Corollary 4.4 (Heat equation). Let X be a UMD space. Let
Proof. Since L p (R d + , w; X) is a UMD space, by Proposition 2.7, d/dt has a bounded H ∞calculus on L q (J, v; L p (R d + , w; X)). Therefore, from Theorem 4.1, Remark 2.5 (1), and Theorem 2.4 Now the result follows from Corollary 4.3 applied pointwise in t.

Remark 4.5.
(i) The same result as in Corollary 4.4 holds for on the whole of R d . For results on elliptic and parabolic equations with A p -weights in space we refer to [35]. (ii) Due to Calderón-Zygmund extrapolation theory one can add A q -weights in time after considering the unweighted case (see [18]). (iii) It would be interesting to extend Corollary 4.4 to spaces of the form L p (R × R d + , w; X) where w depends on time and space. For some result in this direction concerning the maximal regularity estimate we refer to [25].
(iv) The estimates in Corollaries 4.3 and 4.4 also hold for λ = 0. However, solvability does not hold for general f .

Dir on R d + in the non-A p -setting
In this section we will extend the results of Section 4 to weighted L p -spaces with w γ (x) = |x 1 | γ where γ ∈ (p − 1, 2p − 1). This case is not included in the A p -weight and is therefore not accessible through classical harmonic analysis. The reflection argument cannot be applied since the weight is not locally integrable in R d .
Step 1: Reduction to an estimate in the case X = C. In this step we show that it is enough to prove the estimate Having this estimate, we get for all f ∈ C c (R d + ) ⊗ X, from which the analyticity and strong continuity follow. Indeed, note that for g ∈ C c (R + ) ⊗ X, z → T (z)f, g is analytic on φ and continuous on φ by Theorem 4.1 with w = 1. Therefore, in case X = C, the weak continuity of T on φ follows by density in the case p ∈ (1, ∞) and by weak * -sequential density of C c in L ∞ in case p = 1 (see [74,Corollary 2.24]). This in turn implies strong continuity by [ Taking L p (R d−1 )-norms for fixed x 1 ∈ R + , and using Minkowski's inequality and Young's inequality, we obtain In order to prove these estimates, observe that with z = te iδ ,  One can check that |1 − e −4xy | ≤ min{1, 4xy}. Therefore, k satisfies It follows that The first term satisfies where we used 1 − γ +1 p > −1. Since γ > −1, the second term satisfies Next we estimate the integral over the x-variable. For y ∈ (0, 1), we can write where we used 2 − γ +1 p ≥ 0 and γ + 1 ≥ 0. For y ≥ 1, since γ +1 Step 4: The case p = 1: One can still reduce to the case d = 1 by Fubini's theorem. Moreover, instead of using Schur's lemma, by Fubini's theorem it suffices to show that The case γ ∈ [0, 1) can be treated in the same way as in the above proof. In case γ ∈ (−2, 0) we argue as follows: On the other hand, since γ + 2 > 0, we have In the next example we show that the range for γ in Proposition 5.1 is optimal.
Example 5.2. Let p ∈ (1, ∞) and γ / ∈ (−p − 1, 2p − 1). We give an example of a function f ∈ L p (R d + , w γ ) such that for all t > 0, Here T (t)f is defined by (4.1). By duality we only need to consider γ ≥ 2p − 1. Let β ∈ (1/p, 1) and set f ( Then, on the one hand, f ∈ L p (R d + ; w γ ). On the other hand, for x ∈ Q d , Let −A denote the generator of the semigroup (T (z)) z of Proposition 5.1. Then by standard results of analytic semigroups we see that A is sectorial with ω(A) = 0.
In the case of a X is a UMD space, −A even has a bounded H ∞ -calculus: Proof. The case γ ∈ (−1, p − 1) follows from Theorem 4.1. For the other values of γ we use a classical perturbation argument (see [45]).
Fix f ∈ C c (R d + ; X)) and let g = w 1 p γ f and ψ(x, y) = Therefore, The first term on the right-hand side can be estimated by the boundedness of the H ∞ -calculus in the unweighted case (see Theorem 4.1): Therefore, it remains to show x → φ(A)(ψ(x, ·)g)(x) L p (R + ;X) ≤ C g L p (R + ;X) = C f L p (R + ,w γ ;X) . (5.6) Step 2: To prove (5.6) we estimate the integrals over ± in (5.5) separately. By symmetry it suffices to consider + . Let δ = (π − σ )/2. For λ = re iσ with r > 0 and h ∈ L p (R + , w γ ; X), we have the following Laplace transform representation for the resolvent (see [28]): Observe that by (5.4) we can write Below we will write x = (x 1 , x) and y = (y 1 , ỹ). Using the kernel representation of the semigroup we can write Now, Here we use −1 − p < γ < 2p − 1 to obtain Combining the above estimates we obtain the required estimate x → + φ(λ)R(λ, A)(ψ(x, ·)g)(x)dλ L p (R + ;X) ≤ C g L p (R + ;X) cos 2 (δ) .

The Dirichlet Laplacian on R
Therefore, 1 t (T (t)u − u) → Dir u in L p (R + , w γ ; X) by strong continuity of (T (s)) s≥0 . Therefore, u ∈ D(A) with −Au = Dir u. Now for u ∈ W 2,p Dir (R + , w γ ; X), using Proposition 3.8, we can find a sequence (u n ) n≥1 in C ∞ c (R + ; X) such that u n → u in W 2,p Dir (R + , w γ ; X). Then −Au n = Dir u n → Dir u in L p (R + , w; X). Therefore, the closedness of A yields that u ∈ D(A) and −Au = Dir u.
Next we show −A ⊆ Dir . Using Dir ⊆ −A, for this it is enough that 1 + A is injective and 1 − Dir is surjective. Being the generator of a bounded analytic semigroup (see Proposition 5.1), A is sectorial, implying that 1 + A is injective. For the surjectivity of 1 − Dir we consider the equation u − Dir u = f , for f ∈ L p (R d + , w γ ; X). Let us first consider f ∈ C ∞ c (R + ; X). Let f denote the odd extension of f . Clearly, f ∈ C ∞ c (R; X) ⊆ S(R; X). So we can define u ∈ S(R; X) by u : , yielding a solution of the equation u− u = f . Since u is odd, it also satisfies the Dirichlet condition u(0) = 0. By restriction to R + we obtain a solution u := u |R + ∈ W 2,p Dir (R + , w γ ; X) of the equation Proposition 3.8), it suffices to prove the estimate u W 2,p (R d ,w γ ;X) f L p (R d + ,w γ ;X) . To finish, we prove this estimate. (1, A)f . It follows that u L p (R + ,w γ ;X) f L p (R + ,w γ ;X) . Since u = u − f we find that u L p (R + ,w γ ;X) f L p (R + ,w γ ;X) . By interpolation the same estimate holds for u (see Lemma 3.14). Proof. This can be proved in the same way as the second statement in Corollary 4.3.

The Dirichlet Laplacian on
The main result of this section is the following theorem. Note that the case γ ∈ (−1, p − 1) was already considered in Theorem 4.1. See Section 5.5 for the case γ ∈ (−p − 1, −1).
(2) Dir is a closed and densely defined linear operator on L p (R d + , w γ ; X) with with an equivalence of norms only depending on X, p, d, γ .
Below we will frequently use Fubini's theorem in the form of the identification

H t (x, y)f (y)dy
for all f ∈ L p (R d + , w γ ; X). Therefore, Dir is the generator of the heat semigroup from Proposition 5.1.
We now show that with an equivalence of norms. Note that then Dir = Dir and the assertions (1), (2) follow. Since Proof. This can be done in the same way as Corollary 4.3, now using the explicit formula w γ (r · ) = r γ w γ in the scaling argument. Now using Theorem 5.7, as in Corollary 4.4 we obtain the following maximal regularity result for the weights w γ with γ ∈ (p − 1, 2p − 1). The case γ ∈ (−1, p − 1) was already considered in Corollary 4.4. X be a UMD space. Let p, q ∈ (1, ∞), v ∈ A q (R), γ ∈ (p − 1, 2p − 1). Let J ∈ {R + , R}. Then the following assertions hold:

Corollary 5.10 (Heat equation). Let
Moreover, the following estimate holds Given the results of Sections 4 and 5 it would be natural to conjecture that all weights of the form

Extrapolation of functional calculus
As soon as one knows the boundedness of the functional calculus of a generator on a space L 2 (R d + , dμ) for some doubling measure μ, then, if the heat kernel satisfies Gaussian estimates with respect to μ, one can extrapolate the boundedness of the functional calculus to L p (R d + , wdμ) for p ∈ (1, ∞) and w ∈ A p (μ). Here A p (μ) is the weight class associated to the measure μ on R d + . The above is presented in the setting of homogeneous spaces in [27] in the unweighted setting and in [62,Theorem 7.3] in the weighted setting. Extension to the setting without kernel bounds can be found in [12,16].
In order to apply [62,Theorem 7.3] to our setting, we set dμ(x) = x 1 dx. The reason to take this measure is that the kernel H z (x, y) as defined in (4.1) has a zero of order one at x 1 = 0. Then μ is doubling and one can check that w α (x) := x α 1 is in A p (μ) if and only if α ∈ (−2, 2p − 2). From Theorem 5.7 we know that on L 2 (R d + , μ), − Dir has a bounded H ∞ -calculus with ω H ∞ (− Dir ) = 0. So in order to extrapolate the latter to L p (R d + , wdμ) for p ∈ (1, ∞) and w ∈ A p (μ) it suffices to check the kernel condition of [62,Theorem 7.3]. For this (due to (5.4)) it suffices to show that there exist constant C, c > 0 such that Here the nominator y 1 is to correct for the choice of the measure μ. After renormalization for the condition (5.11) it suffices to consider t = 1/4. In the case that x 1 < 1, (5.11) is equivalent to .

Therefore, it remains to check that
where the supremum is taken over all x 1 > 1 and y 1 > 1 4x 1 . Since for x 1 → 0 and x 1 → ∞ this expression tends to zero, optimizing first over x 1 , yields that x 1 = φ(y 1 ) = In the case y 1 ≤ 1/10, the optimal solution x 1 = φ(y 1 ) is not feasible and the maximum is attained at x 1 = 1 4y 1 . In this case we obtain It follows that A ≤ max{B 1 , B 2 } < ∞.
As a consequence we obtain the following result.
Theorem 5.13. Let dμ = x 1 dx, p ∈ (1, ∞) and w ∈ A p (μ). Then the heat semigroup given by (4.1) extends to an analytic semigroup on L p (R d + , w) and its generator −A has the property that A has a bounded H ∞ -calculus with ω H ∞ (A) = 0.
Note that this does not directly imply the same for − Dir because it is unclear whether A = − Dir in the above setting, because we do not know whether the domains coincide. Note that the approach presented in Theorem 5.7 also works for weights of the form w(x) Instead of applying Theorem 5.7 in the above situation one could also apply the simpler Theorem 4.1 with dμ(x) = x β 1 dx with β ∈ (0, 1). Indeed, then w α ∈ A p (μ) if and only if −1 < α + β < βp + p − 1. Again one can check condition (5.11) with left-hand side 1 and for the new measure μ. Therefore, choosing β arbitrary close to 1, we obtain − Dir has a bounded H ∞ -calculus on L p (R d + , w γ ) for γ ∈ (−1, 2p − 1). Finally, let us remark that some work needs to be done in order to obtain Theorem 5.13 in the vector-valued setting using the above approach. Dir (R d + , w γ ; X). In this subsection we will discuss the failure of this domain description for the case γ ∈ (−p − 1, −1).
Let us start with the one-dimensional case. The point where the proof of Proposition 5.4 does not work for the case γ ∈ (−p − 1, −1) is the fact that S odd (R + ; X) W 2,p (R + , w γ ; X) in that case, which is illustrated by the following example.
A duality argument yields that the right-hand side space in (5.12) actually is the "correct" the domain for the Dirichlet Laplacian Dir on L p (R + , w γ ; X) when γ ∈ (−p − 1, −1): Then, viewing L p (R + , w γ ; X) as closed subspace of [L p (R + , w γ ; X * )] * , we have that Dir coincides with the realization of [ Dir ] * in L p (R + , w γ ; X). To see this, denote the latter operator by A. Given v ∈ D( Dir ), we have, for all u in the dense subspace Let us next turn to the d-dimensional case.
Proposition 5. 16. Let X be a UMD space, p ∈ (1, ∞) and γ ∈ (−p − 1, −1). Then Dir , defined as is the generator of the heat semigroup on L p (R d + , w γ ; X) given in Proposition 5.1. Moreover, with an equivalence of norms only depending on X, p, d, γ .
Proof. The first statement can be proved in the same way as Proposition 5.15, using Theorem 5.7 (1) instead of Proposition 5.4. The second statement can be proved using the operator sum method as in Theorem 5.7, using Proposition 5.15 instead of Proposition 5.4.

Dir on bounded domains
In this section we will use standard localization arguments to obtain versions of Theorems 4.1 and 5.7 for bounded C 2 -domains O ⊆ R d . In particular it will be shown that the Dirichlet Laplacian Dir is a closed and densely defined linear operator for which − Dir has a bounded H ∞ -calculus of angle zero. Moreover, (e z Dir ) z∈C + is an exponentially stable analytic C 0 -semigroup. Here, w O γ (x) = dist(x, ∂O) γ . The main result of this section is the following version of Theorems 4.1 and 5.7 for bounded C 2 -domains. In the scalar case Theorem 6.1 implies the following result where we obtain additional information on the value of λ.

Let the Dirichlet Laplacian
Then the following assertions hold: Proof. (3): All assertions follow from Theorem 6.1 except the exponential stability. The latter will follow from (2).
(4): By the sectoriality of − 1 2 δ O − Dir , we have for all λ ≥ 0. On the other hand, and Dir is invertible we can deduce Finally, the estimates for the first order terms follow from Lemma 6.10 below.
Corollary 6.2 has the following consequences similar to Corollaries 4.4 and 5.10. This time we can allow λ = 0 since the semigroup is exponentially stable. A similar maximal regularity consequence can be deduced from Theorem 6.1 in the X-valued case, but this time with additional conditions on λ.

Corollary 6.3 (Heat equation). Let
Moreover, the following estimates hold

The adjoint operator [ Dir ] *
Recall that every UMD space is reflexive. Let X be a reflexive Banach space, p ∈ (1, ∞) and γ ∈ R.
Proof. This can be shown in the same way as in the proof of Proposition 5.15.
Note that in general the above domain is larger than W 2,p (R + , w γ ; X * ) (see (5.12)).

Intermezzo: identification of D((− Dir
In order to transfer the results of the previous sections to smooth domains (and in particular to prove Theorem 6.1) we will use standard arguments. However, in order to use perturbation arguments we need to identify several fractional domain spaces and interpolation spaces. In principle this topic is covered by the literature as well. However, the weighted setting is not available for the class of weights we consider here and requires additional arguments.
We start with a simple interpolation result for general A p -weights. In the next result we extend the definition of (3.1) to all k ∈ N 0 in the following way  We can now prove the two main results of this section.
To identify D((− Dir ) 3/2 ) in the case γ > p − 1 we first consider d = 1. By Theorem 5.7 and the previous case one has Observe that Thus by the d = 1 case, the boundedness of Dir,1 (1 − Dir ) −1 and d−1 (1 − Dir ) −1 (see Corollary 5.8), we obtain that for u ∈ W 2,p with the required norm estimate. Therefore, the required identity for D((− Dir ) 3/2 ) follows.

Localization: the proof of Theorem 6.1
As a first step in the localization we prove the following result for Dir on small deformations of half-spaces. Without loss of generality we can take ε ∈ (0, 1). A simple calculation shows that We first apply perturbation theory to obtain a bounded H ∞ -calculus for Dir + A. By the assumption we have where in the last step we used Corollary 5.8. This proves one of the required conditions for the perturbation theorem. In particular, this part is enough to obtain that for any ϕ > 0 and for ε small enough D( Dir + A) = D( Dir ) and 1 − Dir − A is sectorial of angle ≤ ϕ (see [60,Proposition 2.4.2]).
To obtain the same result for λ − for λ > 0 large enough it remains to apply a lower order perturbation result (see [51,Proposition 13.1]). For this observe The required estimate follows since by Proposition 6.7 and Theorem 6.8, where in the last step we applied Proposition 2.3. The two perturbation arguments give λ > 0 such that λ − Dir has a bounded H ∞ -calculus with ω H ∞ (λ − Dir ) ≤ φ. Moreover, there is an equivalence of norms in D( Dir ) = D( Dir ) = W 2,p (R d + , w γ ; X). The desired results follow.
The following lemma follows from Proposition 6.7 and Theorem 6.8 under a change of coordinates according to the C 2 -diffeomorphism from (2.4) and a standard retraction-coretraction argument using (2.5). The next step in the proof of the above theorem is a localization argument. This localization argument is a modification of the one in [20,Section 8] combined with the one in [48,Ch. 8,Sections 4 & 5] and results in the next lemma. On an abstract level the localization argument takes the following form. Lemma 6.11. Let A be a linear operator on a Banach space X, Ã a densely defined closed linear operator on a Banach space Y such that Ã has a bounded H ∞ -calculus. Assume there exists bounded linear mapping P : Y → X and I : X → Y such that the following conditions hold:  Since Ã +B is closed, the injectivity of I implies that A is closed. Since P is surjective, we have Therefore, A is densely defined. Now we will transfer the functional calculus properties of Ã +B to A. For this we claim that for μ large enough and λ ∈ C \ φ we have λ ∈ ρ(A + μ) and This clearly yields that A + μ has a bounded H ∞ -calculus of angle ≤ φ.
In order to prove the claim we first show that given λ ∈ ρ(Ã +B), for u ∈ D(A) and f ∈ X it holds that To prove the implication ⇐=, define u = PR(λ, Ã +B)If and g = (λ − A)u. Then by the implication =⇒ we find u = PR(λ, Ã +B)Ig and thus by injectivity f = g as required and additionally (6.4) holds.
Note that I maps D(A) into D(Ã) and that IAu =ÃIu + Bu for every u ∈ D(A). Also note that P maps D(Ã) to D(A) and that APũ = PÃũ + Cũ for every ũ ∈ D(Ã). Since each commutator [ , η n ] is a first order partial differential operator with C ∞ c -coefficients, it follows that IA −ÃI extends to a bounded linear operator from W to Y . An application of Lemma 6.11 finishes the proof.

The heat equation with inhomogeneous boundary conditions
In this section we will consider the heat equation on a smooth domain O ⊆ R d with inhomogeneous boundary conditions of Dirichlet type. In particular, Theorem 1.2 is a special case of Theorem 7.16 below. The main novelty is that we consider weights of the form w O γ (x) = dist(x, ∂O) γ with γ ∈ (p − 1, 2p − 1), which allows us to treat the heat equation with very rough boundary data.
In the proof of this theorem we use weighted mixed-norm anisotropic Triebel-Lizorkin spaces as considered in [57, Section 2.4] (see [53] for more details); for definitions and notations we simply refer the reader to these references.
As in the standard isotropic case (see [54]), we have: Proof. We cannot reduce to the R d -case directly since L p (R d , w γ ; X) → L 1 loc (R d ; X) for γ ≥ p − 1 and therefore cannot be seen as a subspace of the distributions on R d . However, we can proceed as follows. An easy direct argument (see [65,Remark 3.13] or [53,Proposition 5.2.31]) shows that and using density of S(R d × R; X) in F 0,( 1 k , 1 ) (p,q),1,(d,1) (R d × R, (w γ , v); X), we find that the restriction operator restricts to a bounded linear operator By [57, Section 2.4] (see [53,Proposition 5.2.29]), this implies that R is also bounded as an operator The desired inclusion now follows.
With a similar argument as in the above proof one can show the following embedding for an arbitrary open set O ⊆ R d : where k ∈ N 0 , γ > −1 and p ∈ [1, ∞).
In the proof of Theorem 7.1 we will furthermore use the following Sobolev embedding, which is a partial extension of Corollary 3.4 to the case k = 0, obtained by dualizing Corollary 3.4.
To prove this embedding we need a simple lemma.
Taking the infimum over all such g and using (7.2), the estimate in (7.5)

Identification of the temporal trace space
For p ∈ (1, ∞), q ∈ [1, ∞], γ ∈ (−1, 2p − 1) and s ∈ (0, 2) we use the following notation: In the case γ ∈ (−1, p − 1) (with general A p -weight) these spaces can be identified with Besov spaces (see [65,Proposition 6.1]). In the case γ ∈ (p − 1, 2p − 1) we only have embedding results (see Lemma 7.9 below). In the next result we identity the temporal trace space. p , then the temporal trace operator Tr t=0 : u → u(0) is a retraction It follows from the trace method (see [61,Section 1.2] or [79, Section 1.8]) that Tr t=0 is a quotient mapping (7.8). The nontrivial fact in the above theorem is to show that it is a retraction. In order to show this we want to apply [66, Theorem 1.1]/ [70, Theorem 3.4.8]. However, these results can only be applied directly in the special case that the boundary condition vanishes in the real interpolation space. In the case γ ∈ (−1, p − 1) this difficulty does not arise because by using a suitable extension operator one can reduce to the case O = R d . To cover the remaining cases we have found a workaround which requires some preparations. The first result is the characterization of the spatial trace of the spaces defined in (7.7). The result will be proved further below. For these spaces we have the following result which will be proved below as well.
for some extension operator E. All statements different from (7.13) directly follow from this.  [59], it follows from the claim that it is also dense in the space on the right hand side. This density implies (7.13).
Proof of Proposition 7.6. The statement simply follows from Lemma 7.13 by a standard localization argument.
Proof of Proposition 7.7. A combination of (7.10) and Lemma 7.13 gives the desired statement for the case O = R d + , from the general case follows by a standard localization argument. If δ < 1+γ p , then W δ p,q,Dir (R d + , w γ ; X) = W δ p,q (R d + , w γ ; X) and we can simply take E Dir as the required coretraction.
As a consequence of the above theorem we obtain the following corresponding result on time intervals J = (0, T ) with T ∈ (0, ∞] in the case of the power weight v = v μ (with μ ∈ (−1, q − 1)), where we need to take initial values into account.
But then v n satisfies v n + (λ − )v n = 0, Tr ∂O v n = g n , so that v n = S R g n = u n by uniqueness of solutions. Therefore, u n ∈ 0 M q,p μ,γ (R). We may thus conclude that u ∈ 0 M q,p μ,γ (R) in view of (7.26).
Remark 7.18. Theorems 7.15 and 7.16 also remain valid in the X-valued setting as long as X is a UMD space and λ ≥ λ 0 , where λ 0 depends on the geometry of X.