Maximal Regularity for Non-Autonomous Evolutionary Equations

We discuss the issue of maximal regularity for evolutionary equations with non-autonomous coefficients. Here evolutionary equations are abstract partial-differential algebraic equations considered in Hilbert spaces. The catch is to consider time-dependent partial differential equations in an exponentially weighted Hilbert space. In passing, one establishes the time derivative as a continuously invertible, normal operator admitting a functional calculus with the Fourier--Laplace transformation providing the spectral representation. Here, the main result is then a regularity result for well-posed evolutionary equations solely based on an assumed parabolic-type structure of the equation and estimates of the commutator of the coefficients with the square root of the time derivative. We thus simultaneously generalise available results in the literature for non-smooth domains. Examples for equations in divergence form, integro-differential equations, perturbations with non-autonomous and rough coefficients as well as non-autonomous equations of eddy current type are considered.


Introduction
If one considers partial differential equations depending on time as an equation in spacetime the following problem of maximal regularity arises naturally. For the sake of the argument, let H be a Hilbert space modelling space-time and let D and A be two closed, densely defined (unbounded) operators, where the former contains the temporal and the latter the spatial derivative(s). Abstractly spoken, the PDE in question then may look like as follows: Du + Au = f for some right-hand side f ∈ H. In particular, when hyperbolic type problems are concerned (think of the transport equation or the wave equation), one cannot expect that for any f ∈ H (usually an L 2 -type space) the solution u to belong to both dom(D) and dom(A). In general one can only hope for u ∈ dom(D + A), thus deeming the above equation to be true only in some generalised sense. A first example, where it is possible to show that u belongs to the individual domains given any f ∈ H is when D = ∂ t and A = −∆ D (Laplacian with Dirichlet boundary conditions on some open Ω ⊆ R n ) and H = L 2 (0, T ; L 2 (Ω)) and u is assumed to satisfy homogeneous initial conditions. Then, the solution u indeed belongs to H 1 (0, T ; L 2 (Ω)) ∩ L 2 (0, T ; dom(∆ D )), see e.g. [11] or below. Traditionally, the method of choice to derive such a regularity result first establishes well-posedness of the equation at hand via bilinear forms and afterwards analysing the problem in terms of the associated generator. Quite naturally and generalising the above situation considerably, Lions raised the following problem (see [11, p. 68 The question now is under which conditions on a, do we actually have u ∈ H 1 (0, T ; H)?
The latter problem indeed fits into the above abstract perspective for D = ∂ t with domain H 1 (0, T ; H) with Dirichlet boundary conditions at 0 and A = A : dom(A) ⊆ L 2 (0, T ; H) → L 2 (0, T ; H), u → (t → A(t)u(t)) with maximal domain. Problem 1.1 has a long history and has rather recently gained some renewed attention. For the latest developments, we refer to the survey article in [2], to [1] and its introduction. We recall here that Hölder continuity for a (and particularly the Hölder exponent 1/2) with respect to time in a suitable sense plays a crucial role, see e.g. [14,10] for a positive and a negative result, respectively. The available results in the literature up to this point consider explicit Cauchy problems similar to the one in Problem 1.1. Thus, in any case, the complexity of the problem is contained in the form a (or in the operator A).
In this article we set a different focus and try to keep the operator containing the spatial derivatives (i.e. A) as simple as possible and move the complexity over to the time derivative. The rationale behind this is the notion of so-called evolutionary equations, invented in [15] and rather self-contained discussed in [17,23]. More precisely, we consider equations of the form where f belongs to an exponentially weighted L 2 -space, A is an unbounded skew-selfadjoint operator solely acting with respect to the spatial variables and M and N are suitable bounded linear operators in space-time. The solution theory developed in [28,27] asserts that under suitable positive definiteness conditions imposed on ∂ t M + N , one indeed has that (∂ t M + N + A) is continuously invertible. In the framework of evolutionary equations, the maximal regularity problem then reads as follows.
Problem 1.2. Given ∂ t M + N satisfies the appropriate positive definiteness conditions and A skew-selfadjoint in order that (∂ t M + N + A) is continuously invertible, what are the additional conditions on M and N (and the right-hand side F ) such that We emphasise that even in the time-independent case Problem 1.1 and Problem 1.2 are rather different types of questions. In fact, since A is skew-selfadjoint in Problem 1.2, the choices M = 1 and N = 0 for F / ∈ H 1 with respect to time do not lead to ∂ t + A −1 F = (∂ t + A) −1 F , if A is unbounded. As it will be obvious in the next example for a solution of Problem 1.2 one is particularly interested in cases where F belongs to spaces not as smooth as H 1 .
In the autonomous case, Problem 1.2 has been addressed in [20]. The conditions derived describe a parabolic type evolutionary equation in an abstract manner. Indeed, one assume that there exists a densely defined closed linear operator C (acting in the spatial variables only) such that Moreover, one has that M = M 00 0 0 0 and N = N 00 N 01 N 10 N 11 with N 11 satisfying an additional positive definiteness condition. The standard case of the heat equation ∂ t u−∆ D u = f mentioned above is then recovered by putting q = − grad 0 u (gradient subject to homogeneous Dirichlet boundary conditions) and considering ∂ t 1 0 0 0 + 0 0 0 1 + 0 div grad 0 0 Then, indeed, by the main result of [20], one has ∂ t 1 0 0 0 leading to the maximal regularity result mentioned at the beginning for F = (f, 0) with f ∈ L 2 (0, T ; L 2 (Ω)) only.
Even though the class of equations treated in [20] particularly contains integro-differential equations rendering rather different equations to enjoy maximal regularity, the case of the heat equation with non-symmetric but time-independent coefficients could not be treated with the methods developed there. In this article we shall enlarge the class of coefficients M and N considerably leading to the equality highlighted in Problem 1.2. In particular, this class will involve nonsymmetric conductivities in the case of the heat equation. What is more, we shall show that the conditions might be weaker than the conditions derived in both [3] or [8] if applied to divergence form problems. Since we do not consider bilinear forms as our central object of study, we do not invoke the Kato square root property explicitly, which proved instrumental in the main result in [1]. In particular, our methods also apply irrespective of the regularity of the considered underlying domains of the exemplarily considered divergence form problems. Another key difference to the results for nonautonomous maximal regularity available in the literature is the possibility of the variable operator coefficient M, which permits us to consider integro-differential equations with the same approach as classical Cauchy problems in divergence form. Moreover, the operator coefficient N permits the introduction of rough (in time) lower order terms. Before we present a plan of our paper, we shortly describe the two main results and instrumental techniques used in the present article. Theorem 4.1, our first main result on maximal regularity of evolutionary equations, in rough terms can be described as follows: Well-posedness in L 2 and H 1/2 together with a parabolic structure of M, N and A imply maximal regularity in the sense of Problem 1.2 for F = (f, g) ∈ L 2 × H 1/2 , which in the standard heat equation case is satisfied as g = 0 anyway. For a proof of Theorem 4.1, the framework of evolutionary equations is particularly helpful since ∂ t is continuously invertible and normal yielding a handy decription of H 1/2 by the functional calculus for ∂ t . The functional calculus is provided with the help of the Fourier-Laplace transformation. Note that the application of this functional calculus naturally leads to the fractional Riemann-Liouville derivative, see also [18,9]. In applications, the conditions on the parabolic structure and the well-posedness in L 2 are rather easy to show. The assumed positive definitenss in H 1/2 leading to the respective well-posedness result might be rather difficult to obtain, though. We emphasise, however, that in addition to the various positive definiteness estimates, we only need to assume that the involved coefficient operators M and N are bounded linear operators in H 1/2 thus leaving this space invariant. In particular, no bounded commutator assumptions need to be imposed suggesting room for improvement along the lines of the low regularity assumed for the coefficients in [1]. We shall not follow up on this but rather assume stronger commutator assumptions on M and N with ∂ −1 t and ∂ 1/2 t , respectively, confirming the particular role of commutator estimates for maximal regularity already observed [3,8]. Our second main theorem on maximal regularity of evolutionary equations (Theorem 5.1) imposes the same parabolic structure assumption and well-posedness-in-L 2 -requirement as Theorem 4.1. The conditions on the commutators then lead to the asked for well-posedness in H 1/2 of Theorem 4.1 via a perturbation argument. For divergence form problems, the assumptions in Theorem 5.1 are implied by the frac-tional Sobolev (or BMO)-regularity properties imposed in either [3] or [8]. This provides a way of classifying the a priori not comparable conditions in [3] and [8]. Furthermore, we recover an analogous regularity phenomenon first observed in [8] and confirmed in [3] of the solution belonging to H 1/2 -time regularity taking values in the form domain. In the next section, we recall the framework of evolutionary equations and highlight the main ingredients of the non-autonomous solution theory in L 2 as well as some facts of the (time) derivative established in vector-valued exponentially weighted L 2 -spaces. This particularly includes the spectral representation and the accompanying functional calculus. In Section 3, we provide a necessary new technical result, Theorem 3.4, which contains a solution theory for evolutionary equations in H 1/2 . Our first main result is presented and proved in Section 4. The corresponding perturbation result with the mentioned commutator assumptions is presented in Section 5. Also, with a focus on operator-valued multiplication operators, we analyse the commutator condition imposed in Theorem 5.1 a bit more closely. We provide a proof of the results in [3,8] for divergence form problems with our methods in Subsection 6.1. An example for integro-differential equations being a non-autonomous variant of some equations considered in [24] is presented in Subsection 6.2. The last application of our abstract findings is concerned with the (non-autonomous) eddy current approximation for Maxwell's equations in Subsection 6.3. We provide a small conclusion in Section 7.

The Framework
We recall the framework of evolutionary equations. For more details and the proofs we refer to [16,23,28]. We start with the underlying Hilbert space setting and the definition of the time derivative operator.
Definition. For ρ ≥ 0 we define the space where we as usual identify functions which are equal almost everywhere. This space is clearly a Hilbert space with respect to the inner product f, g ρ,0 := R f (t), g(t) H e −2ρt dt (f, g ∈ L 2,ρ (R; H)).
Moreover, we define the operator ∂ t,ρ : dom(∂ t,ρ ) ⊆ L 2,ρ (R; H) → L 2,ρ (R; H) as the closure of the operator where C ∞ c (R; H) denotes the space of arbitrarily differentiable functions having compact support attaining values in H. Finally, we define the Fourier-Laplace transformation L ρ : L 2,ρ (R; H) → L 2 (R; H) as the continuous extension of the mapping We collect some properties of the so introduced operators.
(c) As operators in L 2,ρ (R; H) we have With the help of the unitary equivalence of the operators ∂ t,ρ and i m +ρ we can also define derivatives of fractional order (see [18,23,9]).
Then ∂ α t,ρ is densely defined and closed on L 2,ρ (R; H) and if α ≤ 0, it is bounded with ∂ α t,ρ ≤ 1 ρ α . Moreover, for α > 0 we have Re ∂ α t,ρ ≥ ρ α and the operator ∂ −α t,ρ is given by With the help of these operators, we can define the fractional Sobolev spaces with respect to the exponentially weighted Lebesgue-measure.
The following proposition is an immediate consequence of the definitions above. isometrically. For the theory of interpolation spaces we refer to [4,12]. To see (2.1), we consider the unitarily transformed space H θ (i m +ρ) first and show H θ (i m +ρ) = (L 2 (R; H), H 1 (i m +ρ)) θ isometrically. Indeed, for u ∈ H θ (i m +ρ) we define the function Here S := {z ∈ C ; Re z ∈ [0, 1]}. Note that f u is well-defined and bounded, since Moreover, f u is holomorphic in the interior of S, f u (iξ) ∈ L 2 (R; H) with and f u (iξ + 1) ∈ H 1 (i m +ρ) with for each ξ ∈ R. Since f u (θ) = u this implies u ∈ (L 2 (R; H), H 1 ρ (i m +ρ)) θ with For showing the converse inclusion and norm inequality, let u ∈ (L 2 (R; H), H 1 ρ (i m +ρ)) θ and g : S → L 2 (R; H) + H 1 (i m +ρ) continuous, holomorphic in the interior of S and bounded with g(iξ) ∈ L 2 (R; H) and g(iξ + 1) ∈ H 1 (i m +ρ) such that g(θ) = u. To prove that u ∈ H θ (i m +ρ) it suffices to show that defines a bounded functional on H θ (i m +ρ), where H θ c (i m +ρ) denotes the elements in H θ (i m +ρ) having compact support. For this, let v ∈ H θ c (i m +ρ) and set f v : S → L 2 (R; H) as above and consider the function Note that this integral is well-defined since f v (z) ∈ L 2 (R; H) is compactly supported for each z ∈ S. The so defined F is continuous, holomorphic in the interior of S and bounded and thus, by the maximum principle. For ξ ∈ R we estimate Taking now the infimum over all g with the desired properties, we get R u(t), v(t) H |it + ρ| 2θ dt ≤ u (L 2 (R;H),H 1 (i m +ρ)) θ v H θ (i m +ρ) , which yields u ∈ H θ (i m +ρ) with u H θ (i m +ρ) ≤ u (L 2 (R;H),H 1 (i m +ρ)) θ . Finally, using that L ρ : H θ ρ (R; H) → H θ (i m +ρ) is unitary for each θ ∈ [0, 1], we obtain the assertion.
Indeed, this follows from the fact that F : L 2 (R; H) → L 2 (R; H) and F : L 1 (R; H) → L ∞ (R; H) are unitary and continuous, respectively. Note that this applies verbatim to F * .
The next statement contains an approximation result, which has been employed in [27,20] for the particular case α = 0. To have a corresponding result for the case when α > 0 (and particularly when α = 1/2), will turn out to be useful in the next section, where we provide a well-posedness result for evolutionary equations in H Lemma 2.7. Let ρ > 0 and α ≥ 0. We consider the time derivative operator on H α ρ (R; H); that is, Then for each ε > 0 the operator 1 + ε∂ t,ρ is continuously invertible on H α ρ (R; H) and Thus, (1 + ε∂ t,ρ ) is injective, posseses a closed range and its inverse (defined on the range) is continuous with operator norm bounded by 1. Thus, to prove the continuous invertibility, we have to show that ran(1 The latter is equivalent to where both operators are considered as operators on H α ρ (R; H). Thus, and hence, 1 + ε∂ * t,ρ is injective, which shows the density of ran(1 + ε∂ t,ρ ) in H α+1 ρ (R; H). To prove the strong convergence, it suffices to show the convergence for elements in H α+1 We conclude this section, by citing the main result of [27].
. Then the operator ∂ t,ρ M + N + A is closable, and its closure is continuously invertible. Here, A is identified with its canonical extension to a skew-selfadjoint operator on L 2,ρ (R; H) with domain L 2,ρ (R; dom(A)).  [27].
(b) Theorem 2.8 provides a unified solution theory for a broad class of non-autonomous problems. Due to the flexibility of the choice of the operators M and N , which act in space-time, the problem class comprises many different types of differential equations, like delay equations, fractional differential equations, integro-differential equations and coupled problems thereof (see e.g. [19,23] for some survey in the autonomous case and [28,25,21] for some non-autonomous and/or nonlinear examples).
3 The solution theory in H We discuss operators on H In the case that [S, T ] is densely defined in H and extends to a bounded linear operator on H, we omit the closure bar and just write [S, T ] ∈ L(H). Consequently, we also use [S, T ] to denote the (then continuous operator) [S, T ].

Either of the alternative conditions is particularly satisfied if
Moreover, in this case we have If, on the other hand, and, using Proposition 2.2, we get → 0 strongly in L(L 2,ρ (R; H)). For this, we compute the latter tends to 0 since ∂ Hence, we deduce and thus, ∂ t,ρ M, (1 + ε∂ t,ρ ) −1 ∈ L(H 1/2 ρ (R; H)) by (a). Moreover, by Lemma 2.7 strongly in L(L 2,ρ (R; H)), which yields the asserted convergence again by part (a).
In particular H Proof. The proof follows by induction on k. For k = 0 there is nothing to show. Assume now that the assertion holds for k − 1. Then we compute, using Lemma 3.1 (c) and Lemma 2.7 Theorem 3.4. The operator ∂ t,ρ M + N + A considered as an operator on H 1/2 ρ (R; H) is closable and its closure is continuously invertible.
Proof. Recall that all operators are now considered as operators acting on H ). By Lemma 3.3 this positive definiteness extends to all elements in dom(∂ t,ρ M+N +A) and thus, ∂ t,ρ M+ N + A is one-to-one and has a continuous inverse defined on the range of [28,Proposition 2.3.14] or [5,Theorem 4.2.5]). Moreover, it is a standard argument to show that ∂ t,ρ M + N + A is continuously invertible on its range, which is closed. Hence, for showing that ∂ t,ρ M + N + A is onto, it suffices to compute the adjoint and confirm that this adjoint is one-to-one, which in turn would imply the density of the range of ∂ t,ρ M+N +A. For doing so, let Thus, invoking Lemma 3.1 (c), we obtain that Then ψ ∈ dom(∂ t,ρ M) by Proposition 3.2 and we compute Hence, Proof.
Note that such a u exists by Theorem 3.4. Hence, we find a sequence ( for n ∈ N. By Lemma 2.7, Av n,ε → Au n as well as N v n,ε → N u n as ε → 0 and thus, it suffices to show ∂ t,ρ Mv n,ε → ∂ t,ρ Mu n as ε → 0.
This, however, follows from Lemma 3.3 and thus, the assertion follows.
Corollary 3.6. Let H = H 0 ⊕ H 1 for Hilbert spaces H 0 and H 1 . Then for k ≥ 1, the set Proof. By Corollary 3.5, the set is dense in H

Maximal regularity for evolutionary equations
In the following we provide our main result: a criterion for maximal regularity for evolutionary equations. In a nutshell this criterion reads: Well-posedness in both L 2,ρ (R; H) and H Assume, in addition, that M ′ , N ∈ L(H 1/2 ρ (R; H)). We shall assume the positive definiteness conditions and for some c > 0 and all φ ∈ H that is, for f ∈ L 2,ρ (R; H 0 ) and g ∈ H 1/2 ρ (R; H 1 ) and (u, v) ∈ L 2,ρ (R; H) satisfying we have u ∈ H 1 ρ (R; H 0 ) ∩ L 2,ρ (R; dom(C)) and v ∈ L 2,ρ (R; dom(C * )). Moreover, we have u ∈ H 1/2 ρ (R; dom(C)). Remark 4.2. As we shall see in the examples section, the above nutshell description of the Theorem 4.1 is visible as follows: • Well-posedness in L 2,ρ (R; H) is guaranteed by assumption (4.6); see Theorem 2.8. • The parabolic-like structure is visible in the block matrix structure (4.1), (4.2) and the positive definiteness condition (4.5).
Remark 4.3. (a) As in [3,8], we recover the same additional regularity phenomenon u ∈ H 1/2 ρ (R; dom(C)), which is not expected for maximal regularity of evolutionary equations. In fact, as the proof of Theorem 4.1 will show an estimate for Cu ρ, 1 2 is key for obtaining the main result. (b) Note that u ∈ H 1/2 ρ (R; dom(C)) also has a consequence on the time-regularity of v.
Remark 4.5. (a) Note that the regularity statement in the latter result is also accompanied with the corresponding continuity statement; that is, there exists a constant κ ≥ 0 such that for all f ∈ L 2,ρ (R; H 0 ), g ∈ H 1/2 Moreover, note that, as a consequence, the closure bar in the formulation of the evolutionary equation can be omitted so that, indeed, addressing Problem 1.2.

Maximal regularity and bounded commutators
In this section, we will apply our main result Theorem 4.1 to prove maximal regularity for a broad class of evolutionary equations. Note that the second main theorem of the present manuscript is concerned with the case, where well-posedness in H 1/2 ρ is obtained by a bounded commutator assumption involving N and by restricting M to the case commuting with ∂ −1 t,ρ . It turns out that this situation is closer to the applications as we shall outline below. As above, we assume that H 0 and H 1 are two complex Hilbert spaces and we set H := for some densely defined closed linear operator and H). Finally, we assume that there is 0 ≤ c < c and d > 0 such that where S ρ := ∂ t,ρ M + N + A −1 ∈ L(L 2,ρ (R; H)).
t,ρ . Indeed, to start off with, the fact that M commutes with ∂ −1 t,ρ yields Hence, M (∂ t,ρ − 2ρ) ⊆ (∂ t,ρ − 2ρ) M and, thus, using Proposition 2.1 (c), we infer Next, by the approximation theorem of Weierstraß, we have that the polynomials in z and z * as continuous functions on C(V ) are dense in C(V ) endowed with the sup-norm, where V := B C (1/(2ρ), 1/(2ρ)). Hence, we find a sequence of polynomials (z → p n (z, z * )) n in z and z * such that p n → (z → √ z) uniformly on V as n → ∞. In consequence, using the Fourier-Laplace transformation, we obtain that p n ∂ −1 t,ρ , ∂ * t,ρ ρ (R; H)). Thus, we infer using the commutator properties of M shown above Proof of Theorem 5.1. Let f ∈ L 2,ρ (R; H 0 ) and g ∈ H √ ρ and consider the operator It is clear that and hence, to show the claim, it suffices to prove that the operators M and N satisfy the assumptions of Theorem 4.1. We first note that M ′ = 0 and that (4.5) holds by assumption and (4.6) follows from the inequality assumed for ∂ t,ρ M + N and the fact that Re ∂ which shows (4.7).
If the coefficient operators, M and N , act in a 'physically meaningful manner'; that is, if they are causal (see definition below), then the latter result (as well as the other main result Theorem 4.1) also imply a maximal regularity result locally in time. For this we need a closer look into the well-posedness result Theorem 2.8, which in turn prerequisites the following notion. We define S c (R; H) := lin{f : R → H; f simple function with compact support}.
Definition 5.3. Let K 0 , K 1 be Hilbert spaces, ρ 0 ∈ R, and linear. Then we call C evolutionary (at ρ 0 ), if, for all ρ ≥ ρ 0 , C admits a continuous extension C ρ ∈ L(L 2,ρ (R; K 0 ), L 2,ρ (R; K 1 )) satisfying We gather two results important for evolutionary mappings of the type discussed in the latter definition. For the intricacies of the interplay of causality and closure of operators, we refer to [26]. . Let K 0 , K 1 be Hilbert spaces, ρ 0 ∈ R, and C : S c (R; K 0 ) → ρ≥ρ 0 L 2,ρ (R; K 1 ) linear and C evolutionary at ρ 0 . Then C ρ is causal for all ρ ≥ ρ 0 ; that is, for all t ∈ R and f ∈ L 2,ρ (R; K 0 ) we have is causal.
Having presented the remaining technical ingredients for the localisation on bounded time-intervals, we can present the local maximal regularity statement next.
Corollary 5.6. In addition to the assumptions in Theorem 5.1, assume that M and N are evolutionary. Let T ∈]0, ∞[. Then there exists κ ≥ 0 such that for all f ∈ L 2,ρ (R; H 0 ) and g ∈ H Proof. Firstly, observe that H T ); H) continuously, by complex interpolation, see also Remark 2.4. Next, let φ ∈ C ∞ c (R) with 0 ≤ φ ≤ 1 have the following properties Remark 5.7. Note that a prototype of evolutionary operators are operators defined as multiplication by a function, see also [28,Example 2.1.1]. This prototype will be discussed next.

Commutators for multiplication operators
In this subsection we inspect the conditions on the operator N assumed in Theorem 5.1 for the concrete case of N being a multiplication operator. More precisely, we assume the following: Let N : R → L(H) be a strongly measurable bounded mapping. Then N induces an evolutionary operator with N ρ L(L 2,ρ (R;H)) = N ∞ for all ρ ≥ 0. Note that all continuous extensions N ρ , ρ ≥ 0, act as multiplication by N . We start to give a representation for the term ∂ 1/2 t,ρ φ for regular functions φ.
Lemma 5.8. Let ρ > 0 and φ ∈ H 1 ρ (R; H). Then t,ρ φ ′ (see also Proposition 2.2) and thus, by Using this expression, we can prove our first result on commutators with the fractional derivative.
The next proposition is devoted to the limit case ρ 0 = 0, which is the case usually treated in the literature. Proof. Similar to the proof of Proposition 5.9, at first we show For this, let φ ∈ H 1/2 ρ (R; H). Then (i m +ρ) 1/2 L ρ φ = (i m +ρ) 1/2 Fe −ρ· φ ∈ L 2 (R; H). The latter implies (i m) 1/2 Fe −ρ· φ ∈ L 2 (R; H) and hence, e −ρ· φ ∈ H 1/2 (R; H). If, on the other hand, e −ρ· φ ∈ H 1/2 (R; H), then φ ∈ L 2,ρ (R; H) and (i m) 1/2 L ρ φ = (i m) 1/2 Fe −ρ· φ ∈ L 2 (R; H) and hence, and thus, φ ∈ H  H). Following the lines of the proof of Proposition 5.9, we need to find an estimate for The main problem in proving such an estimate is that we do not have an explicit integral representation for ∂ 1/2 t,0 thus far. However, we have t,ρ 0 e ρ 0 · ψ with convergence in L 2 (R; H) for each ψ ∈ H 1/2 (R; H), where we have used dominated convergence in the second line. Thus, for φ ∈ C ∞ c (R; H) we have that where we have used (5.1). Following the lines of the proof of Proposition 5.9 the assertion follows.
Remark 5.11. Note that Theorem 5.1 in combination with Proposition 5.9 or Proposition 5.10 yields maximal regularity of the corresponding evolutionary equation, if N has a bounded commutator for some ρ ≥ 0. In particular, this covers the case treated in [3] (see also Subsection 6.1 below).
Our next goal is to prove the following proposition.
In order to prove Proposition 5.12, we want to apply Lemma 5.8 to derive an integral expression for the commutator. Since Lemma 5.8 just holds for functions in H 1 ρ (R; H), we need to regularise N .

Divergence form equations
In order to treat a first standard example, we consider heat type equations in this section and analyse the relationship to available results in the literature. For this, we need to introduce the following operators.
Let Ω ⊆ R n be open. We define where H 1 (Ω) is the standard Sobolev space of weakly differentiable L 2 (Ω) functions, Similarly, H(div, Ω) is the space of L 2 (Ω)-vector fields with distributional divergence in L 2 (Ω) and H 0 (div, Ω) is the closure of C ∞ c (Ω) n in H(div, Ω).
It is not difficult to see that div * 0 = − grad and grad * 0 = − div, see [23,Chapter 6]. Next, we rephrase a sufficient criterion from [3], which guarantees that N 0 , ∂ 1/2 t,0 is bounded. The result itself is a combination of the techniques used in [3], the BMOcharacterisation by Strichartz and the commutator estimate by Murray [13]. Proof. A direct computation shows that N ε defined in Lemma 5.14 satisfies the same condition imposed on N in the present theorem (with the same C). Thus, using Lemma 5.14 it suffices to treat the case of Lipschitz continuous N . In this case, the arguments in [3,Corollary 7] show that both [N 0 , |∂ t,0 | 1/2 ] and (using [13]) [N 0 , sgn(−i∂ t,0 ) |∂ t,0 | 1/2 ] are bounded. Since and due to linearity of the commutator in the second argument, we infer the assertion.
Remark 6.7. (a) Even though the conditions (5.2) and (6.1) do not compare (see [3,Introduction]), we have established that both of the results in [3] and [8] applied to standard divergence form equations can be obtained by the same overriding principle of suitably bounded commutators with ∂ 1/2 t,ρ . Note that (6.1) implies boundedness as an operator in L 2 , whereas (5.2) yields infinitesimal boundedness relative to ∂ 1/2 t,ρ only. (b) The condition on the regularity of the coefficient N leading to maximal regularity of the considered divergence form equation obtained in [1] seems to be weaker than the one of (infinitesimal) boundedness of the commutator with ∂ 1/2 t,ρ . However, note that in order to apply the maximal regularity theorem in [1], one needs to assume Kato's square root property (potentially) resulting in undue regularity requirements of the boundary of Ω, which we do not want to impose here.
The corresponding theorem for maximal regularity of parabolic-type non-autonomous integro-differential equations, now reads as follows.
Next, as the convolution operator (1 + T * ) commutes with ∂ −1 t,0 , we infer with the help of Theorem 5.1 the desired regularity statement.
Remark 6.12. (a) Note that the coefficients of the lower order terms N 01 and N 00 are not required to satisfy any regularity in time, which is in line with the concluding example in [1]. Moreover, in the theorem presented here the coefficient N 11,ρ may well depend suitably regular on time, i.e., N 11,ρ may be induced by a multiplication operator, which satisfies either (6.1) or (5.2). (b) The results above directly apply to systems of divergence form equations, see [8,Parabolic systems] for examples concerning maximal regularity and [7,Proposition 3.8] for the corresponding formulation as evolutionary equation.

Maxwell's equations
The concluding example is concerned with Maxwell's equations. For this, we introduce the necessary operator from vector analysis: where H(curl, Ω) is the space of L 2 (Ω)-vector fields with distributional curl in L 2 (Ω) 3 and H 0 (curl, Ω) is the closure of C ∞ c (Ω) 3 in H(curl, Ω). It is not difficult to see that curl * 0 = curl . The result on maximal regularity for Maxwell's equations is concerned with the eddy current approximation, which is a parabolic variant of the originial Maxwell's equations. The catch is that in electrically conducting materials like metals the dielectricity ε is negligible compared to the conductivity σ, which we assume to depend on time. This setting has applications to moving domains, see e.g. [6]. The result reads as follows.
Remark 6.15. (a) Again, the commutator condition imposed on σ is satisfied, if σ is a multiplication operator induced by a function satisfying either (6.1) or (5.2). (b) In applications, non-zero terms K can occur, if one considers inhomogeneous boundary values. Note that a result corresponding to Theorem 6.14 is valid also for mixed boundary conditions or with homogeneous boundary conditions for H. (c) There is no condition assumed on the regularity of the boundary of Ω.

Conclusion
We presented a maximal regularity theorem for evolutionary equations. The core assumptions abstracly describe a parabolic type evolutionary equation and lead to well-posedness on L 2,ρ and H 1/2 ρ . For applications, the operator theoretic insight of the need of commutator estimates for the commutator with ∂ 1/2 t found in [8] and [3] showed to be decisive also for evolutionary equations. Moreover, we showed that both conditions on the coefficients imposed in [8] and [3], which are not comparable, imply the well-posedness in H 1/2 ρ and hence, yield the maximal regularity of the problem under consideration within the presented framework. Naturally, the regularity phenomenon for the unknown to belong to H 1/2 with values in the form domain, observed in [8] and [3], resurfaced also in the framework of evolutionary equations. The conditions derived here are deliberately focussed on the coefficients rather than the whole space-time operator in order that it is possible to generate results independent of the regularity of the boundary of the underlying domain, which is needed in [1] in order to warrant some form of the square root property. Due to the view of the time derivative as a normal continuously invertible operator it is possible to use a straightforward functional calculus and to compute fractional powers of the time derivative and to work with them without the need of explicitly invoking the Hilbert transform or other technicalities. It remains to be seen, whether the commutator assumptions or the basic result Theorem 4.1 implying maximal regularity lead to slightly stronger statements also in the situation of divergence form equations.