The maximal regularity property of abstract integrodifferential equations

We provide a convenient framework for the study of the well-posedness of a variety of abstract (integro)differential equations in general Banach function spaces. It allows us to extend and complement the known theory on the maximal regularity of such equations. More precisely, by methods of harmonic analysis, we identify large classes of Banach spaces which are invariant with respect to distributional Fourier multipliers. Such classes include general vector-valued Banach function spaces Φ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varPhi $$\end{document} and/or the scales of Besov and Triebel–Lizorkin spaces defined by Φ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varPhi $$\end{document}. We apply this result to the study of the well-posedness and maximal regularity property of abstract second-order integrodifferential equations, which model various types of elliptic and parabolic problems arising in different areas of applied mathematics.


Introduction with preliminary considerations
The paper continues the well-known research program on the study of the wellposedness and maximal regularity of pseudodifferential equations in a variety of function spaces via techniques of vector-valued Fourier multiplier theory.In a quite general framework, this program was already formulated by Amann in [1, Chapter 3] and was labelled as 'pairs of maximal regularity'.In one of the pioneering papers in this area, [2], Amann showed how the boundedness of translation-invariant operators with operator-valued symbols guarantees the well-posedness of various elliptic and parabolic (integro)differential equations in a scale of vector-valued Besov spaces.His paper constitutes a conceptual background and toolbox for further extensions in the literature.Here, we just mention the breakthrough result due to Weis [44] on the characterization of the L p -maximal regularity of the first-order Cauchy problems and a few others, seminal papers [5], [7], [8], [14], [13], [27] (see also the references therein).In this paper, we provide a convenient framework for such studies.In particular, it allows us to extend and unify several results from the literature.
To describe our main results and explain difficulties arising during their studies, for the transparency, we consider only a special form of a more general abstract second-order integrodifferential equation (20) discussed in Section 5; see Theorem 5.4 with Corollary 5.5 for the corresponding results.Moreover, to keep this presentation compact, we closely follow the notation and terminology from Amann's paper [2], which we only roughly recall below, and refer the reader to [2] for more details in the case.It allows us to keep the novelty of our results more transparent.
Consider the following abstract evolution problem on the line: (1) Here X is a Banach space and by ∂ we denote a distributional derivative, that is, ∂u(φ) := −u(∂φ) for every X-valued distribution u ∈ D ′ (X) := D ′ (R; X) and every test function φ ∈ D := D(R; C).Moreover, A stands for a closed, linear operator on X with domain D A equipped with the graph norm and denoted by Y in this section.
Under suitable assumptions on the operator A one can show that for every tempered distribution f ∈ S ′ (X) := S ′ (R; X) the problem (1) has a unique distributional solution u f ∈ S ′ (Y ).For instance, suppose that iR ⊂ ρ(A), where ρ(A) stands for the resolvent set of A and that the function a(t) := (it+ A) −1 ∈ L(X, Y ), t ∈ R, has a polynomial growth at infinity.Then, one can easily argue that the solution operator U for (1) is given by (2) U : Here, Θ denotes a hypocontinuous, bilinear map from O M (L(X, Y )) × S ′ (X) into S ′ (Y ) (see the kernel theorem, e.g. in [2, Theorem 2.1]), and O M (L(X, Y )) stands for the space of slowly increasing, smooth L(X, Y )-valued functions on R.
One of the main questions of the distributional theory of partial differential equations, in the context of (1), is to specify further assumptions on A and f such that (1) is satisfied in a more classical, strong sense.That is, naturally restricting f to the class of regular distributions, i.e. f ∈ S ′ r (X) := L 1 loc (X)∩S ′ (X), we ask when U f is in the Sobolev class W 1,1 loc (X), which means that U f , ∂U f ∈ L 1 loc (X).Then, in particular, the classical derivative (U f ) ′ of U f exists a.e. on R, ∂U f = (U f ) ′ , U f (t) ∈ Y for a.e.t ∈ R, and (1) is satisfied pointwise a.e on R. In other words, we would like to identify a subspace of S ′ r (X) such that the solution operator U maps it into L 1 loc (Y ) or equivalently the operator AU maps it into L 1 loc (X).The other related question, which is of main importance in the study of nonlinear problems associated with A, concerns the property of maximal regularity with respect to a given Banach space E(X) := E(R; X) ⊂ S ′ (X).That is, under suitable assumptions on A and X, we would like to identify a class of Banach spaces E(X) ⊂ S ′ (X) such that for all f ∈ E(X) each summand on the left side of (1), i.e. ∂U f and AU f , belongs to E(X).Since 0 ∈ ρ(A), it is equivalent to say that the corresponding distributional solution U f of (1) is in E(Y ) (or, equivalently, ∂U f ∈ E(X)).Therefore, this question can be rephrased as the question about the identification of those Banach spaces E(X) ⊂ S ′ (X), which are invariant with respect to AU .Moreover, by the simple algebraic structure of (1), if (in addition) E(X) ⊂ L 1 loc (X), then we get that U f is a strong solution of (1), that is, U f ∈ W 1,1 loc (X) for every f ∈ E(X).The above questions constitute a natural research program, which can be easily reformulated for a variety of different types of (non)autonomous integrodifferential equations.In this paper, following [3,4], we address such questions to an abstract integrodifferential equation (see (20)), which covers various types of particular abstract evolution problems already studied in the literature, e.g. the problem (1); see also Section 6.
Our main result, Theorem 5.4 with Corollary 5.5, in the particular case of the equation (1), reads as follows: Assume that A is an invertible, bisectorial operator on a Banach space X.Let Φ be an arbitrary Banach function space Φ over (R, dt) such that the Hardy-Littlewood maximal operator M is bounded on Φ and its dual Φ ′ .Let Φ(X) denote the X-valued version of Φ, and B s,q Φ (X) and F s,q Φ (X) (s ∈ R, q ∈ [1, ∞]) be the corresponding scales of Besov and Triebel-Lizorkin spaces defined on the basis of Φ.Then, (i) if X is a Hilbert space then for each Consequently, for each f ∈ Φ(X) the distributional solution U f of (1) is a strong one, that is, U (Φ(X)) ⊂ W 1,1 loc (X) and (1) holds a.e. on R. (ii) if X has UMD property and, additionally, A is R-bisectorial, then the conclusion of (i) holds.(iii) for every s ∈ R and q ∈ [1, ∞], the space B s,q Φ (X) is invariant with respect to AU .
We refer to Section 2 for the definition of spaces involved in the above formulation.In the context of our first question, we should also point out a consequence of Theorem 3.1(i), which says that Φ(X) = w∈A2 L 2 (R; wdt; X).
The sum on the left side is taken over all Banach function spaces Φ such that the Hardy-Littlewood maximal operator M is bounded on Φ and its dual Φ ′ , and A 2 stands for the Muckenhoupt class of A 2 -weights on R.
Note that the formulation of the program of 'pairs of maximal regularity' from [1, Chapter III] slightly differs from the one expressed by these two above questions.Here, we do not a priori restrict ourselves to the class of spaces such that E(Y ) ⊂ W 1,1 loc (X) (cf.[1, Chapter III.1.5,p.94]).For the problem (1), as was already pointed out above, if E(X) is invariant with respect to AU and E(X) ⊂ S ′ r (X), then a simply algebraic structure of (1) yields immediately that U (E(X)) ⊂ W 1,1 loc (X).But for equations with a more general structure than that one of (1), the existence of distributional/strong solutions and maximal regularity property usually can be obtained under different conditions.
In any case, the basic idea comes from the study of (1).It relies on the study invariant subspaces with respect to operators corresponding to the solution operator for a given problem via their boundedness properties, i.e., norm estimates they satisfy.More precisely, in the context of (1), the operator U can be expressed as a convolution operator with a distributional kernel F −1 a ∈ S ′ (L(X, Y )), where [2,Section 3]).Next, the methods of the singular integral operator theory allow showing the existence of an operator T in L(E(X), E(Y )), which agrees with the operator U on S(X) (or L 1 c (X)).
For a Banach space E(X) ⊂ S ′ (X), which contains S(X) as a dense subset, it immediately gives that E(X) is invariant with respect to AU .However, in the case when S(X) is not dense in E(X) ⊂ S ′ r (X), e.g., when E(X) is a Lorentz space L p,∞ (X) or a Besov type space B s,q L p,∞ (X) corresponding to it (see Section 2), it is not clear how to show directly that T f = U f for every f ∈ E(X) (cf.[23,Problems 3.2 and 3.3], which arise in the study of the boundedness of Fourier multipliers on the classical Besov spaces B s,q ∞ (X), i.e., defined on the basis of Φ = L ∞ ).Such consistency of the solution operators with their bounded extensions resulting from the convolution representations is one of the main problems we address in the context of general Banach spaces considered in this paper; see Proposition 4.2 for the corresponding result.To solve it, we provide a suitable adaptation of the Rubio de Francia iteration algorithm, see Theorem 3.1, which allows for relaxing some difficulties with density arguments that arise in such and related considerations in the literature; see, e.g.Remarks 3.5(b) and 3.2(b).
The paper is organized as follows.In Section 2 we introduce the classes of Banach spaces with respect to which the questions on well-posedness and maximal regularity are examined.Their basic properties, which are crucial for the proofs of main results, are stated in Lemmas 3.6 and 5.3.Section 3 contains the main extrapolation results, see Theorem 3.8 and its specification to a class of Fourier multipliers, Proposition 4.2.These results are applied in Section 5 to derive the main results on the well-posedness and maximal regularity property of abstract integrodifferential equations (20); see Theorem 5.4 and Corollary 5.5.We conclude with some comments on particular cases of (20), which may be of independent interest.

Function spaces
Throughout, the symbol Φ is reserved to denote a Banach function space over (R, dt).We refer the reader to the monograph by Bennett and Sharpley [10] for the background on Banach function spaces.Here, we mentioned only several facts we use in the sequel.
We define the vector-valued variant of Banach function spaces Φ as follows.Let X be a Banach space with norm Moreover, we introduce a variant of vector-valued Besov and Triebel-Lizorkin spaces corresponding to a Banach function space Φ.
For a ∈ O M (L(X, Y ), where Y denotes another Banach space, we set where Θ denotes a hypocontinuous, bilinear map from O M (L(X, Y )) × S ′ (X) into S ′ (Y ) (see the kernel theorem, e.g.[2, Theorem 2.1]).Here, O M (L(X, Y )) stands for the space of slowly increasing smooth L(X, Y )-valued functions on R. In particular, a(D) ∈ L(S ′ (X), S ′ (Y )) and for every f ∈ S ′ (X) with Note that for every f ∈ S ′ (X) (3) Let Φ be a Banach function space.For all s ∈ R and q ∈ [1, ∞] we set (with usual modification when q = ∞): In the case when Φ = L p with p ∈ [1, ∞] we get the classical vector-valued Besov and Triebel-Lizorkin spaces.For a coherent treatment in this vector setting see, e.g.[1,Chapter VII], where historical remarks are included (see also Triebel [43] for the scalar case).Recall that, if X is a Hilbert space, then F m,2 L p (X) = W m,p (X) for m ∈ N and F s,2 L p (X) = H s,p (X), s ∈ R. The case when Φ = L p w with w in the Muckenhoupt class A ∞ was recently considered, e.g. in [34,35]; see also the references given therein.Here and in the sequel, by L p w we denote the weighted Lebesgue space L p (R, wdt; C).
For general Banach function spaces Φ we need only basic properties of the corresponding spaces B s,q Φ (X) and F s,q Φ (X); see Lemmas 3.6 and 5.3.To keep the presentation of the main result of this paper transparent, we will not study these spaces on their own rights here.

The extrapolation results
The main results of this section, Theorems 3.1 and 3.4, set up a convenient framework for the study of boundedness properties of distributional Fourier multipliers discussed in the next section.
3.1.The iteration algorithm.We start with an adaptation of the Rubio de Francia iteration algorithm, which provides a crucial tool to resolve the consistency issue for solution operators, which was mentioned in Section 1; see also Subsection 4.2.
Theorem 3.1.Let Φ be a Banach function space over (R, dt) such that the Hardy-Littlewood operator M is bounded on Φ and its dual Φ ′ .
(i) Let X be a Banach space.Then, for every p ∈ (1, ∞) (ii) Let X and Y be Banach spaces and p ∈ (1, ∞).Assume that {T j } j∈J is a family of linear operators T j : S(X) → S ′ (Y ) such that for every Then, each T j extends to a linear operator T j on w∈Ap L p w (X) ⊂ L 1 loc (X) and has the restriction to an operator in L(Φ(X), Φ(Y )).Moreover, Proof.Let R and R ′ denote the following sublinear, positive operators defined on Φ and Φ ′ , respectively: Here, M k stands for the k-th iteration of the Hardy-Littlewood operator M , M 0 := I, and for every h ∈ Φ and g ∈ Φ ′ the functions Rg and R ′ h belong to Muckenhoupt's class A 1 := A 1 (R).Consequently, for every p ∈ (1, ∞), g ∈ Φ, g = 0, and h ∈ Φ ′ , h = 0, the function (5) w g,h,p := (Rg) For (ii), first note that w∈Ap L p w (X) is a subspace of L 1 loc (X).Indeed, a direct computation shows that min(w, v) ∈ A p if w, v ∈ A p .Moreover, a standard approximation argument shows that for every w ∈ A p the Schwartz class S(X) is a dense subset L p w (X).Thus, each operator T j (j ∈ J) has an extension to an operator T j,w in L(L p w (X), L p w (Y )).Let T j : w∈Ap L p w (X) → w∈Ap L p w (X) be given by for w ∈ A p such that f ∈ L p w (X).These operators are well-defined and linear.Indeed, if w, v ∈ A p and f ∈ L p w (X) ∩ L p w (X), since T j,w is consistent with T j,v on S(X) and there exists (f N ) N ∈N ⊂ S(X) such that f N → f as N → ∞ both in L p w (X) and L p v (X), T j,w f = T j,v f .By (i), the operators T j , j ∈ J, are well-defined on Φ(X).
We show that T j , j ∈ J, restrict to uniformly bounded operators in L(Φ(X), Φ(Y )).
Fix f ∈ Φ(X).Let 0 = h ∈ Φ ′ and w := w |f |X ,h,p be given by (5).Then, since |f | X ≤ R|f | X , Our assumption on the dependence of the norms of T j and ( 6) yield Consequently, for every h ∈ Φ ′ and f ∈ Φ(X) we have that Therefore, since Φ has Fatou's property, by the Lorentz-Luxemburg theorem This completes the proof.
Remark 3.2.(a) The formulation of Theorem 3.1 corresponds to the idea of the proof of the Rubio de Francia extrapolation result, [39, Theorem A], rather than to a modern framework provided in [18].However, in contrast to the proof given in [39], the above proof is constructive and adapts ideas presented in [18,Chapter 4].
In a slightly less general form (see Corollary 3.3), such modification of the Rubio de Francia extrapolation method was already applied to the study of the abstract Cauchy problems in [15] and [16]) (see also [29]).The scalar variant of (i) (i.e. for X = C) is also proved in [28,Theorem 1.15] by a different argument than the one presented here.
(b) Note that the point (i) shows that if an operator T is defined on L p w (X) for every w ∈ A p , then T f makes sense for every f ∈ Φ(X), where Φ satisfies the assumption of Theorem 3.1.Therefore, no further extension procedure for such operator T is needed.Cf. [22,Section 5.4]; see also Remark 3.5(b) below.
The following result shows that Theorem 3.1 extends [18,Theorem 4.10].We start with some preliminaries.A special class of Banach function spaces is the class of symmetric spaces (or rearrangement invariant Banach function spaces); see [10].For a locally integrable weight w on R, we define , where f * w denotes the decreasing rearrangement of f with respect to wdt.By [10, Theorem 4.9, p. 61], the space Moreover, E w is a Banach function space over (R, dt) in the sense of Section 2.
For the convenience, we define the Boyd indices p E and q E of a symmetric space E following [33] (in [10] the Boyd indices are defined as the reciprocals with respect to [33]).Proposition 3.3.Let E be a symmetric space over (R, dt) with nontrivial Boyd indices p E , q E ∈ (1, ∞).Then, for every Muckenhoupt weight w ∈ A p E , the Hardy-Littlewood operator M is bounded on E w and its dual (E w ) ′ with respect to (R, dt).
Proof.By [18, Lemma 4.12], M is bounded on Φ = E w .By the inverse Hölder inequality, there exist p and q with 1 < p < p E ≤ q E < q < ∞ such that w ∈ A p ⊂ A q .Therefore, by Muckenhoupt's theorem, M is also bounded on L p w and L q w .One can easily check that the operator S(f Hence, it is of joint weak type (q ′ , q ′ ; p ′ , p ′ ) with respect to (R, wdt); see [12,Theorem 4.11,p. 223].Consequently, since p Next, by Boyd's theorem [10, Theorem 5.16, p.153], we get that S is bounded on (E ′ ) w .Moreover, since (R, wdt) is a resonant space (see [10, Theorem 1.6, p.51], we have (E ′ ) w = (E w ) ′ w with equal norms, where (E w ) ′ w denotes the dual space of E w with respect to (R, wdt).Indeed, the inclusion (E ′ ) w ⊂ (E w ) ′ w follows from [10, Proposition 4.2, p. 59] and the reverse one from the Luxemburg representation theorem [10, Theorem 4.10, p. 62].
Finally, note that for every f where S denotes the norm of S on (E ′ ) w .This completes the proof.

3.2.
The singular integral operators.To proceed, we specify a class of operators for which Theorem 3.1 applies.In particular, it allows us to derive the basic properties of generalized Besov and Triebel-Lizorkin spaces introduced in Section 2, which are involved in the proofs of the subsequent results.
We say that a bounded linear operator Here, L ∞ c (X) stands for the space of all X-valued, essentially bounded, measurable functions with compact support in R. If, additionally, k ∈ C 1 ( Ṙ; L(X, Y )) and (7) [k] K1 := max l=0,1 then we say that T is a Calderón-Zygmund operator.The boundedness of Calderón-Zygmund operators on different types of vector-valued Banach function spaces attracted attention in the literature.Here, we mention two pioneering papers [9], and [40], from which some ideas we reproduce below.For instance, by direct analysis of the proof of [40, Theorem 1.6, Part I] each Calderón-Zygmund operator satisfies the assumptions of Theorem 3.1(ii) (it is also true for a larger class of singular integral operators; see [15,Theorem 7]).The following result makes this statement precise.
Corollary 3.4.Let X and Y be Banach spaces.Let {T j } j∈J be a family of Calderón-Zygmund operators with kernels k j , j ∈ J, such that Then, {T j } j∈J satisfies the assumptions of Theorem 3.1(ii), that is, for every q ∈ (1, ∞) and for every W ⊂ A q with sup w∈W [w] Aq < ∞ sup j∈J,w∈W Consequently, the conclusion of Theorem 3.1(ii) holds for {T j } j∈J .
In particular, if {ψ j } j∈N0 is a resolution of the identity on R and Φ satisfies the assumption of Theorem 3.1, then the operators ψ j (D), j ∈ N 0 , restrict to uniformly bounded operators in L(Φ(X)).
Proof.By direct analysis of the constants involved in the main ingredients of the proof of [40, Theorem 1.6, Part I], the norms T L(L q w (X),L q w (Y )) of each Calderón-Zygmund operator T , depend on q ∈ (1, ∞), the constant from the K 1 -condition of its kernel, and the norm of T as an operator in L(L p (X), L p (Y )), and finally they are bounded from above if w varies in a subset of A q on which [w] Aq are uniformly bounded.Cf. also [25], or [32], for the precise dependence of L q w -norms of Calderón-Zygmund operators on the A q -constants [w] Aq of Muckenhoupt's weights w ∈ A q .
For the last statement, since Therefore, by Young's inequality, we get (8) for ψ j (D), j ∈ N 0 .Alternatively and more directly, this statement can be also derived by the argument presented in the proof of Proposition 4.2(i) below.Remark 3.5.(a) In the scalar case, i.e.X = Y = C, the boundedness of Calderón-Zygmund operators on Banach function spaces Φ such that M is bounded on Φ and Φ ′ was already stated in [41,Proposition 6].The proof given in [41] is based on a formal application of Coifman's inequality.Theorem 3.1 allows us to provide an alternative, direct proof of this fact.
(b) The approach via Strömberg's sharp maximal operator.A conceptually different approach to the study of the boundedness of Calderón-Zygmund operators was presented by Jawerth and Torchinsky [26].It relies on the local sharp maximal function operator (α ∈ (0, 1)): for every f ∈ L 1 loc and t ∈ R. It is shown in [26,Theorem 4.6] (in the scalar case, but the vector one follows easily) that, in particular, if T is a Calderón-Zygmund operator, then there exist constants α and µ such that Here, the word 'appropriate' plays a crucial role to apply this inequality for a further study of the boundedness of T on some function spaces.By an analysis of the proof, we see that a priori one can consider the set of all f ∈ L 1 loc (X) such that T f ∈ L 0 (X) with (|T f | Y ) * (+∞) = 0 and for every interval I ⊂ R, T f I ∈ L 1 (I) with f I := f χ R\2I , and Here, for a function g ∈ L 0 , g * (+∞) = 0 if and only if |{t ∈ R : |g(t)| > α}| < ∞ for all α > 0. Denote the above set by D T and set D T,Φ := D T ∩Φ(X) for a Banach function space Φ ⊂ L 1 loc .Of course we have that L 1 c (X) ⊂ D(T ).Moreover, recall Lerner's characterization of boundedness of M on a Φ ′ .Namely, in [31, Corollary 4.2 and Lemma 3.2] Lerner proved that if the maximal operator M is bounded on a Banach function space Φ, then M is bounded on Φ ′ if and only if there exists a constant µ such that for every f ∈ L 0 with f * (+∞) = 0 Note that by the Jawerth-Torchinsky pointwise estimate (9) and Lerner's characterisation, we get for any Banach function space Φ such that M is bounded on Φ and Φ ′ that for every f ∈ D T,Φ (10) T But it is not clear how to show directly (without the use of the Rubio de Francia algorithm) that D T,Φ is dense in Φ.Again, Theorem 3.1(i) solves this density issue (and, in particular, relaxes a density assumption in [41,Proposition 9]).
In fact, let f ∈ Φ(X) and set w (X), we have for some sequence Therefore, the Fatou property of Φ implies that T f ∈ Φ(Y ) and T f Φ(Y ) f Φ(X) .

The basic properties of generalised Besov and Triebel-Lizorkin spaces.
We start with an auxiliary lemma on dense subsets of B s,q Φ (X) and F s,q Φ (X).Lemma 3.6.Let X be a Banach space and let Φ be a Banach function space over (R, dt) such that the Hardy-Littlewood operator M is bounded on Φ and its dual Φ ′ .
Then, for every that is, E(X) embeds continuously into S ′ (X).In particular, E(X) is a Banach space.Furthermore, the set Φ(X)∩E(X) is dense in E(X) for each E with q < ∞, and in the other cases when Φ (X).More precisely, for every f ∈ E(X) we have that for each E with q < ∞, and for Note that the following duality pairing for ))-topology is the weakest topology on B s,∞ Φ (X) for which every g ∈ B −s,1 Φ ′ (X * ) becomes continuous; see also Remark 3.7 below.
The proof of the last assertion makes the use of similar arguments to those applied above.For f ∈ B s,q Φ (X) and N ∈ N set f N := j≤N ψ j (D)f ∈ Φ(X).Since {ψ j (D)} j∈N0 is the resolution of the identity operator on S ′ (X) (see ( 3)), we have that f = f N + j>N ψ j (D)f .Moreover, since the operators {φ j (D)} ∈N0 are uniformly bounded in L(Φ(X), Φ(X)), we get for q < ∞ that For q = ∞, note first that for every j, l ∈ N 0 with |j − l| > 1 Indeed, since ψj (D), ψ(D) ∈ S ′ (X * ) and S(X * ) is dense in S ′ (X * ), we get that Consequently, for N > 2 we have that Therefore, the proof of this case is complete.
The case Let h ∈ Φ ′ with h = 0 and w := w Gf,h = R(Gf ) 1−q R ′ h; see (5).Then, by similar arguments as in the proof of Theorem 3.1, we get Since sup f ∈F s,q Φ (X),h∈Φ ′ [w Gf,h ] Aq < ∞, by the Lorentz-Luxemburg theorem, we get that Gf ∈ Φ(X) and for all r ∈ {±1, 0}, N ∈ N we have The proof of (11) follows the arguments already presented above.Remark 3.7.a) Applying similar arguments to those from the above proof, one can easily show that the definition of Besov and Triebel-Lizorkin type spaces introduced in Section 2 is independent of a particular choice of a function ψ.The same refers to the definition of the duality paring (12).We omit the proof.
b) Direct arguments based on the ideal property of Φ ′ shows that Φ ′ (X * ) is separating for Φ(X), i.e. for each f ∈ Φ(X), if for all g ∈ Φ ′ (X) , and for all l ∈ N 0 and ) is also separating on B s,∞ Φ (X).Now we are in the position to extend Theorem 3.1(ii).This extension is applied in the study of the extrapolation of L p -maximal regularity under additional geometric conditions on the underlying Banach space X; see Remark 4.3(b) and Corollary 5.5.Theorem 3.8.Let X and Y be Banach spaces and p ∈ (1, ∞).Let Φ be a Banach function space such that the Hardy-Littlewood operator M is bounded on Φ and its dual Φ ′ .
Let T : S(X) → S ′ (Y ) be a linear operator such that for all f ∈ S(X) Then the following assertions hold.
then for every s ∈ R and q ∈ [1, ∞] the operator T has an extension to an operator T Φ in L(B s,q Φ (X), B s,q Φ (Y )).Moreover, if Ψ is another Banach function space such that M is bounded on Ψ and Ψ ′ , then T Ψ f = T Φ f for all f ∈ B s,q Φ (X) ∩ B s,q Ψ (X).The same conclusion holds for Triebel-Lizorkin spaces F s,q then for every Moreover, all such extensions are consistent, that is, if Φ i , i = 1, 2, are Banach function spaces such that M is bounded on them and their duals and Proof.(i) By Theorem 3.1(ii) the operators T ψ l (D), l ∈ N 0 , extend to uniformly bounded operators T l := T l,Φ , l ∈ N 0 , in L(Φ(X), Φ(Y )).Let q < ∞.We show that the operators l≤N T l , N ∈ N, are uniformly bounded in L(B s,q Φ (X), B s,q Φ (Y )) and that for every f ∈ Φ(X) ∩ B s,q Φ (X) lim . Since, by Lemma 3.6, Φ(X) ∩ B s,q Φ (X) is a dense subset of B s,q Φ (X), the series l∈N0 T l defines an operator T Φ in L(B s,q Φ (X), B s,q Φ (Y )).First, note that by (13), for all f ∈ S(X) and all j, l ∈ N 0 , we have Since S(X) is a dense subset of L p w (X) for all w ∈ A p , by Lemma 3.1(i), for every f ∈ Φ(X) we get Further, for all f ∈ Φ(X) ∩ B s,q Φ (X) and all integers 0 ≤ N < N ′ we get (with usual modification for q = ∞) , where we set ψ −1 ≡ 0, when N = 0. Therefore, our claim holds.Since {ψ j (D)} j∈N0 is a resolution of the identity operator on S ′ (Y ), that is, for every g ∈ S ′ (Y ) lim Φ (X) and B s,q Ψ (X); see Lemma 3.6.Since, by Theorem 3.1(ii), T l,Φ is consistent with T l,Ψ on Φ(X) ∩ Ψ(X), we get that where C is a constant independent of f .Therefore, This completes the proof of (i) for Besov spaces.
For the last statement of (i), first note that F s,q Ψ = B s,q Ψ for all Ψ = L q w with q ∈ (1, ∞) and w ∈ A q .Therefore, for such Ψ, T extends to an operator T Ψ in L(F s,q Ψ (X), F s,q Ψ (Y ) and its norm is bounded by Moreover, one can check that for every f ∈ F s,q Ψ (X) and j ∈ N 0 we have that ( 14) Fix Φ and f ∈ Φ(X) ∩ F s,q Φ (X), i.e.Gf ∈ Φ, where For h ∈ Φ ′ , h = 0, we set Then Gf ∈ L q w , i.e. f ∈ F s,q L q w (X) and T f := T L q w f ∈ F s,q L q w (X); see (5).Moreover, a similar argument as in the proof of Theorem 3.1 gives . Therefore, since Φ(X) ∩ F s,q Φ (X) is dense in F s,q Φ (X) (see Lemma 3.6), T extends to an operator in L(F s,q Φ (X), F s,q Φ (Y )).Finally, by Lemma 3.6 and similar arguments to those presented in the case of Besov spaces above, one can show the consistency for operators resulting from different underlying Banach function spaces Φ and Ψ.This completes the proof of (i).
(ii) By Theorem 3.1, T extends to an operator in L(Φ(X), Φ(Y )).For instance, the last assertion of Corollary 3.4 shows that T satisfies the assumption of the part (i).Thus, the case when E = B s,q Φ or E = F s,q Φ is already proved above.For the consistency T E with T E , by Theorem 3.1, it holds when E = Φ 1 and E = Φ 2 .By Lemma 3.6 and similar arguments to those presented in the proof of (i), we easily get the general case.

Multipliers and the regularity of their kernels
In this section, we specify Theorem 3.8 for a class of Fourier multipliers in which we are typically interested in the following sections.
is a subspace of S ′ (X) (equipped with the relative topology of S ′ (X)) and a(D) has a unique extension to an operator in L(G(X), S ′ (Y )), then we denote such extension again by a(D).Since X ⊗ F −1 D( Ṙ) is a dense subset of S ′ (X), if X ⊗ F −1 D ⊂ G(X), then such extension of a(D) is unique.
In particular, as was already mentioned in Section 1, if a ∈ O M (L(X, Y )), then a(D) ∈ L(S ′ (X), S ′ (Y )) and ( 15) where Θ denotes a hypocontinuous, bilinear map from O ).The multipliers symbols, which are involved in the study of maximal regularity property of evolution equations usually are of O Mclass; see, e.g.Proposition 6.1.For instance, it is the case of the problem (1) discussed in Section 1, where the solution operator U is given by a multiplier a(D) Further, by the convolution theorem (see e.g.[2,Theorem 3.6]), for all a ∈ L ∞ (L(X, Y )) and f ∈ S(X), ( 16) Here, For more regular symbols a such convolution representation of the corresponding multiplier a(D) allows us to use the basic techniques of harmonic analysis to study of boundedness properties/invariant subspaces of a(D).However, in general, such representation makes sense only for f in a proper subset of a domain of a(D).For instance, even in the case a in the O M -class, in general, what one can say is that ( 16) holds for f ∈ E ′ (X) ∪ S(X), where E ′ (X) denotes the space of distributions with compact support.
Depending on support conditions or integrability conditions of F −1 a this expression can be extended for a larger class of functions; see e.g.[2, Remark 3.2 and Theorem 3.5].It leads to a question on the consistency of a(D) with its bounded extensions resulting from such convolution representation.We address this question for a particular class of symbols below.

The integral representation of a(D).
For symbols a, which arise in the study of abstract evolution equations, usually F −1 a is a tempered distribution which is a regular one only on Ṙ.That is, there exists a function k ∈ L 1 loc ( Ṙ; L(X, Y )) such that for every φ ∈ D( Ṙ; C) for every f ∈ L 1 c (X) and a.e.t / ∈ supp f .Here, L 1 c (X) stands for the space of all X-valued, measurable, Bochner integrable functions with compact support in R. By Young's inequality, this integral is absolutely convergent for a.e.t / ∈ supp f .Note that, even in the case a ∈ O M (L(X, Y )), without any further information on a(D) we only get that (17) a(D)f (t) = Cf (t) for f ∈ L 1 c ⊗ X and a.e.t / ∈ supp f (a priori we know that a(D) is only continuous on S ′ (X); cf. also [23,Example 2.11]).If we know that, e.g.a(D) restricts to an operator in L(L p (X), L p (Y )) for some p ∈ (1, ∞), then of course such representation holds for f ∈ L 1 c (X).In the case when, e.g.X ⊗ D is not dense in a Banach space E(X) under consideration, if there exists an operator T in L(E(X), E(Y )) such that T f (t) = Cf (t) for every f ∈ L 1 c (X) and a.e.t / ∈ supp f , we do not have any ad hoc argument to show that T is consistent with a(D) on E(X), i.e., T is a restriction of a(D) to E(X).The Rubio de Francia iteration algorithm, Theorem 3.1, makes it all rigorous.
The following result is a reformulation of [42,Proposition 4.4.2,p.245].It shows how Mihlin type estimates of a symbol a are reflected by its distributional kernel where the constant C is independent of a.
The proof of this lemma follows the idea of the proof of its scalar counterpart given in Stein [42,Proposition 4.4.2(a)] and is omitted here.
In the sequel, we say that a satisfies the M γ -condition and write and a(0+) = a(0−) (equivalently, a has continuous extension at 0).Moreover, if a is such that F −1 a ∈ C 1 ( Ṙ; L(X, Y )) with [F −1 a] K1 < ∞, then we say that F −1 a satisfies the (standard) Calderón-Zygmund conditions.Consequently, by (17), if a(D) is in L(L p (X), L p (Y )) and F −1 a satisfies the Calderón-Zygmund condition, then a(D) is a Calderón-Zygmund operator according to the definition from Section 3. The boundedness properties of such operators are the subject of the rest of this section.

4.3.
The boundedness of Fourier multipliers.Let {ψ j } j∈N0 be the resolution of the identity on R; see Section 2. The proof of Theorem 3.8 shows that the study of the boundedness of a multiplier a(D) on vector-valued Besov type spaces B s,q E , defined on the basis of Banach space E, reduces to the study of the uniform boundedness of its dyadic parts (ψ j a)(D), j ∈ N 0 , on the underlying space E(X), that is, (18) sup In [2, Proposition 4.5] Amann showed that if a ∈ M 2 (L(X, Y )), then, in particular, the condition (18) holds for E ∈ {BU C, C 0 , L p ; 1 ≤ p ≤ ∞}.See also [23](and the references therein) for a systematic study of the conditions which imply the boundedness of a(D) on the classical vector-valued Besov spaces (corresponding to L p spaces).
The following result extends [2, Proposition 4.5] for a larger class of spaces E. We point out that for symbols which arise from particular types of evolution equations, the M 1 -condition (respectively, the M 1 -condition) implies the M γ -condition (respectively, the M γ -condition) for every γ ∈ N; see, e.g.Propositions 5.1 and 6.1.Proposition 4.2.Let {ψ j } j∈N0 be a dyadic resolution of the identity on R. Let Φ be a Banach function space such that the Hardy-Littlewood operator M is bounded on Φ and its dual Φ ′ .Then the following assertions hold.
Then the operator a(D) extends to a linear operator T Φ in L(B s,q Φ (X), B s,q Φ (Y )).If Ψ is another Banach function space such that M is bounded on Ψ and Ψ ′ , then the corresponding operator T Ψ is consistent with In particular, a(D) restricts to an operator in L(B s,q Φ (X), B s,q Φ (Y )).In the case when s ∈ R and q ∈ (1, ∞), the above statements remain true if we replace the Besov space B s,q Φ with the Triebel-Lizorkin space ))-continuous.Moreover, if a(D) has the extension to an operator in L(G(X), S ′ (Y )), where G(X) is a subspace of S ′ (X) such that E(X) ⊂ G(X), then a(D)f = T E f for every f ∈ E(X).
For the second one, assume that a(D) possesses a continuous extension on a subset G(X) ⊂ S ′ (X) such that B s,q Φ (X) ⊂ G(X).By Lemma 3.6 and (3), B s,q Φ (X) ֒→ S ′ (X) and for every f Φ (X)), we get a(D)f = T Φ f for all f ∈ B s,q Φ (X).The proof of the last statement of (i) regarding the Triebel-Lizorkin spaces is based on the corresponding statement of Theorem 3.8(i) and mimics arguments presented above.This completes the proof of the part (i).
For the proof of (ii), first note that a(D) is a Calderón-Zygmund operator; see Lemma 4.1 and considerations made in Subsection 4.2.Thus, the proof of this part relies on Theorem 3.8(ii) and uses similar arguments to those presented above.(b) We conclude this section with the condition that guarantees a multiplier a(D) is in L(L p (X), L p (Y )) for some p ∈ (1, ∞).We refer the reader, e.g. to [24, Section 1] for a short survey on the theory of operator-valued multipliers on L p spaces, where further references to seminal papers in this area, including [24], can be found.Here, we just mentioned Weis' result [44] who first generalized the Mihlin theorem for operator-valued symbols by requiring R-boundedness instead of norm boundedness in Mihlin's condition (M 1 ).Recall that, if X and Y are UMD Banach spaces and a symbol a then a(D) ∈ L(L p (X), L p (Y )) for every p ∈ (1, ∞).We refer the reader, e.g. to [30] or [19], for the background on R-boundedness and UMD-spaces.Recently, in [21] was proved that the same conditions implies also that a(D) ∈ L(L p w (X), L p w (Y )) for every w ∈ A p and every p ∈ (c) For multiplier symbols arising in the study of evolution equations (e.g., see (31) below), the R-boundedness of their ranges usually implies that they satisfy the RM 1 -condition stated above; see the last statement of Proposition 5.1 and Proposition 6.1.Recall that, by Clément and Prüss [17], the R-boundedness of the range of a symbol a is also the necessary condition for a(D) to be in L(L p (X), L p (Y )).Therefore, if the underlying Banach space X has UMD property, then it leads to the characterisation of the L p -maximal regularity of such equations in the term of R-boundedness; see, e.g.[4, Theorem 4.1].

Maximal regularity of integro-differential equations
In this section, we illustrate our preceding general results by an abstract integrodifferential equation with a general convolution term; see (20) below.We discuss the particular forms of this equation in the following section.
We start with a preliminary observation.Let A, B and P denote densely defined, closed, linear operators on a Banach space X.Set D A , D B and D P for the domains of these operators equipped with the corresponding graph norms.Since A ∈ L(D A , X), the evaluation of A on S ′ (D A ) gives an operator A in L(S ′ (D A ), S ′ (X)), i.e., (Au)(φ) := A(u(φ)) for every u ∈ S ′ (D A ) and φ ∈ S. The symbols B and P have the analogous meaning below.Note that the distributional derivative ∂ commutes with such extensions, i.e. ∂Au = A∂u for all u ∈ S ′ (D A ), and similarly for B and P.
Moreover, let c ∈ S ′ (L(Z, X)), where Z ⊂ X denotes a Banach space.Consider the following problem (20) ∂P∂u For a given f ∈ S ′ (X), by a distributional solution of (20) we mean a distribution u ∈ S ′ (D A ) such that ∂u ∈ S ′ (D B ) ∩ S ′ (D P ) and the convolution of c with u can be (at least formally) well-defined in S ′ (X).Below we assume that c is the inverse where b ′′ (t) = −2P + ĉ′′ (t).
Therefore, our assumptions yield directly that b ′′ a, (•) 2 b ′′ a ∈ L ∞ (L(X)) and, by the step for γ Therefore, the steps for γ = 1 and γ = 2 show that a ∈ M 3 (L(X, Y )).Now we can proceed by induction for γ > 3. Fix 3 ≤ α ≤ γ.Then By the induction step, for every k = 0, ..., α−1, the function ( Moreover, in the case when k = 0, since l + m ≤ α − 1, by the induction, we have that the functions ( In the case l < α, i.e., 1 ≤ m ≤ α − 1, we have Applying the formulas obtained for the function a, one can easily get desired statement for a 0 and a 1 .We omit details.The proof in the case of the M γ -condition follows the one given above.For the last statement, recall that directly from the definition of R-boundedness, for any Banach spaces X, Y, Z, if τ, σ ⊂ L(X, Y ) and ρ ⊂ L(Y, Z) are R-bounded, then the families τ + σ and τ • ρ are R-bounded (see e.g.[30, Fact 2.8, p. 88]).Therefore, the proof of the last statement mimics exactly the argument given already for the usual, norm boundedness.
The following lemmas allow to derive the strong solutions of the problem (20).Lemma 5.2.Let X and Y be Banach spaces such that Y ֒→ X.Let C be a closed, linear operator on X such that Y ֒→ D C .
Let u ∈ L 1 loc (Y ) be such that ∂u and ∂Cu are regular X-valued distribution, i.e. ∂u, ∂Cu ∈ L 1 loc (X).Then, for almost all t ∈ R the strong derivative u ′ (t) of u at t exists in X, u ′ (t) ∈ D C and Cu ′ (t) = (∂Cu)(t).
The proof of this lemma follows easily from the closedness of C, which yields for all φ ∈ S, and Lebesgue's differential theorem.Here, we use the symbol The following result is a counterpart of the characterisation of the classical Besov spaces (X = C, Φ = L p ) in terms of distributional derivatives; see [43,Theorem 2.3.8].
Lemma 5.3.Let s ∈ R, q ∈ [1, ∞] and let Φ denote a Banach function space such that the Hardy-Littlewood operator M is bounded on Φ and its dual Φ ′ .Then, for every distribution f ∈ S ′ (X), f belongs to B s,q Φ (X) if and only if f and ∂f belong to B s−1,q Φ (X).Moreover, the function is an equivalent norm on B s,q Φ (X).The proof of Lemma 5.3 reproduces the ideas of the proof of the classical case when X = C and Φ = L p from [43].Since the proof, even in the classical case, is somewhat complex and based on several ingredients in which formulation the characteristics p and q are involved, we sketch the line of this extension and underline the main supplementary observation should be made.
Proof of Lemma 5.3.First, we show that the lifting operator J given by Φ (X) isomorphically onto B s−1,q Φ (X).Let {ψ j } j∈N0 be the resolution of the identity on R as in Section 2. Set In order to show that J f ∈ B s−1,q Φ (X) for every f ∈ B s,q Φ (X) it is sufficient to show that there exists a constant µ such that for all j ∈ N 0 For note that for every f ∈ S ′ (X) and every t ∈ R We show that there exists a constant µ such that for every f ∈ B s,q Φ (X) and every j ∈ N 0 we have that The second inequality above follows from Proposition 4.2 since and, as it is readily seen, the functions ρ j : R ∋ t → 2 −j (1 + t 2 ) 1/2 χ j (t), j ∈ N 0 , satisfy the M 3 -condition uniformly in j.Therefore, ρ j (D), j ∈ N 0 , restrict to uniformly bounded operators in L(B s,q Φ (X)).To get the first inequality in (29), by the same argument as in the proof of [43,Theorem 1.3.1],we obtain the existence of a constant µ such that for every g ∈ S(X) and j ∈ N 0 φ * j g(t) ≤ µM (|φ j (D)g| X ) (t).Now by an approximation argument similar to that given in the proof of [43, Theorem 1.4.1]one can show that for every Muckenhoupt weight w ∈ A 2 for every f ∈ L 2 w (X) we have that φ A2 (see [11,Theorem 2.5]), by an argument similar to that from the proof of Lemma 3.6(the case E = F s,q Φ ), we get the first inequality in (29).Therefore, J maps B s,q Φ (X) into B s−1,q and, in the case when s > 0, the function u is a strong solution of (20), ∂u = u ′ and (P u ′ ) ′ ∈ B s,q Φ (X).(ii) Assume that for every t = 0 b is invertible and let ĉ ∈ M 3 (L(Y, X)).If (20) has L p -maximal regularity property, then for every Furthermore, if E ⊂ L 1 loc , then the function u is a strong solution of (20), ∂u = u ′ and (P u ′ ) ′ ∈ E(X).
In addition, if we know that u ∈ W 1,1 loc (X), then by Lemma 5.2, B∂u = Bu ′ and ∂P∂u = (P u ′ ) ′ and (20) takes the form (22).One can easily check that our additional assumption that the function (•)a(•) is in M 2 (L(X)) yields u ∈ W 1,1 loc (X).It completes the proof of (i).
By similar arguments to those presented in the proof of the part (i), first one can show that (20) has Φ-maximal regularity.If E = Φ, then relying on the fact that E(X) ∩ Φ(X) is a dense subset of E(X) (see Lemma 3.6) and the fact that T E is consistent with T Φ (see Theorem 3.8(ii)), we get that for every f ∈ E(X) the distribution u := T E,a f is a solution of (20) in E(X) ⊂ S ′ (X) with desired properties, i.e. ∂P∂u, B∂u, Au, c * u ∈ E(X).
For the second statement, if (•)a(•) ∈ L ∞ (L(X)), then by Propositions 5.1 and 4.2, one can argue that T E,(•)a(•) f = ∂T E f ∈ E(X) for every f ∈ E(X).The closedness of B and P implies that If, additionally, E ⊂ L 1 loc , then again by Lemma 5.2, we get that the function u = T E f is a strong solution of (20), ∂u = u ′ and ∂P∂u = (P u ′ ) ′ ∈ E(X).This completes the proof.
Under additional assumptions on the geometry of the underlying Banach space X one can relax the regularity conditions imposed on the function ĉ.We refer to Remark 4.3 for the notion of the RM 1 -condition, which is involved in the formulation of the following result.

4. 1 .
The Fourier multipliers.Let X and Y be Banach spaces.For a ∈ L ∞ (L(X, Y )) we set D a := {f ∈ S ′ (X) : F f ∈ S ′ r (X) and a(•)F f ∈ S ′ (Y )} to denote the initial domain of a Fourier multiplier a(D) associated with a which is defined by a(D)f :

Remark 4 . 3 .
(a)  The additional statements of the points (i) and (ii) of Proposition 4.2 are mainly address to the case when a ∈ O M (L(X, Y )).Then, a(D) has a canonical extension to an operator in L(S ′ (X), S ′ (Y )); see, e.g.Proposition 6.1 (cf.also[23, Problems 3.2 and 3.3]).

( 1 ,
∞); see[21, Theorem  3.5(a)].Moreover, it is readily seen from the proof that for everyW ⊂ A p with sup w∈W [w] Ap < ∞ sup w∈W a(D) L(L p w (X),L p w (Y )) < ∞.Therefore, T := a(D) |S(X)satisfies the assumptions of Theorem 3.8(ii).Consequently, if X and Y have UMD property, then it allows us to relax the assumptions of Theorem 4.2(ii) to get the same conclusion stated therein.