Maximal regularity for non-autonomous equations with measurable dependence on time

In this paper we study maximal $L^p$-regularity for evolution equations with time-dependent operators $A$. We merely assume a measurable dependence on time. In the first part of the paper we present a new sufficient condition for the $L^p$-boundedness of a class of vector-valued singular integrals which does not rely on H\"ormander conditions in the time variable. This is then used to develop an abstract operator-theoretic approach to maximal regularity. The results are applied to the case of $m$-th order elliptic operators $A$ with time and space-dependent coefficients. Here the highest order coefficients are assumed to be measurable in time and continuous in the space variables. This results in an $L^p(L^q)$-theory for such equations for $p,q\in (1, \infty)$. In the final section we extend a well-posedness result for quasilinear equations to the time-dependent setting. Here we give an example of a nonlinear parabolic PDE to which the result can be applied.


Introduction
In this paper we study maximal L p -regularity of the Cauchy problem: Here (A(t)) t∈(0,T ) is a family of closed operators on a Banach space X 0 . We assume the operators have a constant domain D(A(t)) = X 1 for t ∈ [0, T ]. In recent years there has been much interest in maximal regularity techniques and their application to nonlinear PDEs. Maximal regularity can often be used to obtain a priori estimates which give global existence results. For example, using maximal regularity it is possible to solve quasi-linear and fully nonlinear PDEs by elegant linearization techniques combined with the contraction mapping principle [4,6,14,16,64,77]. This has found numerous applications in problems from mathematical physics (e.g. fluid dynamics, reaction-diffusion equations, material science, etc. see e.g. [1,14,21,32,41,60,67,66,75,77,80,81,92]). For maximal Hölder-regularity we refer the reader to [2,64] and references therein. In this paper we focus on maximal L p -regularity as this usually requires the least regularity of the data in PDEs.
An important step in the theory of maximal L p -regularity was the discovery of an operator-theoretic characterization in terms of R-boundedness properties of the differential operator A due to Weis (see [88,89]). This characterization was proved for the class of Banach spaces with the UMD property. About the same time Kalton and Lancien discovered that not every sectorial operator A on X = L q of angle < π/2 has maximal L p -regularity (see [50,51] and [28]).
In the case t → A(t) is (piecewise) continuous, one can study maximal L pregularity using perturbation arguments (see [5,7,78]). In particular, in [78], it was shown that maximal L p -regularity of (1.1) is equivalent to the maximal L pregularity for each operator A(t 0 ) for t 0 ∈ [0, T ] fixed. This, combined with the characterization of [89] yields a very precise condition for maximal L p -regularity. The case where the domains D(A(t)) vary in time will not be considered in this paper. In that setting maximal L p -regularity results can be obtained under certain Hölder regularity assumptions in the time variable (see [76] and references therein).
In many real-life models, the differential operator A has time-dependent coefficients, and the dependence on time can be rather rough. If this is the case, the operator-theoretic characterization of maximal regularity just mentioned does not apply. In the present paper we develop a functional analytic approach to maximal L p -regularity in the case t → A(t) is only measurable (see Theorems 1.1 and 4.9 below). Our approach is based on the L p -boundedness of a new class of vector-valued singular integrals of non-convolution type (see Theorem 3.4). It is important to note that we do not assume any Hörmander conditions in the time variable. For discussion and references on (vector-valued) singular integrals we refer the reader to Section 3.
When the time-dependence is just measurable, an operator-theoretic condition for maximal L p -regularity is known only in the Hilbert space setting for p = 2 (see [62,63] and [84,Section 5.5]). The assumption here is that A arises from a coercive form a(t, ·, ·) : V × V → C and V ֒→ X 0 ֒→ V ′ . Unfortunately, this only yields a theory of maximal L 2 -regularity on V ′ . In many situations one would like to have maximal L p -regularity on X 0 and also for any p ∈ (1, ∞). Results of this type have been obtained in [8,23,38] using regularity conditions on the form in the time variable.
Most results will be presented in the setting of weighted L p -spaces. This will be important for several reasons. Maximal L p -regularity with a power weight t α in time (e.g. see [56,67]) allows one to consider rather rough initial values. It can also be used to prove compactness properties which can be used to obtain global existence of solutions. Another advantage of using weights comes from a harmonic analytic point of view. The theory of Rubio de Francia (see [18] and references therein) enables one to extrapolate from weighted L p -estimates for a single p ∈ (1, ∞), to any p ∈ (1, ∞). In Section 5 A p -weights in space will be used to check R-boundedness of certain integral operators. We refer to Theorem 2.6 and Step 1 of the proof of Theorem 5.4 for details. Weights in time will be used for extrapolation arguments more directly. For instance in step 4 of the proof of Theorem 5.4 and also the proof of Theorem 1.1 at the end of Section 4.4.
In the special case X 0 is a Hilbert space, our main result Theorem 4.9 implies the following result. Theorem 1.1. Let X 0 be a Hilbert space. Assume A : (0, τ ) → L (X 1 , X 0 ) is such that for all x ∈ X 1 , t → A(t)x is measurable and Assume there is an operator A 0 on X 0 with D(A 0 ) = X 1 which generates a contractive analytic semigroup (e −zA0 ) z∈Σ θ which is such that (A(t) − A 0 ) t∈(0,τ ) generates an evolution system (T (t, s)) 0≤s≤t≤τ on X 0 which commutes with (e −rA0 ) r≥0 . e −rA0 T (t, s) = T (t, s)e −rA0 , 0 ≤ s ≤ t ≤ τ, r ≥ 0.
As an application we prove maximal L p -regularity for the following class of parabolic PDEs: Here . For such concrete equations with coefficients which depend on time in a measurable way, maximal L p -regularity results can be derived using PDE techniques. Our results enable us to give an alternative approach to several of these problems. Moreover, we obtain a full L p (0, T ; L q (R d ))theory, whereas previous papers usually only give results for p = q or q ≤ p (see Remark 5.7 for discussion).
In the next result we will use condition (C) on A which will be introduced in Section 5. It basically says that A is uniformly elliptic and the highest order coefficients are continuous in space.
Theorem 1.2. Let T ∈ (0, ∞). Assume condition (C) on the family of operators (A(t)) t∈(0,T ) . Let p, q ∈ (1, ∞). Then the operator A has maximal L p -regularity on (0, T ), i.e. for every f ∈ L p (0, T ; L q (R d )) and 2) holds a.e. and there is a C > 0 independent of u 0 and f such that The conditions on f and u 0 are also necessary in the above result. Here B s q,p (R d ) denotes the usual Besov space (see [87] for details). The proof of Theorem 1.2 is given at the end of Section 5. It will be derived from Theorem 5.4 which is a maximal regularity result with weights in time and space.
Overview In Section 2 we discuss preliminaries on weights, R-boundedness and functional calculus. In Section 3 we prove the L p -boundedness of a new class of singular integrals. The main result on maximal L p -regularity is presented in Section 4. Finally in Section 5 we show how to use our new approach to derive maximal L p -regularity for (1.2).
Notation Throughout this paper we will write L (X, Y ) for the space of all bounded linear operators mapping X into Y . In the estimates below, C can denote a constant which varies from line to line.
Acknowledgement The authors thank Doyoon Kim for pointing out the references [55,57]. We also thank Nick Lindemulder and Jan Rozendaal for careful reading and useful comments.

2.1.
A p -weights. Details on A p -weights can be found in [35,Chapter 9] and [83, Here the supremum is taken over all cubes Q ⊆ R d with axes parallel to the coordinate axes and Q = 1 |Q| Q . The extended real number [w] Ap is called the A p -constant. The Hardy-Littlewood maximal operator is defined as with Q ⊆ R d cubes as before. Recall that w ∈ A p if and only if the Hardy-Littlewood maximal operator M is bounded on L p (R d , w).
The following extrapolation result from [18, Theorem 3.9] will be needed. .
Below estimates of the form (2.1) with increasing function α p0 will appear frequently. In this situation we say there is an A p0 -consistent constant C such that Note that the L p -estimate obtained in Theorem 2.1 is again A p -consistent for all p ∈ (1, ∞).
The following simple observation will be applied frequently. For a bounded Borel set A ⊂ R d and for every f ∈ L p (R d , w; X) one has 1 A f ∈ L 1 (R d ; X) and by Hölder's inequality A linear subspace Y ⊆ X * is said to be norming for X if for all x ∈ X, x = sup{| x, x * | : x * ∈ Y, x * ≤ 1}. The following simple duality lemma will be needed.
Let Y ⊆ X * be a subspace which is norming for X. Then setting

R-boundedness and integral operators.
In this section we recall the definition of R-boundedness (see [15,22,59] for details). A sequence of independent random variables (r n ) n≥1 on a probability space The least possible constant C is called the R-bound of T and is denoted by R(T ).
Every R-bounded family of operators is uniformly bounded. A converse holds for Hilbert spaces X and Y : every uniform bounded family of operators is automatically R-bounded.
The R-boundedness of a certain family of integral operators plays a crucial role in this paper. Let K be the class of kernels k ∈ L 1 (R) for which |k| * f ≤ M f for all simple functions f : R → R + , where M denotes the Hardy-Littlewood maximal operator. The next example gives an important class of kernels which are in K.
Consider the family of integral operators I := {I kT : k ∈ K} ⊆ L (L p (R; X)).
The R-boundedness of such families I of operators will play an important role in Section 3.
Proposition 2.4. If {T (t, s) : s, t ∈ R} is uniformly bounded on X, then I is uniformly bounded on L p (R, v; X) for every p ∈ (1, ∞) and v ∈ A p . Moreover, it is also uniform bounded on L 1 (R; X).
Proof. For any p ∈ (1, ∞), note that for a.e. t ∈ R. Therefore the uniform boundedness of I kT follows from the boundedness of the maximal operator. The case v ≡ 1 and p = 1 follows from Fubini's theorem and the fact that k L 1 (R) ≤ 1 (see [73,Lemma 4.3]).
The R-boundedness of (2.4) has the following simple extrapolation property: Proof. The special structure of I will not be used in this proof. Let I 1 , . . . , I N ∈ I , f 1 , . . . , f N ∈ L p (R, v; X) and let .
Then the assumption combined with Fubini's theorem yields that for all v ∈ A p0 , where C is a constant which is A p0 -consistent. Therefore, by Theorem 2.1 we find that for each p ∈ (1, ∞), there is an A p -consistent constant C ′ (depending only on C) such that Now by (2.3), F p ≤ κ p,p0 F p0 , G p0 ≤ κ p0,p G p , and the result follows from (2.5) and another application of Fubini's theorem.
In [30] the following simple sufficient condition for R-boundedness of such families was obtained in the case X = L q . Assume that for all A q0 -weights w, where C is A q0 -consistent and independent of t, s ∈ R. Then the family of integral operators I = {I kT : k ∈ K} ⊆ L (L p (R, v; L q (O, w))) as defined in (2.4) is Rbounded for all p, q ∈ (1, ∞) and all v ∈ A p and w ∈ A q . Moreover, in this case the R-bounds R(I ) are A p -and A q -consistent.
The proof of this result is based on extrapolation techniques of Rubio de Francia. As for fixed t, s ∈ R, T (t, s) on L q (O) is usually defined by a singular integral of convolution type in R d , one can often apply Calderón-Zygmund theory and multiplier theory to verify (2.6). In this case it is usually not more difficult to prove the boundedness for all A q -weights, than just w = 1. The reason for this is that for large classes of operators, boundedness implies weighted boundedness (see [31,Theorem IV.3.9], [35,Theorem 9.4.6] and [42,Corollary 2.10]). Another situation where weights are used to obtain R-boundedness can be found in [40,29].
Example 2.7. For a bounded measurable function θ : . Then (2.6) holds and hence Theorem 2.6 implies that I ⊆ L (L p (R, v; L q (R d , w))) is R-bounded for all p, q ∈ (1, ∞) and all v ∈ A p and w ∈ A q .
Many differential operators on L q -spaces with q ∈ (1, ∞) are known to have a bounded H ∞ -calculus (see [22,59] and the survey [90]). The case A = −∆ on L p (R d , w) has a bounded H ∞ -calculus of arbitrary small angle σ ∈ (0, π) for every w ∈ A p and p ∈ (1, ∞). This easily follows from the weighted version of Mihlin's multiplier theorem (see [59,Example 10.2] and [31, Theorem IV.3.9]). For instance, it includes all sectorial operators A of angle < π/2 for which e −tA is a positive contraction (see [53]).

A class of singular integrals with operator-valued kernel
Let X be a Banach space. In this section we will study a class of singular integrals of the form There is a natural generalization of the theory of singular integrals of convolution type to the vector-valued setting (see [44]). In the case the singular integral is of nonconvolution type, the situation is much more complicated. An extensive treatment can be found in [45,47,46], where T 1-theorems [19] and T b-theorems [20] have been obtained in an infinite dimensional setting. Checking the conditions of these theorems can be hard. For instance, from [86] it follows that the typical BMO conditions one needs to check, have a different behavior in infinite dimensions. Our motivation comes from the application to maximal L p -regularity of (1.2). At the moment we do not know whether the T 1-theorem and T b-theorem can be applied to study maximal L p -regularity for the time dependent problems we consider. Below we study a special class of singular integrals with operator-valued kernel for which we prove L p -boundedness. The assumptions on K are formulated in such a way that they are suitable for proving maximal L p -regularity of (1.2) later on.

3.1.
Assumptions. The assumptions in the main result of this section are as follows.
(H3) Assume X has finite cotype. Assume A j has a bounded H ∞ (Σ σj )-calculus with σ j ∈ [0, π) for j = 0, 1. (H4) Assume the family of integral operators I : is R-bounded. The class of kernels K is as defined in Section 2.2. Recall from (3.1) that Since T is uniformly bounded, the operator I kT is bounded on L p (R, v; X) by Proposition 2.4.
(1) The class of Banach spaces with finite cotype is rather large. It contains all L p -spaces, Sobolev, Besov and Hardy spaces as long as the integrability exponents are in the range [1, ∞). The spaces c 0 and L ∞ do not have finite cotype. The cotype of X will be applied in order to have estimates for certain continuous square functions (see (3.7)). Details on type and cotype can be found in [24]. (2) In the theory of singular integrals in a vector-valued setting one usually assumes X is a UMD space. Note that every UMD has finite cotype and nontrivial type by the Maurey-Pisier theorem (see [24]). (3) A sufficient condition for the R-boundedness condition in the case X = L q can be deduced from Theorem 2.6.
On the other hand, one can also consider T (t, s)1 {s<t} and T (t, s)1 {t<s} separately. Indeed, the hypothesis (H1)-(H4) holds for these operators as well whenever they hold for T (t, s).
This kernel satisfies K(t, s) ∼ (t−s) −1 for t close to s. If one takes T (t, s) varying in t and s one might view it as a multiplicative perturbation of the above kernel.
The following simple observation shows that I K as given in (3.1) can be defined

Lemma 3.3. Under the assumptions (H1) and (H2), I K is bounded as an operator
The proof is inspired by the recent solution to the stochastic maximal L pregularity problem given in [72].
Before we turn to the proof, we have some preliminary results and remarks.
Example 3.5. Assume (H2) and (H3). If T (t, s) is as in Example 2.7 then (H4) holds. Therefore, I K is bounded by Theorem 3.4. Surprisingly, we do not need any smoothness of the mapping (t, s) → K(t, s) in this result. In particular we do not need any regularity conditions for K(t, s) (such as Hörmander's condition) in (t, s).

Recall the following Poisson representation formula (see [72, Lemma 4.1]).
Lemma 3.6. Let α ∈ (0, π) and α ′ ∈ (α, π] be given, let E be a Banach space and let f : Σ α ′ → E be a bounded analytic function. Then, for all s > 0 we have where k α : R + × R + → R is given by Remark 3.7. In the special case X = L q (S) with q ∈ (1, ∞), we present some identification of spaces which can be used to simplify the proof below. This might be of use to readers who are only interested in L q -spaces. First of all one can use the usual adjoint * instead of the moon adjoint # in the proof below. In this case one can take ). The γ-multiplier theorem which is applied below in (3.6) can be replaced by [88, 4a] in this case. Finally, the estimates in (3.7) can be found in [61] in this special case.

Proof of Theorem 3.4.
Step 1: By density it suffices to prove ). Note that by Lemma 3.3, I K is well defined on this subspace.
Step 2: Together with (H2) this yields the following representation of K(t, s)x for x ∈ D(A 1 ) ∩ R(A 1 ): Therefore, the following representation holds for the singular integral Step 3: In this way Y 2 can be identified with an isometric closed subspace of Y * 1 . Note that by [59,Proposition 15.4], X # is norming for X and hence Lemma 2.2 implies that Y 2 is norming for Y 1 . For fixed g ∈ Y 2 it follows from Fubini's theorem and γ-duality (see [ Here Φ 0,j (u) # := φ 0 (ue ijα A # 0 ). By (H4) the family {I Su : u > 0} is R-bounded by some constant C T . Therefore, by the Kalton-Weis γ-multiplier theorem (see [52,Proposition 4.11] and [70,Theorem 5 Here we used that X does not contain an isomorphic copy of c 0 as it has finite cotype (see (H3)). The remaining two square function norms can be estimated by the square function estimates of Kalton and Weis. Indeed, by (H3) and [37,Theorem 4.11] or [52, Section 7] (here we again use the finite cotype of X) and the γ-Fubini property (see [70,Theorem 13.6]), we obtain Combining all the estimates yields Taking the supremum over all Remark 3.8. One can also apply standard extrapolation techniques to obtain weighted boundedness results for singular integrals from the unweighted case (see [13,42]). However, for this one needs Hörmander conditions on the kernel. As our proof gives a result in the more general setting, we can avoid smoothness assumptions on the kernel.

Maximal L p -regularity
In this section we will apply Theorem 3.4 to obtain maximal L p -regularity for the following evolution equation on a Banach space X 0 .
As explained in the introduction no abstract L p -theory is available for (4.1) outside the case where t → A(t) is continuous.
The following assumption will be made throughout this whole section.
(A) Let X 0 be a Banach space and assume the Banach space X 1 embeds densely and continuously in X 0 . Let p ∈ [1, ∞) and v ∈ A p with the convention x is strongly measurable, and there is a constant C > 0 such that The above implies that each A(t) is a closed operator on X 0 with D(A(t)) = X 1 . Note that whenever A is given on an interval I ⊆ R, we may always extend it constantly or periodically to all of R.
Before we state the main result we will present some preliminary results on evolution equations with time-dependent A.
(3) Assume b = ∞. The function u is said to be a strong solution of (4.2) if for every T > a the restriction to [a, T ] or (a, T ] yield strong solutions in the sense of (1) and (2) respectively. Note the following simple embedding result for general A p -weights. p ((a,b),v;X0) .
Proof. Since L p ((a, b), v; X 0 ) ֒→ L 1 (a, b; X 0 ), and u(t) − u(s) = t s u ′ (r) dr, the continuity of u is immediate. Moreover, Taking L p ((a, b), v)-norms with respect to the s-variable yields the result.
There is a correspondence between the evolution problem (4.2) and evolution families as defined below. Note that (iii) says that u is a strong solution of (4. 2) with f = 0.  Here we use the strong operator topology to define the integral. Assume there exist φ, M and N such that all B ∈ A are all sectorial of angle φ < π/2 and for all λ ∈ Σ φ , λ(λ + B) −1 ≤ M and x X1 ≤ N ( x X0 + Bx X0 ) Assume for every B 1 , B 2 ∈ A and λ, µ ∈ Σ φ , the operators (λ + B 1 ) −1 and (µ + is an evolution family for A. Here the exponential operator is defined by the usual Cauchy integral (see [64,Chapter 2]). Usually, no simple formula for S is available if the operators in A do not commute.

Proposition 4.5. Let S be an evolution family for
where we allow s = a and t = b whenever these are finite numbers. In particular, strong solutions are unique if a > −∞. In the case a = −∞ this remains true if lim s→−∞ S(t, s) = 0.
A partial converse is used and proved in Theorem 4.9.
If a > −∞, then we may take s = a in the above proof and hence we can replace u(s) = u(a) by the initial value x. If a = ∞, the additional assumption on S allows us to let s → −∞ to obtain

4.2.
Assumptions on A. The following condition can be interpreted as an abstract ellipticity condition.
(E) Assume that X 0 has finite cotype and assume that there exists A 0 ∈ L (X 1 , X 0 ) which has a bounded H ∞ -calculus of angle σ < π/2 and there exists a strongly continuous evolution system (T (t, s)) s≤t for (A(t)−A 0 ) t∈R such that e −rA0 commutes with T (t, s) for every t ≥ s and r ∈ R + and assume there exists an ω ∈ R such that Set T (t, s) = 0 for t < s. The following R-boundedness condition will be used.
(1) By (A) and (E) there is a constant C such that and both norms are equivalent to x X1 . (2) For m even, if the A(t) are m-th order elliptic operators with x-independent coefficients one typically takes A 0 = δ(−∆) m with δ > 0 small enough. (3) For p, q ∈ (1, ∞), v ∈ A p and X = L q , the R-boundedness assumption follows from the weighted boundedness of T (t, s) for all w ∈ A q (see Theorem 2.6). (4) Although we allow p = 1 and v = 1 in the above assumptions, checking the assumption (Rbdd) seems more difficult in this limiting case. Proof. The second identity follows from (E). To prove the first identity, we check that S(t, s) given by (4.6) is an evolution family for A. By Corollary 4.6 this would complete the proof. It is simple to check properties (i) and (ii) of Definition 4.2 and it remains to check (iii) and (iv). Let x ∈ X 1 . By the product rule for weak derivatives and (E) we find Similarly, one checks that d ds S(t, s)x = S(t, s)A(s)x. The fact that S(t, s) satisfies the same exponential estimate as T (t, s) follows from the estimate (2.7) applied to A 0 .
By assumptions, for every x ∈ X 1 , e −rA0 S(t, s)x = S(t, s)e −rA0 x. Thus, by differentiation we find −A 0 S(t, s)x = −S(t, s)A 0 x and therefore

4.3.
Main result on maximal L p -regularity. Next we will present our main abstract result on the regularity of the strong solution to the problem Theorem 4.9. Assume (A), (E), and (Rbdd). For any λ > ω and for every f ∈ L p (R, v; X 0 ) there exists a unique strong solution u ∈ W 1,p (R, v; X 0 ) ∩ L p (R, v; X 1 ) of (4.7). Moreover, there is a constant C independent of f and λ such that Remark 4.10. Parts of the theorem can be extended to λ = ω, but we will not consider this in detail. The constant in the estimate (4.8) for u ′ can be improved if one knows A(t)x X0 ≤ C A 0 x X0 or when taking λ ≥ ω + 1 for instance.
Before, we turn to the proof of Theorem 4.9 we introduce some shorthand notation. Let S λ (t, s) = e −λ(t−s) S(t, s) and T λ (t, s) = e −λ(t−s) T (t, s). Since by Lemma 4.8, S is an evolution family for A, also S λ is the evolution family for A(t) + λ. Similarly, T λ (t, s) is an evolution family for A(t) − A 0 + λ. By (4.3) if the support of f ∈ L 1 (R; X 0 ) is finite, a strong solution of (4.7) satisfies Proof. Replacing A(t) and T (t, s) by A(t) + ω and e −(t−s)ω T (t, s) one sees that without loss of generality we may assume ω = 0 in (E) and (Rbdd). We first prove that u given by (4.9), is a strong solution and (4.8) holds. First let f ∈ L p (R, v; X 1 ) and such that f has support on the finite interval [a, b]. Later on we use a density argument for general f ∈ L p (R, v; X 0 ). Let u be defined as in (4.9). Note that u = 0 on (−∞, a]. Step 1: By Lemma 4.8 the function u defined by (4.9) satisfies We show that u is a strong solution of (4.2). Observe that from Fubini's Theorem and d ds S λ (s, r) Therefore, u is a strong solution of (4.7).
Step 3: In this step we show there exists a C ≥ 0 independent of λ and f such that Using (4.9) and S(t, s) ≤ C we find where r λ (t) = λe −λ|t| and g(s) = f (s) X0 As r 1 ∈ L 1 (R) is radially decreasing by [ in the case p > 1. The case p = 1 follows from Fubini's theorem and the convention v ≡ 1. This estimate yields (4.11).
Step 4: To prove the estimate for u ′ note that u ′ = −λu − Au + f , and hence writing Z = L p (R, v; X 0 ), by (4.5) and (4.8), we obtain This finishes the proof of (4.8) for f ∈ L p (R; X 1 ) with support in [a, b] Step 5: Now let f ∈ L p (R, v; X 0 ). Choose for n ≥ 1, f n ∈ L p (R, v; X 1 ) with compact support and such that f n → f in L p (R + , v; X 0 ). For each n ≥ 1 let u n be the corresponding strong solution of (4.7) with f replaced by f n . From (4.8) applied to u n − u m we can deduce that (u n ) n≥1 is a Cauchy sequence and hence convergent to some u in L p (R, v; X 1 ) ∩ W 1,p (R, v; X 0 ). On the other hand, for u defined as in (4.9) one can show in the same way as in Step 3 that for almost all t ∈ R, where M is the Hardy-Littlewood maximal operator. Taking L p (v)-norms and using the boundedness of the maximal operator we find u n → u in L p (R, v; X 0 ) and hence u = u if p ∈ (1, ∞). Taking limits (along a subsequence), (4.7) and (4.8) follow if p ∈ (1, ∞). The case p = 1 is proved similarly using Young's inequality.
It will be convenient to restate our results in terms of maximal L p v -regularity. p ((a, b), v; X 0 ) ∩ L p ((a, b), v; X 1 ).
has a unique strong solution u : (a, b) → X 0 and there is a constant C independent of f such that Here we omit the condition u(a) = 0 if a = −∞.
Of course, the reverse estimate of (4.13) holds trivially. Note that maximal L p v -regularity on (a, b) implies maximal L p v -regularity on (c, d) ⊆ (a, b). It is also easy to check that if |b − a| < ∞, the maximal L p v -regularity on (a, b) for A and λ + A are equivalent. Indeed, the solutions of u ′ (t) + (λ + A(t))u(t) = f (t) and w ′ (t) + A(t)w(t) = e λt f (t) are connected by the identity u(t) = e −λt w(t).
The result of Theorem 4.9 immediately implies that Actually the constant in the estimate can be taken uniformly in λ. Indeed, for fixed λ 0 > ω by (4.8) and Remark 4.10, there is a constant C such that for all λ ≥ λ 0 and for all f ∈ L p (R + , v; X 0 ), This is a maximal regularity estimate with constant which is uniform in λ.
Remark 4.13. If A is time independent and has an H ∞ -calculus of angle < π/2, then setting A 0 = A, and T (t, s) = I, in Theorem 4.9 yields a maximal regularity result for autonomous equations. There are much more suitable ways to derive maximal L p -regularity results in the autonomous case (see [53,59,88,89]), using less properties of the operator A. Indeed, only R-sectoriality of A is needed, but the Banach space X 0 is assumed to be a UMD space. We assume more on the operator but less on the space as we only require finite cotype of X 0 and the Rboundedness of a certain integral operator. Another theory where no assumptions on the Banach space are made but even more on the operator, can be found in [49].
In the above mentioned works only maximal L p -regularity on R + is considered, but  ((a, b), v; X 0 ) has a continuous version. We introduce certain interpolation spaces in order to give a more precise description of traces. Let X v,p be the space of all x ∈ X 0 for which there is a u ∈ MR p (R + , v) such that u(0) = x. Let Spaces of this type have been studied in the literature (see [10,12,48] and references therein). Obviously, one has X 1 ֒→ X v,p ֒→ X 0 . For t ∈ R and a weight v, let v t = v(· − t). The following trace estimate on R + is a direct consequence of the definitions. A similar assertions holds for u ∈ MR p (R, v) for all t ∈ R. Proposition 4.14 (Trace estimate). For u ∈ MR p (R + , v), one has A simple application of maximal regularity is that one can automatically consider nonzero initial values. Note that without loss of generality we can let a = 0.
In this case there is a constant c v,p such that the following estimate holds: Proof. (1) ⇒ (2): Let w ∈ MR p (R + , v) be such that w(0) = x. Let g(t) = −(w ′ (t) + A(t)w(t)). Then g ∈ L p ((0, T ), v; X 0 ). Letũ be the solution to (4.15) with zero initial value and with f replaced by f + g. Now u(t) =ũ(t) + w(t) is the required strong solution of (4.15). Indeed, clearly u(0) = x and Moreover, Taking the infimum over all w ∈ MR p (R + , v) with w(0) = x also yields the second part of (4.16).
(2) ⇒ (1): As u ′ and Au are both in L p ((0, T ), v; X 0 ), the identity in (4.15) yields that f ∈ L p ((0, T ), v; X 0 ) with the estimate as stated. To obtain the required properties for x note that u ∈ MR p ((0, T ), v) can be extended to a function u ∈ ((0,T ),v) . In the case T = ∞ we can take c v,p = 1.
It can be difficult to identify X v,p . For power weights this is possible. Including a power weight has become an important standard technique to allow non-smooth initial data and to create compactness properties. At the same time, the regularity properties of the solution to (4.15) for t > 0 are unchanged. For more details and applications to evolution equations we refer to [36,56,64,68,69,79]. [87,Theorem 1.8.2]). Here (X 0 , X 1 ) θ,p stands for the real interpolation space between X 0 and X 1 . In the limiting cases α ↑ p − 1 and α ↓ −1, one sees that the endpoint X 1 and X 0 can almost be reached.
Proof of Theorem 1.1. First of all we may use a constant extension of A to an operator family on R. Clearly, we can do this in such a way that T (t, s) is uniformly bounded in −∞ < s ≤ t < ∞ say by a constant M . For instance one can take A(t) = A 0 for t / ∈ (0, τ ). Assumption (A) is clearly satisfied. Note that by the assumption and [59,Theorem 11.13], A 0 has a bounded H ∞ -calculus of angle < π/2 and hence (E) is satisfied.
By Proposition 2.4 {I kT : k ∈ K} is uniformly bounded. For p = 2, this implies R-boundedness of {I kT : k ∈ K} ⊆ L (L 2 (R, v; X 0 )), because L 2 (R, v; X 0 ) is a Hilbert space. By Proposition 2.5 this implies that {I kT : k ∈ K} ⊆ L (L p (R, v; X 0 )) is R-bounded as well and hence condition (Rbdd) holds. Therefore, all the conditions of Theorem 4.9 are satisfied, and we find that A has maximal L p v -regularity on R. This implies that A has maximal L p v -regularity on (0, τ ), and hence the required result follows from Proposition 4.15 and Example 4.16.
Here B : [0, T ] × X 1 → X 0 is such that there exists a constant ε > 0 small enough and constants C, L ≥ 0 such that for all x, y ∈ X 1 and t ∈ (0, T ), Proposition 4.17. Assume T < ∞. Assume (A) holds and assume there is a λ 0 such that for all λ ≥ λ 0 , λ + A has maximal L p v -regularity on (0, T ) and there is a constant C A > 0 such that for all λ ≥ λ 0 and f ∈ L p ((0, T ), v; X 0 ), the strong solution u to (4.12) satisfies Assume the constant from (4.18) satisfies ε < 1 CA . Then for every f ∈ L p ((0, T ), v; X 0 ) and x ∈ X v,p , there exists a unique strong solution u ∈ MR p ((0, T ), v) of (4.17) and where C is independent of f and x.
The proof of this proposition is a standard application of the regularity estimate (4.19) combined with the Banach fixed point theorem. A similar result holds on infinite time intervals if one assumes Proof. Let λ > 0 be so large that CALB λ < C A ε := 1 − θ and define the following equivalent norm on MR p ((0, T ), v): We will prove that for all g ∈ L p ((0, T ), v; X 0 ) and x ∈ X v,p there exists a unique strong solution w ∈ MR p ((0, T ), v) of (4. 21) w ′ (t) + (A(t) + λ)w(t) +B(t, w(t)) = g(t), w(0) = x.
and that w satisfies the estimate (4.20) with (u, f ) replaced by (w, g). Herẽ B(t, x) = e −λt B(t, e λt x) and note thatB satisfies the same Lipschitz estimate (4.18) as B. To see that the required result for (4.17) follows from this, note that there is a one-to-one correspondence between both problems given by u(t) = e λt w(t) and f = e λt g. Therefore, from now it suffices to consider (4.21). In order to solve (4.21) we use the maximal regularity estimate (4.19) combined with Proposition 4.15 and the special choice of λ.
Then for φ 1 , φ 2 ∈ MR p ((0, T ), v), by (4.19) one has Hence L is a contraction on MR p ((0, T ), v) with respect to the norm · λ . Therefore, by the Banach fixed point theorem there is a unique w ∈ MR p ((0, T ), v) such that L(w) = w. It is clear that w is the required strong solution of (4.21). To prove the required estimate note that by (4.19) and Proposition 4.15 one has Subtracting (1 − θ) w λ on both sides, and rewriting the estimate in terms of f and u gives the required result.
Consider the sequence of problems: Here we omit the initial condition if a = −∞.
Recall that v a = v(· − a). The following approximation result holds.
Proposition 4.18. Assume (A) holds for A and A n for n ≥ 1 with uniform estimates in n. Assume A and A n for n ≥ 1 have maximal L p v -regularity on (a, b) with uniform estimates in n. Let f n , f ∈ L p ((a, b), v; X 0 ) and x n , x ∈ X va,p for n ≥ 1. Then if u and u n are the solutions to (4.2) and (4.23) respectively, then there is a constant C only dependent on the maximal L p v regularity constants and the constants in (A) such that In particular if x n → x in X va,p , for all z ∈ X 1 , A n (t)z → A(t)z in X 0 a.e. and f n → f in L p ((a, b), v; X 0 ), then u n → u in MR p ((a, b), v).
Typically, one can take A n = ϕ n * A where (ϕ n ) n≥1 is an approximation of the identity. If ϕ n are smooth functions, then A n will also be smooth and therefore, A n will generate an evolution system with many additional properties (see [64,84]).
Proof. The last assertion follows from (4.24) and the dominated convergence theorem. To prove the estimate (4.24) note that w n = u n − u satisfies the following equation w ′ n + A n w n = (f n − f ) + (A n − A)u, w n (a) = x n − x. Therefore, the (4.24) follows immediately from the maximal L p v -regularity estimate.

An example: m-th order elliptic operators
In this section let p, q ∈ (1, ∞), m ∈ {1, 2, ...} and consider the usual multi-index w) and for all |α| ≤ m, D α f L q (R d ,w) < ∞. In this case we let The weights in space will be used in combination with Theorem 2.6 to obtain R-boundedness of the integrals operators arising in (Rbdd).
Consider an m-th order elliptic differential operator A given by where D j := −i ∂ ∂j and a α : R + × R d → C. In this section we will give conditions under which there holds maximal L p vregularity for A or equivalently we will prove optimal L p v -regularity results for the solution to the problem A function u will be called a strong L p v (L q w )-solution of (5.2) if u ∈ MR p ((a, b), v) and (5.2) holds almost everywhere.
With slight abuse of notation we write A for the realization of A on X 0 = L q (R d , w) with domain D(A) = X 1 . In this way (5.2) can be modeled as a problem of the form (4.15). Also, we have seen in Section 4 (and in particular Proposition 4.15) that it is more general to study maximal L p v -regularity on R. Therefore,we will focus on this case below.

Preliminaries on elliptic equations.
In this section we introduce notation and present some results for elliptic equations which will be needed below. Let with a α ∈ C constant. The principal symbol of A is defined as We say that A is uniformly elliptic of angle θ ∈ (0, π) if there exists a constant κ ∈ (0, 1) such that and there is a constant K such that a α ≤ K for all |α| ≤ m. In this case we write A ∈ Ell(θ, κ, K).
The following result is on the sectoriality of the operator in the x-independent case. The proof is an application of the Mihlin multiplier theorem.
Theorem 5.1. Let 1 < q < ∞ and w ∈ A q . Assume A ∈ Ell(θ 0 , κ, K) with θ 0 ∈ (0, π). Then for every θ > θ 0 there exists an A q -consistent constant C depending on the parameters m, d, θ 0 − θ, κ, K, q such that In particular, there is a constantC depending only on θ and C such that e −tA ≤ C.
The case of x-dependent coefficients can be derived by standard localization arguments, but we will not need this case below (see [40,Theorem 3.1] and [58,Section 6]).
Proof. For (5.3) we need to check that for every λ ∈ Σ π−θ , and |β| ≤ m, the symbol M : R d → C given by satisfies the following: for every multiindex α ∈ N d , there is a constant C α which only depends on d, α, θ − θ 0 , K, κ such that Indeed, as soon as this is checked, the result is a consequence of the weighted version of Mihlin's multiplier theorem (see [31,Theorem IV.3.9]). In order to check the condition for ℓ ≥ 0 let F ℓ be the span of functions of the form We claim that for f ∈ F ℓ the mapping ξ → |ξ| ℓ f (ξ) is uniformly bounded. In order to prove this it suffices to consider f = λ η gh −1 with g and h as before, and ℓ = m(µ − η) − ν. As ξ → |ξ| −ν g(ξ) is bounded it remains to estimate where ξ * = ξ/|ξ| and s = λ|ξ| −m .
In order to check (5.4) note that M ∈ F 0 and hence by the above D α M(ξ) ∈ F |α| . Therefore, the bound follows from the claim about F ℓ and the observation that the functions g arising in the linear combinations of the form λ η gh −1 satisfy |g(ξ)| ≤ C K,d,α .
The assertion for e −tA follows from (2.7) and the estimate (5.3) with β = 0.
As a consequence we obtain the following: where c −1 is A q -consistent and only depends on m, d, θ 0 − θ, κ, K, q and λ 0 .
Corollary 5.2 for x-dependent coefficients will be derived from Theorem 5.4 in Remark 5.6. Corollary 5.3. Let m ≥ 1, 1 < q < ∞ and w ∈ A q . If m ≥ 2, then there is an A q -consistent constant C depending only on d, q and m such that for all |β| ≤ m−1 Proof. Note that for |β| = 1, ) and x 0 ∈ R d and t 0 ∈ R let us introduce the notation: for the operator with constant coefficients.
(C) Let A be given by (5.1) and assume each a α : R × R d → C is measurable.
As θ 0 < π/2, the above ellipticity condition implies that m is even in all the results below.
The set of parameters on which all constant below will depend is given by Moreover, all the dependence on the weights will be in an A p and A q -consistent way.
Theorem 5.4. Let p, q ∈ (1, ∞). Let v ∈ A p (R) and w ∈ A q (R d ). Assume condition (C) on A. Then there exists a λ 0 ∈ R depending on the parameters in P such that for all λ ≥ λ 0 the operator λ + A has maximal L p v -regularity on R. Moreover, for every λ ≥ λ 0 and for every f ∈ L p (R, v; X 0 ) there exists a unique and there is constant C depending on the parameters in P such that Recall that MR p (R, v) = W 1,p (R, v; X 0 ) ∩ L p (R, v; X 1 ). Also note that the estimate (5.6) also holds if one replaces R by (−∞, T ) for some T ∈ R. The above result also implies that λ + A has maximal L p v -regularity on (0, T ) for every T < ∞ and every λ ∈ R.
The proof of the above result is a based on Theorem 4.9, standard PDE techniques and extrapolation arguments. The proof of Theorem 5.4 is divided in several steps of which some are standard, but we prefer to give a complete proof for convenience of the reader. In Steps 1 and 2 we assume a α = 0 for |α| < m and show how to include these lowers order terms later on.
Step 2: Next we consider the case where the coefficients of A are also x-dependent, but still with a α = 0 for α < m. We start with a standard freezing lemma.
In order to obtain (5.9) from (5.14) and (5.15) note that it follows from the interpolation inequality from Corollary 5.3 that for all ν > 0 Therefore, choosing ν small enough we can combine the latter with (5.15) to obtain λ u L q (R,v;X0) + u L q (R,v;X1) ≤ C f L q (R,v;X0) + 1 2 u L q (R,v;X1) + C ν u L q (R,v;X0) .
Step 4: To extrapolate the estimate from the previous step to p = q, let u : R → X 1 be a Schwartz function. Then by (5.9) we have for all v ∈ A q F L q (R,v) ≤ C G L q (R,v) , where F = u X1 , G = (λ + A)u + u ′ X0 and where C only depends on [v] Aq (and the additional parameters). Therefore, by the extrapolation result Theorem 2.1 it follows that for all v ∈ A p where C ′ depends on [v] Aq , p, and q (and the additional parameters). This yields u L p (R,v;X1) ≤ C ′ (λ + A)u + u ′ L p (R,v;X0) . Similarly, one proves the estimate for λ u L p (R,v;X0) . As u ′ = (λ + A)u + u ′ − (λ + A)u, (5.6) with righthand side f = (λ + A)u + u ′ follows.
Step 5: Let A be as in the theorem. For s ∈ [0, 1] let A s = sA + (1 − s)(−∆) m/2 , where we recall that m is even. Then A s satisfies condition (C) with constants κ and K replaced by min{κ, 1} and max{K, 1}, respectively. Therefore, for all λ ≥ λ 0 , (5.6) holds with right-hand side f = (λ + A s )u + u ′ with a constant C which does not dependent on s. For s = 0 for all λ ≥ λ 0 , for every f ∈ L p (R, v; X 0 ), one has existence and uniqueness of a strong solution u ∈ MR p (R, v) to u ′ +(λ+A s )u = f by step 1. Therefore, the method of continuity (see [33,Theorem 5.2]) yields existence and uniqueness of a strong solution for every s ∈ [0, 1]. Taking s = 1, the required result follows and this completes the proof of Theorem 5.4.
Remark 5.6. In the proof of Theorem 5.4 we applied Corollary 5.2 only for the case of x-independent coefficients. It is rather simple to derive Corollary 5.2 with x-dependent coefficients from Theorem 5.4 (cf. [58,Exercise 4.3.13]). Indeed, let A be t-independent but such that (C) holds and let u ∈ W m,q (R d , w). Applying (5.6) toũ : R → X 1 given byũ(t) = e −µ|t| u with µ > 0 and v = 1, and letting µ ↓ 0, we find λ u L q (R d ,w) + u W m,q (R d ,w) ≤ C (λ + A)u L q (R d ,w) .
Finally we show how to derive Theorem 1.2 from Theorem 5.4.
Proof of Theorem 1.2. By Theorem 5.4 there is a λ ∈ R such that λ + A has maximal L p -regularity on R and hence on (0, T ) as well. By the observation after Definition 4.11 this implies that A has maximal L p -regularity on (0, T ) and hence we can find a unique solution u ∈ W 1,p (0, T ; L q (R d )) ∩ L p (0, T ; W m,q (R d )) of (1.2) with u 0 = 0. By Proposition 4.15 with v ≡ 1, we can allow nonzero initial values u 0 ∈ X v,p = (L q (R d ), W m,q (R d )) 1− In the next remark we compare Theorems 1.2 and 5.4 to part of the literature on such equations.
(1) In the case A is time-independent and v ≡ 1, Theorem 5.4 reduces to [40,Theorem 3.1] in case of scalar equations. (2) In a series of papers by Kim and Krylov several maximal L p -regularity results for (1.2) have been derived under VMO conditions on the coefficients. In [58,Theorem 4.3.8] the case p = q and m = 2 has been considered under the same assumptions under the coefficients. Extensions to the case 1 < q ≤ p < ∞ have been given in [54] and [58,Chapter 7]. Here only VMO conditions in the space variable are required. In the x-independence case results for p, q ∈ (1, ∞) can be found in [55,57]. For further results and references in the case p = q we refer to [25].
In future works we will consider other examples and applications of the above methods: Remark 5.8.
(1) One can extend Theorem 5.4 to systems of equations. This is more complicated as the evolution family is not explicitly given in this situation. This will be addressed in future works. (2) With standard methods one can extend the result of Theorem 5.4 to half spaces and domains. This will be presented elsewhere.
(3) The same method gives new information on stochastic maximal L p -regularity for SPDEs with coefficients which depend on time in a measurable way. The case of continuous dependence on time was considered in [71].