On the Behaviour of Singular Semigroups in Intermediate and Interpolation Spaces and Its Applications to Maximal Regularity for Degenerate Integro-Differential Evolution Equations

Abstract

For those semigroups, which may have power type singularities and whose generators are abstract multivalued linear operators, we characterize the behaviour with respect to a certain set of intermediate and interpolation spaces. The obtained results are then applied to provide maximal time regularity for the solutions to a wide class of degenerate integro- and non-integro-differential evolution equations in Banach spaces.

1. Introduction

Let 𝑋 be a complex Banach space and let {𝒯_𝐴(𝑡)}_𝑡≥0 be a semigroup of operators on 𝑋, which is generated by a multivalued linear operator 𝐴∶𝒟(𝐴)⊆𝑋→𝑋 and which may have a power type singularity at the origin 𝑡=0, that is, 𝒯_𝐴(𝑡)_ℒ(𝑋)≤𝐶₀𝑡^𝜈𝒯,∀𝑡>0,_𝐴(0)𝑥=𝑥,∀𝑥∈𝑋, (1) for some nonnegative constant 𝐶₀ and nonpositive exponent 𝜈, where ℒ(𝑋) denotes the Banach algebra of all endomorphisms of 𝑋 endowed with the uniform operator norm. In this context our aim here is twofold. The first is to characterize the behaviour of {𝒯_𝐴(𝑡)}_𝑡≥0 with respect to some intermediate and interpolation spaces between 𝑋 and the domain 𝒟(𝐴) of 𝐴. The second is to investigate how this behaviour reflects on the question of maximal time regularity for the solutions to a class of degenerate integro- and non-integrodifferential initial value problems in 𝑋.

The class of operators we will deal with consists precisely of those multivalued linear operators 𝐴 whose single-valued resolvents satisfy the following estimate: (𝜆𝐼−𝐴)⁻¹_ℒ(𝑋)≤𝐶(|𝜆|+1)^−𝛽,∀𝜆∈Σ_𝛼. (2) Here, 𝐼 is the identity operator, 𝐶 is a positive constant, 𝛽∈(0,1], and Σ_𝛼 is the complex region {𝑧∈𝐂∶ℜe𝑧≥−𝑐(|ℑm𝑧|+1)^𝛼,ℑm𝑧∈𝐑}, 𝑐>0, 𝛼∈[𝛽,1]. It thus happens (cf. [1–3]) that 𝐴 is the infinitesimal generator of a semigroup of linear bounded operators in 𝑋 satisfying (1) with 𝜈=𝜈_𝛼,𝛽, where 𝜈_𝛼,𝛽=(𝛽−1)/𝛼.

To outline the motivations of our research, let us assume for a moment that 𝐴 is a single-valued linear operator satisfying (2). It is well known that if 𝛽=1, then 𝐴 is the infinitesimal generator of a bounded analytic semigroup. For this case, an extensive literature exists concerning the behaviour of {𝒯_𝐴(𝑡)}_𝑡≥0 with respect to the real interpolation spaces (𝑋,𝒟(𝐴))_𝛾,𝑝, 𝛾∈(0,1), 𝑝∈[1,∞], and its application to questions of maximal regularity for the solutions to nondegenerate (possibly nonautonomous) integro- and non-integrodifferential abstract Cauchy problems. See, for instance, [4–11]. Due to (1) with 𝜈=𝜈_1,𝛽, the case of 𝛼=1 and 𝛽∈(0,1) is definitely worsened and the literature for it is considerably less conspicuous, although estimate of type (2), with (ℜe𝜆+|ℑm𝜆|^𝛽)⁻¹ in place of (|𝜆|+1)^−𝛽, goes back even to [12, Remark p. 383] in the ambit of Abel summable semigroups admitting uniform derivatives of all orders. One of the main problems with the case 𝛽∈(0,1) is that some equivalent characterizations of (𝑋,𝒟(𝐴))_𝛾,𝑝 begin to fail (cf. [13]), so that some spaces which were just real interpolation spaces between 𝑋 and 𝒟(𝐴) in the case 𝛽=1 become only intermediate spaces in the case 𝛽∈(0,1). However, avoiding questions of interpolation theory and of maximal regularity, a quite satisfactorily semigroup theory for the single-valued case with 𝛽∈(0,1) and its application to the unique solvability of some concrete partial (non-integro-) differential equations have been developed in [14–18]. Since the multivalued case embraces the single-valued one, our contribution in this field is to fill this gap, supplying a theory for the behaviour of singular semig 𝑛 intermediate and interpolation spaces which, in the case 𝛽=1, reduces to that in [9, 11]. As an effect of this theory, there is the possibility of investigating questions of maximal time regularity for an entire class of nondegenerate evolution equations which does not fall within the case 𝛽=1.

The case when 𝐴 is really a multivalued linear operator arises naturally when we shift our attention to degenerate evolution equations of the type considered in [1–3]. There, a semigroup theory for multivalued linear operators was introduced as a tool to handle degenerate equations by means of analogous techniques of the nondegenerate ones. Such a theory has been then successfully applied to questions of maximal regularity for the solutions to a wide class of degenerate integro- and non-integrodifferential equations. We quote [2, 19–23] where, in general and unless 𝛽=1, it is shown that the time regularity of the solutions decreases with respect to that of the data. In this respect, we mention the recent results in [20] where, under an additional condition of space regularity on the data and provided that 𝛼 and 𝛽 are large enough, the loss of time regularity is restored. Regrettably (cf. the appendix below), we have found some inaccuracies in [20, Section 4], and for this reason we must indicate some changes to that paper. On the other side, fortunately, the basic idea in [20] is correct and remedy can be applied to all the inappropriate items. Furthermore, unexpectedly, we will see that the more delicate approach followed in this paper not only corrects the mistakes in [20], but also gives rise to an effective improvement of the achievable results. In fact, here, we will straighten out, refine, and extend [20], enlarging the class of the admissible spaces to which the data may belong, weakening the assumption for the pair (𝛼,𝛽), and complicating the structure of the underlying equations. This is why we will first analyze the behaviour of the semigroup generated by 𝐴 with respect to some intermediate and interpolation spaces which turn out to be equivalent only in the case 𝛽=1. Indeed, the phenomena exhibited in [13] for the single-valued case extend to the multivalued one (cf. [24]), and, until now, for the mentioned behaviour there exist no more than some partial results obtained in [2, 19, 24].

We now give the detailed plan of the paper. In Section 2, for a multivalued linear operator 𝐴 having domain 𝒟(𝐴) and satisfying (2), we introduce the corresponding generated semigroup {e^𝑡𝐴}_𝑡≥0. This leads us to define also the linear bounded operators [(−𝐴)^𝜃]^∘e^𝑡𝐴, ℜe𝜃≥0, 𝑡>0, ([(−𝐴)⁰]^∘e^𝑡𝐴=e^𝑡𝐴) and to recall the fundamental estimates for their ℒ(𝑋)-norm. For the operators [(−𝐴)^𝜃]^∘e^𝑡𝐴 a semigroup type property is proven in Proposition 1. We then introduce the spaces we will deal with in this paper, that is, the interpolation spaces (𝑋,𝒟(𝐴))_𝛾,𝑝 and the spaces𝑋_𝐴^𝛾,𝑝, 𝛾∈(0,1), 𝑝∈[1,∞]. Special attention is given to the embeddings linking these two classes of spaces which, in general, are equivalent only in the case 𝛽=1. Some relations existing between the spaces 𝑋_𝐴^𝛾,𝑝 for different values of 𝛾 and 𝑝 are proven in Proposition 2 and discussed in Remarks 3–5. We conclude the section recalling the estimates proven in [19, 24] for the norms ‖[(−𝐴)^𝜃]^∘e^𝑡𝐴‖_{ℒ(𝑋;(𝑋,𝒟(𝐴))𝛾,𝑝)}, ℜe𝜃≥0, and ‖[(−𝐴)¹]^∘e^𝑡𝐴‖_ℒ(𝑌^𝑝_𝛾;𝑋), 𝑌^𝑝_𝛾∈{(𝑋,𝒟(𝐴))_𝛾,𝑝,𝑋_𝐴^𝛾,𝑝}. In Remarks 7 and 8 we explain why, unless we renounce to optimality, in the case 𝛽<1 these estimates can not be directly extended to the norms ‖[(−𝐴)^𝜃]^∘e^𝑡𝐴‖_{ℒ(𝑋;𝑋𝐴}^𝛾,𝑝₎ and ‖[(−𝐴)^𝜃]^∘e^𝑡𝐴‖_ℒ(𝑌^𝑝_𝛾;𝑋), ℜe𝜃≥1, respectively.

In Section 3, we investigate the behaviour of the operators [(−𝐴)^𝜃]^∘e^𝑡𝐴 with respect to both of the spaces (𝑋,𝒟(𝐴))_𝛾,𝑝 and 𝑋_𝐴^𝛾,𝑝. First, in Proposition 9, we deal with the norms ‖[(−𝐴)^𝜃]^∘e^𝑡𝐴‖_{ℒ(𝑋;𝑋𝐴}^𝛾,𝑝₎, ℜe𝜃≥0, and we show that, except for replacing (𝑋,𝒟(𝐴))_𝛾,𝑝 with 𝑋_𝐴^𝛾,∞ if 𝑝=∞ and with 𝑋_𝐴^{𝛽𝛾,𝑝} if 𝑝∈[1,∞), the same estimates of [19] for the norms ‖[(−𝐴)^𝜃]^∘e^𝑡𝐴‖_{ℒ(𝑋;(𝑋,𝒟(𝐴))𝛾,𝑝)} continue to hold. The second significant result is Proposition 12 where, extending those in [24] to values of 𝜃 other than one, we establish estimates for the norms ‖[(−𝐴)^𝜃]^∘e^𝑡𝐴‖_ℒ(𝑌^𝑝_𝛾;𝑋), ℜe𝜃≥1, 𝑌^𝑝_𝛾∈{(𝑋,𝒟(𝐴))_𝛾,𝑝,𝑋_𝐴^𝛾,𝑝}. As a byproduct we deduce the basic Corollary 14, which in Section 5 will be a key tool in proving the equivalence between the following problem (3) and the fixed-point equation (179). The estimates in Proposition 12 are then merged together with those in [19] to achieve estimates for the norms ‖[(−𝐴)^𝜃]^∘e^𝑡𝐴‖_{ℒ((𝑋,𝒟(𝐴))𝛾,𝑝,(𝑋,𝒟(𝐴))𝛿,𝑝)}, ℜe𝜃≥1. In particular, two different estimates are obtained, if 𝛾+𝛿<1 or not. For if 𝛾+𝛿<1, then (cf. the proof of Proposition 16) we can take advantage of the reiteration theorem for interpolation spaces and obtain estimates that, unless 𝛽=1, are better than those rougher estimates derived in the general case 𝛾,𝛿∈(0,1) (see Remarks 17 and 18). We stress that if 𝛽=1, 𝜃∈𝐍 and 𝐴 is single-valued, then we restore the estimates in [9]. Finally, in Proposition 20, a combination of Propositions 9 and 12 yields the estimate for the norms ‖[(−𝐴)^𝜃]^∘e^𝑡𝐴‖_ℒ(𝑋𝐴^𝛾,𝑝_,𝑋𝐴^𝛿,𝑝₎, ℜe𝜃≥1. Since 𝛽<1, the spaces 𝑋_𝐴^𝜎,𝑞 are, in general, only intermediate spaces between 𝑋 and 𝒟(𝐴) for 𝜎∈(0,𝛽); here the reiteration theorem does not apply and a weaker result is obtained (cf. (101)–(103)).

The estimates of Section 3 are applied in Section 4 to study the time regularity of those operator functions 𝑄_𝑗, 𝑗=1,…,6, that we will need in Section 5. In particular (cf. formula (106)), we modify the definition of 𝑄₂ in [20, Section 4] in order that it is well defined, at least when acting on functions 𝑔∈𝐶^𝛿([0,𝑇];𝑋), 𝛿∈((2−𝛼−𝛽)/𝛼,1) (cf. Corollary 26). Consequently, operators 𝑄₃ and 𝑄₄ in [20] change too, and the new 𝑄₅ and 𝑄₆ should be introduced (cf. formulae (107)–(110)). The Hölder in time regularity of the 𝑄_𝑗’s is characterized in Lemmas 22, 24, 30, and 32 and Propositions 29 and 36. The main feature of these results is to show that the loss of regularity produced by 𝑄₂ and 𝑄₅ can be restored, in 𝑄₃ and 𝑄₆ respectively, employing the regularization property established in [20, Section 3] for a wide range of general convolution operators.

In Section 5 we analyze the maximal time regularity of the strict solutions 𝑣 to the following class of degenerate integrodifferential equations in a complex Banach space 𝑋: d((𝑡=𝜆d𝑡𝑀𝑣))₀𝑀+𝐿𝑣(𝑡)+^𝑛1_𝑖1=1𝒦𝑘_𝑖1,𝐿_𝑖1𝑣(𝑡)+^𝑛2_𝑖2=1ℎ_𝑖2(𝑡)𝑦_𝑖2+𝑓(𝑡),𝑡∈𝐼_𝑇,𝑀𝑣(0)=𝑀𝑣₀. (3) Here, 𝐼_𝑇=[0,𝑇], 𝜆₀∈𝐂, 𝑛₁,𝑛₂∈𝐍, ℎ_𝑖2∶𝐼_𝑇→𝐂, 𝑦_𝑖2∈𝑋, 𝑖₂=1,…,𝑛₂, whereas, 𝑍 being another complex Banach space and 𝒫∶𝑍×𝑋→𝑋 being a bilinear bounded operator, 𝑘_𝑖1∶𝐼_𝑇→𝑍, and 𝒦(𝑘_𝑖1,𝐿_𝑖1𝑣)(𝑡)=∫^𝑡₀𝒫(𝑘_𝑖1(𝑡−𝑠),𝐿_𝑖1𝑣(𝑠))d𝑠, 𝑖₁=1,…,𝑛₁. Of course, if 𝑍=𝐂, then 𝒫 may be the scalar multiplication in 𝑋. As 𝑀, 𝐿, and 𝐿_𝑖1, 𝑖₁=1,…,𝑛₁, we take closed single-valued linear operators from 𝑋 to itself, whose domains fulfill the relation 𝒟(𝐿)⊆⋂^𝑛1_𝑖1=1[𝒟(𝑀)∩𝒟(𝐿_𝑖1)], and we require 𝐿 to have a bounded inverse, allowing 𝑀 to be not invertible. Hence, in general, 𝐴=𝐿𝑀⁻¹ is only a multivalued linear operator in 𝑋 having domain 𝒟(𝐴)=𝑀(𝒟(𝐿)). Assuming that 𝐴 satisfies (2) and that the data 𝑘_𝑖1, ℎ_𝑖2, 𝑦_𝑖2 and 𝑓, 𝑖_𝑙=1,…,𝑛_𝑙, 𝑙=1,2, are suitably chosen, problem (3) is then reduced to an equivalent fixed point-equation for the new unknown 𝑤=𝐿(𝑣−𝑣₀), 𝑣₀∈𝒟(𝐿). It is here that the results of Sections 3 and 4 play their role, leading us to Theorem 48. In that theorem, provided that 5𝛼+2𝛽>6, we will prove that if 𝑘_𝑖1∈𝐶^𝜂𝑖1(𝐼_𝑇;𝑍), ℎ_𝑖2∈𝐶^𝜎𝑖2(𝐼_𝑇;𝐂), 𝑦_𝑖2∈𝑌^𝑟_𝛾𝑖2, 𝑌^𝑟_𝛾𝑖2∈{(𝑋,𝒟(𝐴))_{𝛾𝑖2,𝑟},𝑋^𝛾𝑖2_𝐴^,𝑟}, and 𝑓∈𝐶^𝜇(𝐼_𝑇;𝑋) for opportunely chosen 𝜂_𝑖1, 𝜎_𝑖2, 𝛾_𝑖2, and 𝜇, 𝑖_𝑙=1,…,𝑛_𝑙, 𝑙=1,2, then problem (3) has a unique strict solution 𝑣∈𝐶^𝜏(𝐼_𝑇;𝒟(𝐿)) satisfying 𝑣(0)=𝑣₀ and 𝐿𝑣,d𝑀𝑣/d𝑡∈𝐶^𝜏(𝐼_𝑇;𝑋), where 𝜏=min_{𝑖𝑙=1,…,𝑛𝑙,𝑙=1,2}{𝜂_𝑖1,𝜎_𝑖2} (cf. Remark 51). Section 5 concludes with applications of Theorem 48 to integral and nonintegral subcases of (3), (cf. Theorems 52–54 and 56). We stress that Theorem 48 repairs, generalizes, and improves [20, Theorems 5.6 and 5.7], where similar results were proven only for the case (𝑛₁,𝑛₂,𝑌^𝑝_𝜓)=(1,1,𝑋_𝐴^𝜓,𝑝) and under the stronger condition 3𝛼+8𝛽>10.

In Section 6, we give an application of Theorem 48 to a concrete case of problem (3) arising in the theory of heat conduction for materials with memory. In particular, we show how Theorem 48 characterizes the appropriate functional framework where to search for the solution of the inverse problem of recovering both 𝑣 and the vector (𝑘₁,…,𝑘_𝑟1), 𝑟₁≤𝑛₁, in (3) with (𝑖₂,𝑛₂)=(𝑖₁,𝑛₁) and ℎ_𝑖1=𝑘_𝑖1, 𝑖₁=1,…,𝑛₁.

Finally, in the Appendix we explain how to amend [20, Theorems 5.6 and 5.7] in accordance to Theorem 48.

2. Multivalued Linear Operators, Singular Semigroups, and the Spaces (𝑋,𝒟(𝐴))_𝛾,𝑝 and 𝑋_𝐴^𝛾,𝑝

Let 𝑋 be a complex Banach space endowed with norm ‖⋅‖_𝑋 and let 𝒫(𝑋) be the collection of all the subsets of 𝑋. For a number 𝜆∈𝐂 and elements 𝒰,𝒱,𝒲∈𝒫(𝑋)⧵∅, 𝜆𝒰, and 𝒱+𝒲 denote the subsets of 𝑋 defined by {𝜆𝑢∶𝑢∈𝒰} and {𝑣+𝑤∶𝑣∈𝒱,𝑤∈𝒲}, respectively. Then, a mapping 𝐴 from 𝑋 into 𝒫(𝑋) is called a multivalued linear operator in 𝑋 if its domain 𝒟(𝐴)={𝑥∈𝑋∶𝐴𝑥≠∅} is a linear subspace of 𝑋 and 𝐴 satisfies the following: (i) 𝐴𝑥+𝐴𝑦⊂𝐴(𝑥+𝑦), for all 𝑥,𝑦∈𝒟(𝐴); (ii) 𝜆𝐴𝑥⊂𝐴(𝜆𝑥), for all 𝜆∈𝐂, for all 𝑥∈𝒟(𝐴). From now on, the shortening m. l. will be always used for multivalued linear.

The set ℛ(𝐴)=⋃_{𝑥∈𝒟(𝐴)}𝐴𝑥 is called the range of 𝐴. If ℛ(𝐴)=𝑋, then 𝐴 is said to be surjective. The following properties of a m. l. operator 𝐴 are immediate consequences of its definition (cf. [1, Theorems 2.1 and 2.2]): (iii) 𝐴𝑥+𝐴𝑦=𝐴(𝑥+𝑦), for all𝑥,𝑦∈𝒟(𝐴); (iv) 𝜆𝐴𝑥=𝐴(𝜆𝑥), for all𝜆∈𝐂⧵{0}, for all𝑥∈𝒟(𝐴); (v) 𝐴0 is a linear subspace of 𝑋 and 𝐴𝑥=𝑦+𝐴0 for any 𝑦∈𝐴𝑥, 𝑥∈𝒟(𝐴). In particular, 𝐴 is single-valued if and only if 𝐴0={0}.

If 𝐴 is an m. l. operator in 𝑋, then its inverse 𝐴⁻¹ is defined to be the operator having domain 𝒟(𝐴⁻¹)=ℛ(𝐴) such that 𝐴⁻¹𝑦={𝑥∈𝒟(𝐴)∶𝑦∈𝐴𝑥}, 𝑦∈𝒟(𝐴⁻¹). 𝐴⁻¹ is an m. l. operator in 𝑋 too, and (𝐴⁻¹)⁻¹=𝐴. The set 𝐴⁻¹0={𝑥∈𝒟(𝐴)∶0∈𝐴𝑥} is called the kernel of 𝐴 and denoted by 𝒩(𝐴). If 𝒩(𝐴)={0}; that is, if 𝐴⁻¹ is single-valued, then 𝐴 is said to be injective. Observe that (v) yields 𝐴𝑥=𝐴0 if and only if 𝑥∈𝒩(𝐴).

Given 𝒰∈𝒫(𝑋)⧵∅, we write 𝐴(𝒰)=⋃_{𝑢∈𝒰∩𝒟(𝐴)}𝐴𝑢, so that, in particular, 𝐴(𝑋)=𝐴(𝒟(𝐴))=ℛ(𝐴). If 𝐴_𝑗, 𝑗=1,2 are m. l. operators in 𝑋 and 𝜆∈𝐂, then the scalar multiplication 𝜆𝐴₁, the sum 𝐴₁+𝐴₂, and the product 𝐴₁𝐴₂ are defined by 𝒟𝜆𝐴₁=𝒟𝐴₁,𝜆𝐴₁𝑥=𝜆𝐴₁𝑥,𝑥∈𝒟𝜆𝐴₁,𝒟𝐴₁+𝐴₂=𝒟𝐴₁∩𝒟𝐴₂,𝐴₁+𝐴₂𝑥=𝐴₁𝑥+𝐴₂𝑥,𝑥∈𝒟𝐴₁+𝐴₂,𝒟𝐴₁𝐴₂=𝑥∈𝒟𝐴₂∶𝐴₁𝐴₂𝑥,𝐴≠∅₁𝐴₂𝑥=𝐴₁𝐴₂𝑥,𝑥∈𝒟𝐴₁𝐴₂, (4) where 𝜆𝐴₁, 𝐴₁+𝐴₂ and 𝐴₁𝐴₂ are m. l. operators in 𝑋 and (𝐴₁𝐴₂)⁻¹=𝐴₂⁻¹𝐴₁⁻¹.

Let 𝐴 and 𝐵 be m. l. operators in 𝑋. We write 𝐴⊂𝐵 if 𝒟(𝐴)⊆𝒟(𝐵) and 𝐴𝑥⊆𝐵𝑥 for every 𝑥∈𝒟(𝐴). Clearly, 𝐴⊂𝐵⊂𝐴 if and only if 𝐴=𝐵. If 𝐴⊂𝐵 and 𝐴𝑥=𝐵𝑥 for every 𝑥∈𝒟(𝐴), then 𝐵 is called an extension of 𝐴. If a linear single-valued operator 𝑆 has domain 𝒟(𝑆)=𝒟(𝐴) and 𝑆⊂𝐴, that is, 𝑆𝑥∈𝐴𝑥 for every 𝑥∈𝒟(𝐴), then 𝑆 is called a section of 𝐴. With an arbitrary section 𝑆, it holds 𝐴𝑥=𝑆𝑥+𝐴0, 𝑥∈𝒟(𝐴), and ℛ(𝐴)=ℛ(𝑆)+𝐴0, but this latter sum may or may not be direct (cf. [25, p. 14]). A method for constructing sections is provided in [25, Proposition I.5.2].

If 𝑋_𝑗, 𝑗=1,2, are two complex Banach spaces, then the linear space of all bounded single-valued linear operators 𝐿 from 𝑋₁=𝒟(𝐿) to 𝑋₂ is denoted by ℒ(𝑋₁;𝑋₂) (ℒ(𝑋₁) if 𝑋₁=𝑋₂) and it is equipped with the uniform operator norm ‖𝐿‖_{ℒ(𝑋1;𝑋2)}=sup_{‖𝑥‖𝑋1≤1}‖𝐿𝑥‖_𝑋2=inf_𝐾≥0{‖𝐿𝑥‖_𝑋2≤𝐾‖𝑥‖_𝑋1∶𝑥∈𝑋₁}. Then the resolvent set 𝜌(𝐴) of a m. l. operator 𝐴 is defined to be the set {𝑧∈𝐂∶(𝑧𝐼−𝐴)⁻¹∈ℒ(𝑋)}, with 𝐼 being the identity operator in 𝑋. The basic properties of the resolvent set of single-valued linear operators hold the same for m. l. operators. First, if 𝜌(𝐴)≠∅, then 𝐴 is closed; that is, its graph {(𝑥,𝑦)∈𝑋×𝑋∶𝑥∈𝒟(𝐴),𝑦∈𝐴𝑥} is closed (cf. [25, p. 43]). Further (cf. [1, Theorem 2.6]), 𝜌(𝐴) is an open set and the operator function 𝑧∈𝜌(𝐴)→(𝑧𝐼−𝐴)⁻¹∈ℒ(𝑋) is holomorphic. Finally (cf. [1, formula (2.1)]), the resolvent equation (𝜆₂−𝜆₁)(𝜆₁𝐼−𝐴)⁻¹(𝜆₂𝐼−𝐴)⁻¹=(𝜆₁𝐼−𝐴)⁻¹−(𝜆₂𝐼−𝐴)⁻¹, 𝜆₁,𝜆₂∈𝜌(𝐴), is satisfied, too. Unlike the single-valued case, instead, for 𝑧∈𝜌(𝐴) the following inclusions hold (cf. [1, Theorem 2.7]): (𝑧𝐼−𝐴)⁻¹𝐴⊂𝑧(𝑧𝐼−𝐴)⁻¹−𝐼⊂𝐴(𝑧𝐼−𝐴)⁻¹. (5) Then, in general, 𝑧(𝑧𝐼−𝐴)⁻¹−𝐼, 𝑧∈𝜌(𝐴), is only a bounded section of the m. l. operator 𝐴(𝑧𝐼−𝐴)⁻¹. Throughout this paper, we denote this bounded section by 𝐴^∘(𝑧𝐼−𝐴)⁻¹, but we warn the reader that here 𝐴^∘ does not necessarily denote a section of 𝐴 itself. Of course, if 𝐴 is single-valued, then 𝐴^∘(𝑧𝐼−𝐴)⁻¹ reduces to 𝐴(𝑧𝐼−𝐴)⁻¹. Notice that (5) implies that (𝑧𝐼−𝐴)⁻¹𝐴, 𝑧∈𝜌(𝐴), is single-valued on 𝒟(𝐴) and (𝑧𝐼−𝐴)⁻¹𝐴𝑥=(𝑧𝐼−𝐴)⁻¹𝑦 with any 𝑦∈𝐴𝑥, 𝑥∈𝒟(𝐴). Another difference with the single-valued case is that for every 𝑧∈𝜌(𝐴) it holds 𝒩((𝑧𝐼−𝐴)⁻¹)=𝐴0. Indeed, ((𝑧𝐼−𝐴)⁻¹)⁻¹0=(𝑧𝐼−𝐴)0=𝐴0. Therefore, in the m. l. case, {0}⊊𝒩((𝑧𝐼−𝐴)⁻¹), 𝑧∈𝜌(𝐴). However (cf. [24, Lemma 2.1]), if 0∈𝜌(𝐴), then 𝒩(𝐴^∘(𝑧𝐼−𝐴)⁻¹)={0}, and, in addition, 𝑥∉𝐴0 if and only if 𝐴^∘(𝑧𝐼−𝐴)⁻¹𝑥∉𝐴0, 𝑧∈𝜌(𝐴). We also recall that for every 𝜆₁,𝜆₂∈𝜌(𝐴) the following slight variants of the resolvent equation hold (cf. [24, Lemma 2.2]): 𝜆₂−𝜆₁𝜆₁𝐼−𝐴⁻¹𝐴^∘𝜆₂𝐼−𝐴⁻¹=𝐴^∘𝜆₁𝐼−𝐴⁻¹−𝐴^∘𝜆₂𝐼−𝐴⁻¹,𝜆₂−𝜆₁𝐴^∘𝜆₁𝐼−𝐴⁻¹𝜆₂𝐼−𝐴⁻¹=𝐴^∘𝜆₁𝐼−𝐴⁻¹−𝐴^∘𝜆₂𝐼−𝐴⁻¹. (6) In particular, if 0∈𝜌(𝐴), then, since 𝐴^∘(0𝐼−𝐴)⁻¹=−𝐼, the first in (6) with (𝜆₁,𝜆₂)=(0,𝜆) yields 𝜆(−𝐴)⁻¹𝐴^∘(𝜆𝐼−𝐴)⁻¹=−𝐼−𝐴^∘(𝜆𝐼−𝐴)⁻¹=−𝜆(𝜆𝐼−𝐴)⁻¹; that is, 𝐴⁻¹𝐴^∘(𝜆𝐼−𝐴)⁻¹=(𝜆𝐼−𝐴)⁻¹,𝜆∈𝜌(𝐴). (7) Let (𝐴,𝒟(𝐴)) be a m. l. operator in 𝑋 satisfying the following resolvent condition: (H1)𝜌(𝐴) contains a region Σ_𝛼={𝑧∈𝐂∶ℜe𝑧≥−𝑐(|ℑm𝑧|+1)^𝛼,ℑm𝑧∈𝐑}, 𝛼∈(0,1], 𝑐>0, and for some exponent 𝛽∈(0,𝛼] and constant 𝐶>0 the following estimate holds: (𝜆𝐼−𝐴)⁻¹_ℒ(𝑋)≤𝐶(|𝜆|+1)^−𝛽,∀𝜆∈Σ_𝛼. (8) Introduce the family {e^𝑡𝐴}_𝑡≥0∈ℒ(𝑋) defined by e^0𝐴=𝐼 and e^𝑡𝐴=12𝜋𝑖_Γe^𝑡𝜆(𝜆𝐼−𝐴)⁻¹d𝜆,𝑡>0, (9) where Γ⊊Σ_𝛼⧵{𝑧∈𝐂∶ℜe𝑧≥0} is the contour parametrized by 𝜆=−𝑐(|𝜂|+1)^𝛼+𝑖𝜂, 𝜂∈(−∞,∞). Then (cf. [1, pp. 360, 361]), {e^𝑡𝐴}_𝑡≥0 is a semigroup on 𝑋, infinitely many times strongly differentiable for 𝑡>0 with 𝐷^𝑘_𝑡e^𝑡A=12𝜋𝑖_Γ𝜆^𝑘e^𝑡𝜆(𝜆𝐼−𝐴)⁻¹{},d𝜆,𝑡>0,𝑘∈𝐍=1,2,… (10) where 𝐷^𝑘_𝑡=d^𝑘/d𝑡^𝑘. In general, no analyticity should be expected for e^𝑡𝐴. For if 𝛼<1 in (H1), then Σ_𝛼 does not contain any sector Λ_𝜔+𝜋/2={𝑧∈𝐂⧵{0}∶|arg𝑧|<𝜔+𝜋/2}, 𝜔∈(0,𝜋/2), and [15, Theorem 5.3], which extends e^𝑡𝐴 analytically to the sector Λ_𝜔 containing the positive real axis, is not applicable. We stress that (9) and 𝒩((𝑧𝐼−𝐴)⁻¹)=𝐴0, 𝑧∈𝜌(𝐴), imply 𝐴0⊆𝒩(e^𝑡𝐴) for every 𝑡>0, whereas 𝒩(e^0𝐴)=𝒩(𝐼)={0}. Hence, if 𝐴 is really an m. l. operator, then {0}⊊𝐴0⊆⋂_𝑡>0𝒩(e^𝑡𝐴). From the semigroup property it also follows that 𝒩(e^𝑡0𝐴)⊆𝒩(e^𝑡1𝐴) for 𝑡₁≥𝑡₀≥0.

Now, for every 𝜃∈𝐂 such that ℜe𝜃≥0 we set (−𝐴)^𝜃∘e^𝑡𝐴=12𝜋𝑖_Γ(−𝜆)^𝜃e^𝑡𝜆(𝜆𝐼−𝐴)⁻¹d𝜆,𝑡>0. (11) Here, for the multivalued function (−𝜆)^𝜃=e^{𝜃Ln(−𝜆)} we choose the principal branch holomorphic in the region 𝐂⧵{𝑧∈𝐂∶ℜe𝑧≥0}, where for principal branch we mean the principal determination ln|𝑧|+𝑖arg(𝑧) of Ln(𝑧). We briefly recall the main properties of operators [(−𝐴)^𝜃]^∘e^𝑡𝐴. Of course, [(−𝐴)⁰]^∘e^𝑡𝐴=e^𝑡𝐴, 𝑡>0. As shown in [26, p. 426], [(−𝐴)^𝑘]^∘e^𝑡𝐴, 𝑘∈𝐍, 𝑡>0, is a section of (−𝐴)^𝑘e^𝑡𝐴, so that from (10) we get (−1)^𝑘𝐷^𝑘_𝑡e^𝑡𝐴=(−𝐴)^𝑘∘e^𝑡𝐴⊂(−𝐴)^𝑘e^𝑡𝐴,𝑡>0,𝑘∈𝐍. (12) Moreover (cf. [19, formula (22)] with 𝜃≥0 being replaced by ℜe𝜃≥0), we get (−𝐴)^𝜃∘e^𝑡𝐴−(−𝐴)^𝜃∘e^𝑠𝐴=−^𝑡_𝑠(−𝐴)^𝜃+1∘e^𝜉𝐴d𝜉,ℜe𝜃≥0,0<𝑠<𝑡. (13) Finally, (H1) implies the following estimates (cf. [1, 24]): (−𝐴)^𝜃∘e^𝑡𝐴_ℒ(𝑋)≤𝑐̃_{𝛼,𝛽,𝜃}𝑡^{(𝛽−ℜe𝜃−1)/𝛼},ℜe𝜃≥0,𝑡>0, (14) where the 𝑐̃_{𝛼,𝛽,𝜃}’s are positive constants depending on 𝛼, 𝛽, and 𝜃. Thus, letting 𝜃=0 in (14), we see that if 𝛽∈(0,1), then the operator function 𝑡∈(0,∞)→e^𝑡𝐴∈ℒ(𝑋) may be singular at the origin and the semigroup is not necessarily strongly continuous in the 𝑋-norm on the closure 𝒟(𝐴) of 𝒟(𝐴) in 𝑋. Notice that if 𝛼+𝛽>1, then the singularity is a weak one, in the sense that {e^𝑡𝐴}_𝑡≥0 is integrable in norm in any interval [0,𝜏], 𝜏>0. Further (cf. [24, Lemma 3.9]), if 𝛼+𝛽>1, then 𝐴0=⋂_𝑡>0𝒩(e^𝑡𝐴), and if 𝛼=1, then 𝐴0=𝒩(e^𝑡𝐴) for every 𝑡>0.

Observe that 𝐴0⊆𝒩([(−𝐴)^𝜃]^∘e^𝑡𝐴), ℜe𝜃≥0, 𝑡>0, so that 𝐴0⊆⋂_𝑡>0𝒩([(−𝐴)^𝜃]^∘e^𝑡𝐴), ℜe𝜃≥0. The operators [(−𝐴)^𝜃]^∘e^𝑡𝐴 satisfy the following semigroup type property.

Proposition 1. Let 𝜃_𝑗∈𝐂, ℜe𝜃_𝑗≥0, and let 𝑡_𝑗>0, 𝑗=1,2. Then (−𝐴)^𝜃1∘𝑒^𝑡1𝐴(−𝐴)^𝜃2∘𝑒^𝑡2𝐴=(−𝐴)^{𝜃1+𝜃2∘}𝑒^{(𝑡1+𝑡2)𝐴}. (15)

Proof. First, the function 𝜆∈𝜌(𝐴)→(−𝜆)^𝜃e^𝑡𝜆(𝜆𝐼−𝐴)⁻¹∈ℒ(𝑋) being holomorphic for every ℜe𝜃≥0 and 𝑡>0, and the contour Γ in (11) with (𝜃,𝑡)=(𝜃₂,𝑡₂) can be replaced with the contour Γ^′⊊Σ_𝛼⧵{𝑧∈𝐂∶ℜe𝑧≥0} parametrized by 𝜇=−𝑐^′(|𝜂|+1)^𝛼+𝑖𝜂, 𝜂∈(−∞,∞), 𝑐^′∈(0,𝑐), and lies to the right of Γ. Then, for every 𝑥∈𝑋, from the resolvent equation we obtain (−𝐴)^𝜃1∘e^𝑡1𝐴(−𝐴)^𝜃2∘e^𝑡2𝐴𝑥=12𝜋𝑖²_Γ(−𝜆)^𝜃1e^𝑡1𝜆×_Γ′−𝜇^𝜃2e^𝑡2𝜇(𝜆𝐼−𝐴)⁻¹𝜇𝐼−𝐴⁻¹=1𝑥d𝜇d𝜆2𝜋𝑖²_Γ(−𝜆)^𝜃1e^𝑡1𝜆(𝜆𝐼−𝐴)⁻¹×_Γ′−𝜇^𝜃2e^𝑡2𝜇𝜇−𝜆⁻¹𝑥−1d𝜇d𝜆2𝜋𝑖²_Γ^′−𝜇^𝜃2e^𝑡2𝜇𝜇𝐼−𝐴⁻¹×_Γ(−𝜆)^𝜃1e^𝑡1𝜆𝜆−𝜇⁻¹d𝜆𝑥d𝜇. (16) Now, after having enclosed Γ and Γ^′ on the left with an arc Δ_𝑅 of the circle {𝑧∈𝐂∶|𝑧+𝑐^′|=𝑅}, 𝑅>𝑐−𝑐^′, we apply the residue theorem and let 𝑅 go to infinity. To this purpose, we observe that since the contours Γ and Γ^′ both lie in the half-plane {𝑧∈𝐂∶ℜe𝑧≤−𝑐^′}, the arc Δ_𝑅 may be parametrized in polar coordinates by ℜe𝑧=−𝑐^′+𝑅cos𝜑, ℑm𝑧=𝑅sin𝜑, 𝜑∈(𝜋/2,3𝜋/2). Then, for every 𝑧∈Δ_𝑅 we have (−𝑧)^𝜃e^𝑡𝑧=|𝑧|^ℜe𝜃e^{−ℑm𝜃arg(−𝑧)}e^𝑡ℜe𝑧≤𝑅+𝑐^′ℜe𝜃e^{(𝜋/2)|ℑm𝜃|}e^{−𝑡𝑐′}e^{𝑡𝑅cos𝜑}. (17) Since 𝑡>0 and 𝜑∈(𝜋/2,3𝜋/2), the right-hand side of the latter inequality goes to zero as 𝑅 goes to infinity, so that lim_{𝑅→∞,𝑧∈Δ𝑅}(−𝑧)^𝜃e^𝑡𝑧=0 for every ℜe𝜃≥0 and 𝑡>0. The residue theorem together with the fact that Γ^′ lies to the right of Γ thus yields ∫_Γ′(−𝜇)^𝜃2e^𝑡2𝜇(𝜇−𝜆)⁻¹d𝜇=2𝜋𝑖(−𝜆)^𝜃2e^𝑡2𝜆 and ∫_Γ(−𝜆)^𝜃1e^𝑡1𝜆(𝜆−𝜇)⁻¹d𝜆=0. Replacing these identities in (16) and using the equality (−𝜆)^𝜃1(−𝜆)^𝜃2=(−𝜆)^𝜃1+𝜃2 which is satisfied for the principal branch of the function (−𝜆)^𝜃=e^{𝜃Ln(−𝜆)}, we finally find (−𝐴)^𝜃1∘e^𝑡1𝐴(−𝐴)^𝜃2∘e^𝑡2𝐴𝑥=12𝜋𝑖_Γ(−𝜆)^𝜃1+𝜃2e^{(𝑡1+𝑡2)𝜆}(𝜆𝐼−𝐴)⁻¹𝑥d𝜆. (18) The right-hand side being precisely [(−𝐴)^𝜃1+𝜃2]^∘e^{(𝑡1+𝑡2)𝐴}𝑥, the proof is complete.

For an m. l. operator 𝐴 satisfying (H1) we introduce now the spaces (𝑋,𝒟(𝐴))_𝛾,𝑝 and 𝑋_𝐴^𝛾,𝑝. We first specify a topology on 𝒟(𝐴) equipping it with the norm ‖𝑥‖_𝒟(𝐴)=inf_{𝑦∈𝐴𝑥}‖𝑦‖_𝑋, 𝑥∈𝒟(𝐴). Since 𝐴⁻¹∈ℒ(𝑋), this norm is equivalent to the graph norm and makes 𝒟(𝐴) a complex Banach space (cf. [2, Proposition 1.11]). As 𝑋₁ and 𝑋₂ being given normed complex linear spaces, we will write 𝑋₁↪𝑋₂ if 𝑋₁⊆𝑋₂ and there exists a positive constant 𝐶₀ such that ‖𝑥‖_𝑋2≤𝐶₀‖𝑥‖_𝑋1 for every 𝑥∈𝑋₁. If 𝑋₁↪𝑋₂↪𝑋₁, that is, if 𝑋₁=𝑋₂ and the norms ‖⋅‖_𝑋1 and ‖⋅‖_𝑋2are equivalent, then we will write 𝑋₁≅𝑋₂. Of course, 𝒟(𝐴) with the norm ‖⋅‖_𝒟(𝐴) satisfies 𝒟(𝐴)↪𝑋. In fact, if 𝑥∈𝒟(𝐴), then for every 𝑦∈𝐴𝑥 we have 𝑥=𝐴⁻¹𝑦, so that ‖𝑥‖_𝑋≤‖𝐴⁻¹‖_ℒ(𝑋)‖𝑦‖_𝑋≤𝐶‖𝑦‖_𝑋. Taking the infimum with respect to 𝑦∈𝐴𝑥, we thus find ‖𝑥‖_𝑋≤𝐶‖𝑥‖_𝒟(𝐴) for every 𝑥∈𝒟(𝐴). If 𝑌 is a Banach space, we denote by 𝐶((0,∞);𝑌) the set of all continuos functions from (0,∞) to 𝑌, and for a 𝑌-valued strongly measurable function 𝑔(𝜉), 𝜉∈(0,∞), we set ‖𝑔(𝜉)‖_{𝐿∗𝑞(𝑌)}=(∫^∞₀‖𝑔(𝜉)‖^𝑞_𝑌(d𝜉/𝜉))^1/𝑞, 𝑞∈[1,∞), and ‖𝑔(𝜉)‖_{𝐿∗∞(𝑌)}=sup_{𝜉∈(0,∞)}‖𝑔(𝜉)‖_𝑌. Let 𝑝₀,𝑝₁∈[1,∞) or let 𝑝₀=𝑝₁=∞, and for 𝛾∈(0,1) define 𝑝⁻¹=(1−𝛾)𝑝₀⁻¹+𝛾𝑝₁⁻¹ if 𝑝₀,𝑝₁∈[1,∞) and 𝑝=∞ if 𝑝₀=𝑝₁=∞. Let us set (𝑋,𝒟(𝐴))_𝛾,𝑝=𝑥∈𝑋∶𝑥=𝑣₀(𝜉)+𝑣₁(𝜉)(),𝑣,𝜉∈0,∞₀∈𝐶((0,∞);𝑋),𝑣₁)(𝐴,𝜉∈𝐶((0,∞;𝒟))_𝛾𝑣₀(𝜉)_{𝐿∗𝑝0(𝑋)}+𝜉^𝛾−1𝑣₁(𝜉)_{𝐿∗𝑝1(𝒟(𝐴))},‖𝑥‖<∞_{(𝑋,𝒟(𝐴))𝛾,𝑝}=inf_𝑣0,𝑣1𝜉^𝛾𝑣₀(𝜉)_{𝐿∗𝑝0(𝑋)}+𝜉^𝛾−1𝑣₁(𝜉)_{𝐿∗𝑝1(𝒟(𝐴))}. (19) This characterization of the spaces (𝑋,𝒟(𝐴))_𝛾,𝑝 is that obtained by the so-called “mean-methods”, and it is equivalent to that performed by the “K-method" (cf. [27, Theorem 1.5.2 and Remark 1.5.2/2]) and the “trace-method” (cf. [27, Theorem 1.8.2]). Then, due to [27, Theorem 1.3.3], for every 𝛾∈(0,1) and 𝑝∈[1,∞] the space (𝑋,𝒟(𝐴))_𝛾,𝑝 is an exact real interpolation space of exponent 𝛾 between 𝑋 and 𝒟(𝐴). Observe that by exchanging the role of 𝑋 and 𝒟(𝐴) and performing the transformation 𝜉=𝜏⁻¹, we get (𝑋,𝒟(𝐴))_𝛾,𝑝=(𝒟(𝐴),𝑋)_{1−𝛾,𝑝}. Also, if 𝒟(𝐴)=𝑋, then (𝑋,𝒟(𝐴))_𝛾,𝑝≅𝑋 (cf. [27, Theorem 1.3.3(f)]). The definition of the spaces (𝑋,𝒟(𝐴))_𝛾,𝑝 is meaningful even for the limiting cases (𝛾,𝑝)=(𝑖,∞), 𝑖=0,1, whereas (𝑋,𝒟(𝐴))_𝑖,𝑝, 𝑖=0,1, 𝑝∈[1,∞), reduces to the zero element of 𝑋. In particular (cf. [28, pp. 10–15]), denoting by 𝑌^𝑋 the completion of 𝒟(𝐴) relative to 𝑋 and endowing it with the norm 𝑌‖⋅‖^𝑋 in [28, p. 14], we get (𝑋,𝒟(𝐴))_0,∞≅𝑋 and (𝑋,𝒟(𝐴))_1,∞≅𝑌^𝑋. Let 𝛾₁∈(0,1) and let 𝑝_𝑗∈[1,∞], 𝑗=1,2. Then, for 𝛾₂∈(0,𝛾₁) and 𝑞_𝑗∈[1,𝑝_𝑗], 𝑗=1,2, the following chain of embeddings holds: 𝒟(𝐴)↪(𝑋,𝒟(𝐴))_1,∞↪(𝑋,𝒟(𝐴))_𝛾1,1↪(𝑋,𝒟(𝐴))_𝛾1,𝑞1↪(𝑋,𝒟(𝐴))_𝛾1,𝑝1↪(𝑋,𝒟(𝐴))_𝛾2,1↪(𝑋,𝒟(𝐴))_𝛾2,𝑞2↪(𝑋,𝒟(𝐴))_𝛾2,𝑝2↪𝒟(𝐴). (20) Let 𝛾∈[0,1]. Recall that a Banach space 𝐸 is said to be of class 𝐽(𝛾,𝑋,𝒟(𝐴))∩𝐾(𝛾,𝑋,𝒟(𝐴)) and shortened to 𝐸∈𝐽(𝛾)∩𝐾(𝛾), if 𝐸 is an intermediate space between (𝑋,𝒟(𝐴))_𝛾,∞ and (𝑋,𝒟(𝐴))_𝛾,1, that is, if (𝑋,𝒟(𝐴))_𝛾,1↪𝐸↪(𝑋,𝒟(𝐴))_𝛾,∞. From (20) it thus follows that (𝑋,𝒟(𝐴))_𝛾,𝑝∈𝐽(𝛾)∩𝐾(𝛾), for every 𝛾∈(0,1) and 𝑝∈[1,∞]. Moreover, since (𝑋,𝒟(𝐴))_𝑖,1={0}, 𝑖=0,1, and (𝑋,𝒟(𝐴))_0,∞≅𝑋, we have 𝒟(𝐴)∈𝐽(1)∩𝐾(1) and 𝑋∈𝐽(0)∩𝐾(0). Then (cf. [28, p. 12], [27, Theorem 1.10.2], and [9, Section 1.2.3]), for 𝛾_𝑗∈(0,1) and 𝑝_𝑗∈[1,∞], 𝑗=0,1,2, the reiteration theorem yields (𝑋,𝒟(𝐴))_𝛾1,𝑝1,(𝑋,𝒟(𝐴))_{𝛾2,𝑝2𝛾0,𝑝0}≅(𝑋,𝒟(𝐴))_{(1−𝛾0)𝛾1+𝛾0𝛾2,𝑝0},(𝑋,𝒟(𝐴))_𝛾1,𝑝1,𝒟(𝐴)_𝛾0,𝑝0≅(𝑋,𝒟(𝐴))_{(1−𝛾0)𝛾1+𝛾0,𝑝0},𝑋,(𝑋,𝒟(𝐴))_{𝛾2,𝑝2𝛾0,𝑝0}≅(𝑋,𝒟(𝐴))_{𝛾0𝛾2,𝑝0}. (21) Finally (cf. [29, Theorem 1.II and Remark 1.III]), we recall that if 𝑋₁ and 𝑋₂ are two complex Banach spaces and 𝑇∈ℒ(𝑋₁;𝑋₂) is such that 𝑇∈ℒ(𝑌_1𝑘;𝑌_2𝑘), 𝑌_𝑗𝑘⊆𝑋_𝑗, 𝑗,𝑘=1,2, then 𝑇∈ℒ((𝑌₁₁,𝑌₁₂)_𝛾0,𝑝0;(𝑌₂₁,𝑌₂₂)_𝛾0,𝑝0), 𝛾₀∈(0,1), 𝑝₀∈[1,∞], and ‖𝑇‖_{ℒ((𝑌11,𝑌12)𝛾00,𝑝;(𝑌21,𝑌22)𝛾00,𝑝)}≤‖𝑇‖^1−𝛾0_{ℒ(𝑌11;𝑌21)}‖𝑇‖^𝛾0_{ℒ(𝑌12;𝑌22)}. (22) As a consequence of this general result and the identity (𝑋,𝒟(𝐴))_𝛾1,𝑝1,𝑋_𝛾0,𝑝0=𝑋,(𝑋,𝒟(𝐴))_{𝛾1,𝑝11−𝛾0,𝑝0}, (23) from the third in (21) we find that if 𝑇∈ℒ(𝑋) is such that 𝑇∈ℒ(𝑋;(𝑋,𝒟(𝐴))_𝛾1,𝑝1) and 𝑇∈ℒ((𝑋,𝒟(𝐴))_𝛾2,𝑝2;𝑋), then 𝑇∈ℒ((𝑋,𝒟(𝐴))_{𝛾0𝛾2,𝑝0};(𝑋,𝒟(𝐴))_{(1−𝛾0)𝛾1,𝑝0}), 𝛾_𝑗∈(0,1), 𝑝_𝑗∈[1,∞], 𝑗=0,1,2, and the following estimate holds: ‖𝑇‖_{ℒ((𝑋,𝒟(𝐴))𝛾0𝛾20,𝑝;(𝑋,𝒟(𝐴))010(1−𝛾)𝛾,𝑝)}≤‖𝑇‖^1−𝛾0_{ℒ(𝑋;(𝑋,𝒟(𝐴))𝛾11,𝑝)}‖𝑇‖^𝛾0_{ℒ((𝑋,𝒟(𝐴))𝛾22,𝑝;𝑋)}. (24) Notice that here 𝛾₀𝛾₂+(1−𝛾₀)𝛾₁∈(min{𝛾₁,𝛾₂},max{𝛾₁,𝛾₂})⊊(0,1) for every 𝛾₀∈(0,1). Therefore, if we let 𝛾=𝛾₀𝛾₂ and let 𝛿=(1−𝛾₀)𝛾₁, then 𝛾+𝛿<1, 𝛾₁=𝛿/(1−𝛾₀)>𝛿, and 𝛾₂=𝛾/𝛾₀>𝛾. Hence, in order that the additional inequalities 𝛾_𝑗<1, 𝑗=1,2, are satisfied, we have to choose 𝛾₀∈(𝛾,1−𝛿). As we will see this simple observation will be the key for the proof of the second estimates (90) in the following Proposition 16.

We recall that for every fixed 𝑥∈𝒟(𝐴) the map 𝑇(𝜆)=𝜆𝑥 satisfies ‖𝑇‖_{ℒ(𝐂;𝑋)}=‖𝑥‖_𝑋, ‖𝑇‖_{ℒ(𝐂;𝒟(𝐴))}=‖𝑥‖_𝒟(𝐴) and ‖𝑇‖_{ℒ(𝐂,(𝑋,𝒟(𝐴))𝛾,𝑝)}=‖𝑥‖_{(𝑋,𝒟(𝐴))𝛾,𝑝}. Then (22) with 𝑋₁=𝑌₁₁=𝑌₁₂=𝐂, 𝑋₂=𝑌₂₁=𝑋 and 𝑌₂₂=𝒟(𝐴) yields the interpolation inequality: ‖𝑥‖_{(𝑋,𝒟(𝐴))𝛾,𝑝}≤𝑐₀‖𝑥‖_𝑋^1−𝛾‖𝑥‖^𝛾_𝒟(𝐴),𝑥∈𝒟(𝐴),𝛾∈(0,1),𝑝∈[1,∞], (25) with 𝑐₀ being the positive constant depending on 𝛾 and 𝑝 such that ‖𝜆‖_{(𝐂,𝐂)𝛾,𝑝}≤𝑐₀|𝜆|.

As another application of (22) and for further needs, we also recall that if 𝐴 satisfies (H1), then 𝐴^∘(𝑧𝐼−𝐴)⁻¹ satisfies the estimate (cf. [24, formulae (4.16) and (4.17)]).

Consider 𝐴^∘(𝑧𝐼−𝐴)⁻¹_ℒ(𝑋)≤(𝐶+1)(|𝑧|+1)^1−𝛽,∀𝑧∈Σ_𝛼,𝐴^∘(𝑧𝐼−𝐴)⁻¹_{ℒ(𝒟(𝐴);𝑋)}≤𝐶(|𝑧|+1)^−𝛽,∀𝑧∈Σ_𝛼. (26) From (26), using (22) with 𝑋_𝑗=𝑌_𝑗1=𝑌₂₂=𝑋, 𝑗=1,2, and 𝑌₁₂=𝒟(𝐴), it then follows for every 𝛾∈(0,1) and 𝑝∈[1,∞]𝐴^∘(𝑧𝐼−𝐴)⁻¹_{ℒ((𝑋,𝒟(𝐴))𝛾,𝑝;𝑋)}≤𝑐₁(𝐶+1)^1−𝛾𝐶^𝛾(|𝑧|+1)^{1−𝛽−𝛾},∀𝑧∈Σ_𝛼, (27) where 𝑐₁ is the positive constant depending on 𝛾 and 𝑝 such that ‖𝑥‖_𝑋≤𝑐₁‖𝑥‖_{(𝑋,𝑋)𝛾,𝑝}.

For 𝛾∈(0,1) and 𝑝∈[1,∞] we now define the Banach spaces 𝑋_𝐴^𝛾,𝑝 by 𝑋_𝐴^𝛾,𝑝=𝑥∈𝑋∶[𝑥]_𝑋𝐴^𝛾,𝑝∶=𝜉^𝛾𝐴^∘(𝜉𝐼−𝐴)⁻¹𝑥_{𝐿∗𝑝(𝑋)},‖𝑥‖<∞_{𝑋𝐴𝛾,𝑝}=‖𝑥‖_𝑋+[𝑥]_𝑋𝐴^𝛾,𝑝. (28) It is a well-known fact that if 𝐴 is single-valued and 𝛽=1 in (H1), then (𝑋,𝒟(𝐴))_𝛾,𝑝≅𝑋_𝐴^𝛾,𝑝 (cf. [30, Theorem 3.1] and [27, Theorem 1.14.2]). On the contrary, if 𝛽∈(0,1), then such an equivalence is no longer true, as first observed in [13, Theorem 2] for single-valued operators and, in the case 𝑝=∞, in [2, Theorem 1.12] for the m. l. ones. Recently, extending [13] to m. l. operators and [2] to 𝑝∈[1,∞], in [24, Proposition 4.3] it has been shown that the following embedding relations hold: 𝑋_𝐴^𝛾,𝑝↪(𝑋,𝒟(𝐴))_𝛾,𝑝()[],(),𝛾∈0,1,𝑝∈1,∞𝑋,𝒟(𝐴)_𝛾,𝑝↪𝑋_𝐴^{𝛾+𝛽−1,𝑝},𝛾∈1−𝛽,1,𝑝∈[1,∞]. (29)(30) Then, as in the single-valued case, (𝑋,𝒟(𝐴))_𝛾,𝑝≅𝑋_𝐴^𝛾,𝑝 if 𝛽=1 in (H1). More precisely (see the proof of [24, Proposition 4.3]), if 𝑥∈𝑋_𝐴^𝛾,𝑝, 𝛾∈(0,1), 𝑝∈[1,∞], then ‖𝑥‖_{(𝑋,𝒟(𝐴))𝛾,𝑝}≤2‖𝑥‖_𝑋𝐴^𝛾,𝑝, (31) whereas if 𝑥∈(𝑋,𝒟(𝐴))_𝛾,𝑝, 𝛾∈(1−𝛽,1), 𝑝∈[1,∞], then ‖𝑥‖_𝑋𝐴^{𝛾+𝛽−1,𝑝}≤𝑐₂‖𝑥‖_{(𝑋,𝒟(𝐴))𝛾,𝑝}, (32) with 𝑐₂ being a positive constant depending on 𝛽, 𝛾 and 𝑝.

By setting 𝛿=𝛾+𝛽−1, 𝛾∈(1−𝛽,1), from (30) it follows 𝒟(𝐴)↪(𝑋,𝒟(𝐴))_{1+𝛿−𝛽,𝑝}↪𝑋_𝐴^𝛿,𝑝[].↪𝑋,𝛿∈0,𝛽,𝑝∈1,∞ (33) Then, if 𝛽∈(0,1), the spaces 𝑋_𝐴^𝛿,𝑝, 𝛿∈(0,1), 𝑝∈[1,∞], are intermediate spaces between 𝑋 and 𝒟(𝐴) only for 𝛿∈(0,𝛽), whereas, when 𝛿∈[𝛽,1), they may be smaller than 𝒟(𝐴). In any case, when 𝛽∈(0,1), it is not known if the spaces 𝑋_𝐴^𝛿,𝑝, 𝛿∈(0,𝛽), 𝑝∈[1,∞], are only intermediate or just interpolation spaces between 𝑋 and 𝒟(𝐴).

Notice that [𝑋_𝐴^𝛾,𝑝∩𝐴0]={0}, 𝛾∈(0,1), 𝑝∈[1,∞]. Indeed, assume that there exists 𝑥≠0 such that 𝑥∈[𝑋_𝐴^𝛾,𝑝∩𝐴0] for some 𝛾∈(0,1) and 𝑝∈[1,∞]. Then, since 𝑥∈𝐴0=𝒩((𝑧𝐼−𝐴)⁻¹), 𝑧∈𝜌(𝐴), we have 𝐴^∘(𝜉𝐼−𝐴)⁻¹𝑥=𝜉(𝜉−𝐴)⁻¹𝑥−𝑥=−𝑥 for every 𝜉>0 and [𝑥]_𝑋𝐴^𝛾,𝑝=‖𝜉^𝛾‖_𝐿^∗_𝑝(𝑋)‖𝑥‖_𝑋=∞, contradicting 𝑥∈𝑋_𝐴^𝛾,𝑝. This property plays a key role in the proof of many of the results in [24]. Further, due to (30), it implies that [𝒟(𝐴)∩𝐴0]=[(𝑋,𝒟(𝐴))_𝛾,𝑝∩𝐴0]={0}, 𝛾∈(1−𝛽,1), 𝑝∈[1,∞]. On the contrary, since {0} may be a proper subset of [(𝑋,𝒟(𝐴))_𝛾,𝑝∩𝐴0] for 𝛾∈(0,1−𝛽], 𝛽<1, in general it is not true that [𝒟(𝐴)∩𝐴0]={0}. This is true, instead, if 𝛽=1. In this case the topological direct sum 𝑋₀=𝒟(𝐴)⊕𝐴0 is a closed subspace of 𝑋, and if 𝑋 is reflexive, it coincides with the whole 𝑋 (cf. [3, Theorems 2.4 and 2.6]).

For every 𝛾∈(0,1) and 𝑝∈[1,∞] from (27), (29), and (31) it follows 𝐴^∘(𝑧𝐼−𝐴)⁻¹_{ℒ(𝑋𝐴𝛾,𝑝;𝑋)}≤2𝑐₁(𝐶+1)^1−𝛾𝐶^𝛾(|𝑧|+1)^{1−𝛽−𝛾},∀𝑧∈Σ_𝛼. (34) Hence, for 𝛾∈(0,1) and 𝑝∈[1,∞] we may rewrite (27) and (34) more compactly as 𝐴^∘(𝑧𝐼−𝐴)⁻¹_{ℒ(𝑌𝑝𝛾;𝑋)}≤𝑐₃(|𝑧|+1)^{1−𝛽−𝛾},∀𝑧∈Σ_𝛼, (35) where 𝑌^𝑝_𝛾∈{(𝑋,𝒟(𝐴))_𝛾,𝑝,𝑋_𝐴^𝛾,𝑝} and 𝑐₃ is equal to 𝑐₁(𝐶+1)^1−𝛾𝐶^𝛾 or 2𝑐₁(𝐶+1)^1−𝛾𝐶^𝛾 according that 𝑌^𝑝_𝛾=(𝑋,𝒟(𝐴))_𝛾,𝑝 or 𝑌^𝑝_𝛾=𝑋_𝐴^𝛾,𝑝.

With the exception of the case 𝛽=1, in general it is not clear if embeddings analogous to (20) hold even for the spaces 𝑋_𝐴^𝛾,𝑝. In fact, using (20), (29), and (30) we can only prove that if 𝛾∈(1−𝛽,1) and 1≤𝑞≤𝑝≤∞, then 𝑋_𝐴^𝛾,𝑞↪(𝑋,𝒟(𝐴))_𝛾,𝑞↪(𝑋,𝒟(𝐴))_𝛾,𝑞↪𝑋_𝐴^{𝛾+𝛽−1,𝑝}, (36) whereas if 1−𝛽<𝛾₂<𝛾₁<1 and 𝑝₁,𝑝₂∈[1,∞], then 𝑋^𝛾1,𝑝1_𝐴↪(𝑋,𝒟(𝐴))_𝛾1,𝑝1↪(𝑋,𝒟(𝐴))_𝛾2,𝑝2↪𝑋^{𝛾2+𝛽−1,𝑝2}_𝐴. (37) What can be proved without invoking (20), (29), and (30) and using only the definition of the norm ‖⋅‖_𝑋𝐴^𝛾,𝑝 is instead the following result, which extends to the spaces 𝑋_𝐴^𝛾,𝑝 the embeddings (𝑋,𝒟(𝐴))_𝛾1,𝑝↪(𝑋,𝒟(𝐴))_𝛾2,𝑝, and (𝑋,𝒟(𝐴))_𝛾1,∞↪(𝑋,𝒟(𝐴))_𝛾2,𝑝, 0<𝛾₂<𝛾₁<1, 𝑝∈[1,∞] (cf. (20) with (𝑝₁,𝑝₂)=(𝑝,𝑝) and (𝑝₁,𝑝₂)=(∞,𝑝)).

Proposition 2. Let 𝐴 be an m. l. operator satisfying the resolvent condition (H1). Then the following embeddings hold for every 0<𝛾₂<𝛾₁<1 and 𝑝∈[1,∞]: 𝑋^𝛾1_𝐴^,𝑝↪𝑋^𝛾2_𝐴^,𝑝,𝑋^𝛾1_𝐴^,∞↪𝑋^𝛾2_𝐴^,𝑝. (38)(39)

Proof. If 𝛽=1 in (H1), then there is nothing to prove since (𝑋,𝒟(𝐴))_𝛾,𝑝≅𝑋_𝐴^𝛾,𝑝 and both (38) and (39) follow from (20). Therefore, without loss of generality, we assume that 𝛽∈(0,𝛼] is such that 𝛽<𝛼 if 𝛼=1. We begin by proving (38). Let first 𝑝∈[1,∞). For every 𝑥∈𝑋^𝛾1_𝐴^,𝑝, 0<𝛾₂<𝛾₁<1, we write [𝑥]^𝑝_{𝑋𝛾2𝐴,𝑝}=𝐼₁+𝐼₂, (40) where 𝐼_𝑗=^𝑏𝑗_𝑎𝑗𝜉^𝛾2𝐴^∘(𝜉𝐼−𝐴)⁻¹𝑥^𝑝_𝑋d𝜉𝜉,𝑗=1,2, (41)(𝑎₁,𝑏₁,𝑎₂,𝑏₂)=(0,1,1,∞). Using the first inequality in (26) we find 𝐼₁≤(𝐶+1)^𝑝‖𝑥‖^𝑝_𝑋¹₀𝜉^{𝛾2𝑝−1}(𝜉+1)^{(1−𝛽)𝑝}d𝜉≤2^{(1−𝛽)𝑝}(𝐶+1)^𝑝‖𝑥‖^𝑝_𝑋¹₀𝜉^{𝛾2𝑝−1}d𝜉≤𝑐₄‖𝑥‖_𝑋^𝛾_1𝐴^,𝑝𝑝, (42) where 𝑐₄=2^1−𝛽(𝐶+1)(𝛾₂𝑝)^−1/𝑝. Concerning 𝐼₂, instead, using 𝛾₂−𝛾₁<0, we get 𝐼₂=^∞₁𝜉^{(𝛾2−𝛾1)𝑝}𝜉^𝛾1𝐴^∘(𝜉𝐼−𝐴)⁻¹𝑥^𝑝_𝑋d𝜉𝜉≤^∞₁𝜉^𝛾1𝐴^∘(𝜉𝐼−𝐴)⁻¹𝑥^𝑝_𝑋d𝜉𝜉≤[𝑥]^𝑝_{𝑋𝛾1𝐴,𝑝}≤‖𝑥‖^𝑝_{𝑋𝛾1𝐴,𝑝}. (43) Summing up (40)–(43) and setting 𝑐₅=[(𝑐₄)^𝑝+1]^1/𝑝, it thus follows ‖𝑥‖_𝑋^𝛾_2𝐴^,𝑝=‖𝑥‖_𝑋+[𝑥]_𝑋^𝛾_2𝐴^,𝑝≤(1+𝑐₅)‖𝑥‖_𝑋^𝛾_1𝐴^,𝑝, completing the proof of (38) in the case 𝑝∈[1,∞). Let 𝑝=∞. For every 𝑥∈𝑋^𝛾1_𝐴^,∞, 0<𝛾₂<𝛾₁<1, we write [𝑥]_𝑋^𝛾_2𝐴^,∞=max𝐼₃,𝐼₄, (44) where 𝐼_𝑗=sup_{𝜉∈𝑈𝑗}‖𝜉^𝛾2𝐴^∘(𝜉𝐼−𝐴)⁻¹𝑥‖_𝑋, 𝑗=3,4, 𝑈₃=(0,1), 𝑈₄=[1,∞). Again, the first inequality in (26) yields 𝐼₃≤(𝐶+1)‖𝑥‖_𝑋sup_()𝜉∈0,1𝜉^𝛾2(𝜉+1)^1−𝛽≤2^1−𝛽(𝐶+1)‖𝑥‖_𝑋^𝛾_1𝐴^,∞. (45) Instead, using 𝛾₂−𝛾₁<0, we have 𝐼₄=sup_{𝜉∈[1,∞)}𝜉^{𝛾2−𝛾1}𝜉^𝛾1𝐴^∘(𝜉𝐼−𝐴)⁻¹𝑥_𝑋≤[𝑥]_𝑋^𝛾_1𝐴^,∞≤‖𝑥‖_𝑋^𝛾_1𝐴^,∞. (46) Summing up (44)–(46) and setting 𝑐₆=2^1−𝛽(𝐶+1), we thus find ‖𝑥‖_𝑋^𝛾_2𝐴^,∞=‖𝑥‖_𝑋+[𝑥]_𝑋^𝛾_2𝐴^,∞≤(1+𝑐₆)‖𝑥‖_𝑋^𝛾_1𝐴^,∞. This completes the proof of (38) for the case 𝑝=∞. We now prove (39). Due to (38) with 𝑝=∞, it suffices to assume that 𝑝∈[1,∞). As above, for every 𝑥∈𝑋^𝛾1_𝐴^,∞, 0<𝛾₂<𝛾₁<1, we write [𝑥]^𝑝_{𝑋𝛾2𝐴,𝑝}=𝐼₁+𝐼₂, where 𝐼₁ and 𝐼₂ are defined by (41). Hence, the same computations as in (42) yield 𝐼₁≤𝑐₄‖𝑥‖_𝑋^𝛾_1𝐴^,∞𝑝. (47) As far as 𝐼₂ is concerned, instead, we have 𝐼₂≤[𝑥]^𝑝_{𝑋𝛾1𝐴,∞}^∞₁𝜉^{(𝛾2−𝛾1)𝑝−1}d𝜉≤𝑐₇‖𝑥‖_𝑋^𝛾_1𝐴^,∞𝑝, (48) where 𝑐₇=[(𝛾₁−𝛾₂)𝑝]^−1/𝑝. Summing up (47) and (48) and setting 𝑐₈=[(𝑐₄)^𝑝+(𝑐₇)^𝑝]^1/𝑝, we deduce ‖𝑥‖_𝑋^𝛾_2𝐴^,𝑝≤(1+𝑐₈)‖𝑥‖_𝑋^𝛾_1𝐴^,∞. The proof is complete.

Remark 3. Notice that (37) with 𝑝₁=𝑝₂=𝑝 yields 𝑋^𝛾1_𝐴^,𝑝↪𝑋^𝛾2_𝐴^{+𝛽−1,𝑝}, 1−𝛽<𝛾₂<𝛾₁<1, and this latter embedding is less accurate than (38).

Remark 4. The main problem for extending (20) to the spaces 𝑋_𝐴^𝛾,𝑝 in the case 𝛽<1 is that it is not clear if it holds 𝑋_𝐴^𝛾,𝑞↪𝑋_𝐴^𝛾,𝑝, 1≤𝑞<𝑝≤∞. In fact, the embedding (𝑋,𝒟(𝐴))_𝛾,𝑞↪(𝑋,𝒟(𝐴))_𝛾,𝑞,𝛾∈(0,1),1≤𝑞<𝑝≤∞, (49) is a consequence of the property of the functional 𝐾 entering the definition of the interpolation spaces (𝑋,𝒟(𝐴))_𝛾,𝑝 through the “𝐾-method”, and in particular of its monotonicity (see the proof of [27, Theorem 1.3.3(c),(d)]). With embedding (49) at hands, to derive (20) it thus suffices to prove that (𝑋,𝒟(𝐴))_𝛾1,∞↪(𝑋,𝒟(𝐴))_𝛾2,1, 0<𝛾₂<𝛾₁<1 (see the proof of [27, Theorem 1.3.3(e)] taking there (𝐴₀,𝐴₁,𝜃,𝜃)=(𝒟(𝐴),𝑋,1−𝛾₁,1−𝛾₂) and using (𝒟(𝐴),𝑋)_{1−𝛾,𝑝}=(𝑋,𝒟(𝐴))_𝛾,𝑝). If we try to repeat the proof of (49) for the spaces 𝑋_𝐴^𝛾,𝑝, we will be faced with two problems. The first is that we do not know if the function 𝑔(𝜉)=‖𝐴^∘(𝜉𝐼−𝐴)⁻¹𝑥‖_𝑋, 𝜉∈(0,∞), 𝑥∈𝑋, is monotone decreasing, which would allow us to prove 𝑋_𝐴^𝛾,𝑝↪𝑋_𝐴^𝛾,∞, 𝛾∈(0,1), 𝑝∈[1,∞). For if 𝑔(𝜉) was monotone decreasing, then for every 𝜉∈(0,∞) and 𝑥∈𝑋_𝐴^𝛾,𝑝, 𝛾∈(0,1), 𝑝∈[1,∞), we would find 𝜉^𝛾𝑔(𝜉)=𝑐₉^𝜉₀𝜇^𝛾𝑝d𝜇𝜇^1/𝑝𝑔(𝜉)≤𝑐₉^𝜉₀𝜇^𝛾𝑔𝜇^𝑝d𝜇𝜇^1/𝑝≤𝑐₉[𝑥]_𝑋𝐴^𝛾,𝑝, (50) where 𝑐₉=(𝛾𝑝)^−1/𝑝. Taking the supremum with respect to 𝜉∈(0,∞) in the latter inequality, we would get [𝑥]_𝑋𝐴^𝛾,∞≤𝑐₉[𝑥]_𝑋𝐴^𝛾,𝑝, proving 𝑋_𝐴^𝛾,𝑝↪𝑋_𝐴^𝛾,∞, 𝛾∈(0,1), 𝑝∈[1,∞). The second problem is that the function 𝜉^𝛾𝑔(𝜉) is not necessarily bounded for 𝑥∈𝑋_𝐴^𝛾,𝑝, 𝛾∈(0,1), 𝑝∈[1,∞), precluding us to prove 𝑋_𝐴^𝛾,𝑞↪𝑋_𝐴^𝛾,𝑝, 𝛾∈(0,1), 𝑞∈[1,𝑝). Indeed, from (35) we can only find 𝜉^𝛾𝑔(𝜉)≤𝑐₃𝜉^𝛾(𝜉+1)^{1−𝛽−𝛾}‖𝑥‖_𝑋𝐴^𝛾,𝑝, and when 𝛽<1, the right-hand side of this inequality goes to infinity as 𝜉 goes to infinity. On the contrary, if 𝜉^𝛾𝑔(𝜉) were bounded, then for every 1≤𝑞<𝑝<∞ we would obtain [𝑥]^𝑝_{𝑋𝐴𝛾,𝑝}=^∞₀𝜉^𝛾𝑔(𝜉)^𝑝d𝜉𝜉≤sup_{()𝜉∈0,∞}𝜉^𝛾𝑔(𝜉)^{𝑝−𝑞∞}₀𝜉^𝛾𝑔(𝜉)^𝑞d𝜉𝜉=[𝑥]_𝑋^𝑝−𝑞_𝐴𝛾,∞[𝑥]^𝑝_{𝑋𝐴𝛾,𝑝}. (51) If now in addition 𝑔(𝜉) were also monotone decreasing, in order that [𝑥]_𝑋𝐴^𝛾,∞≤𝑐₉[𝑥]_𝑋𝐴^𝛾,𝑞, from the latter inequality we would get [𝑥]_𝑋𝐴^𝛾,𝑝≤(𝑐₉)^{(𝑝−𝑞)/𝑝}[𝑥]_𝑋𝐴^𝛾,𝑞, completing the proof of 𝑋_𝐴^𝛾,𝑞↪𝑋_𝐴^𝛾,𝑝, 𝛾∈(0,1), 1≤𝑞<𝑝<∞. Due to the former computations, we can thus conclude that in the case 𝛽<1 the quoted problems are the main obstacles which prevent us to extend (49) and, as its consequence, (20) to the spaces 𝑋_𝐴^𝛾,𝑝.

Remark 5. Let 0<𝛾₂<𝛾₁<1 be fixed and for every 𝑝∈[1,∞] and let us set 𝐴_𝑝=𝑋^𝛾2_𝐴^,𝑝 and 𝐵_𝑝=𝑋^𝛾1_𝐴^,𝑝. We thus have the two families of sets 𝒜={𝐴_𝑝}_{𝑝∈[1,∞]} and ℬ={𝐵_𝑝}_{𝑝∈[1,∞]}. Let first 𝛽=1. In this case, since (𝑋,𝒟(𝐴))_𝛾,𝑝≅𝑋_𝐴^𝛾,𝑝, from (20) we deduce that the sets 𝐴_𝑝 and 𝐵_𝑝 are related by the following inclusions in which 1<𝑞₁<𝑞₂<∞: 𝐵₁⊆𝐵_𝑞1⊆𝐵_𝑞2⊆𝐵_∞⊆𝐴₁⊆𝐴_𝑞1⊆𝐴_𝑞2⊆𝐴_∞. (52) Now let 𝛽<1. As observed in Remark 4, in this case the embedding 𝑋_𝐴^𝛾,𝑞↪𝑋_𝐴^𝛾,𝑝, 1≤𝑞<𝑝≤∞, may be not satisfied and the chain of inclusions (52) could not take place. However, (38) and (39) hold true and for every 𝑝∈[1,∞], and we have 𝐵_𝑝⊆𝐴_𝑝 and 𝐵_∞⊆𝐴_𝑝.

We have already pointed out that {e^𝑡𝐴}_𝑡≥0 may be not strongly continuous in the 𝑋-norm on 𝒟(𝐴). On the contrary, the following result (cf. [24, Proposition 5.2] for the proof) shows thatthe things are finer on (𝑋,𝒟(𝐴))_𝛾,𝑝 and 𝑋_𝐴^𝛾,𝑝. Later, we will need this fact.

Proposition 6. Let 𝐴 be as in Proposition 2. If 𝛾∈(1−𝛽,1); then {𝑒^𝑡𝐴}_𝑡≥0 is strongly continuous in the 𝑋-norm on 𝑌^𝑝_𝛾∈{(𝑋,𝒟(𝐴))_𝛾,𝑝,𝑋_𝐴^𝛾,𝑝} for every 𝑝∈[1,∞].

We conclude the section listing some estimates for the operators [(−𝐴)^𝜃]^∘e^𝑡𝐴 defined by (11) with respect to the spaces (𝑋,𝒟(𝐴))_𝛾,𝑝 and 𝑋_𝐴^𝛾,𝑝. First, in [19, Lemma 3.1] it is shown that [(−𝐴)^𝜃]^∘e^𝑡𝐴𝑥∈𝒟(𝐴) for every 𝑥∈𝑋 and that the estimate ‖[(−𝐴)^𝜃]^∘e^𝑡𝐴𝑥‖_𝒟(𝐴)≤‖[(−𝐴)^𝜃+1]^∘e^𝑡𝐴𝑥‖_𝑋 is satisfied. Hence, using (14), we get (−𝐴)^𝜃∘e^𝑡𝐴_{ℒ(𝑋;𝒟(𝐴))}≤𝑐̃_{𝛼,𝛽,𝜃+1}𝑡^{(𝛽−ℜe𝜃−2)/𝛼},ℜe𝜃≥0,𝑡>0. (53) Combining (14) and (53) with (25) and letting 𝑐₁₀=𝑐₀(𝑐_{𝛼,𝛽,𝜃})^1−𝛾(𝑐_{𝛼,𝛽,𝜃+1})^𝛾, it thus follows (cf. [19, Proposition 3.1]) that for every 𝛾∈(0,1) and 𝑝∈[1,∞] the following estimate holds: (−𝐴)^𝜃∘e^𝑡𝐴_{ℒ(𝑋;(𝑋,𝒟(𝐴))𝛾,𝑝)}≤𝑐₁₀𝑡^{(𝛽−𝛾−ℜe𝜃−1)/𝛼},ℜe𝜃≥0,𝑡>0. (54)

Remark 7. We stress that if 𝛽<1, then we can not derive an estimate for the ℒ(𝑋;𝑋_𝐴^𝛾,𝑝)-norm of [(−𝐴)^𝜃]^∘e^𝑡𝐴 simply by replacing (𝑋,𝒟(𝐴))_𝛾,𝑝 with 𝑋_𝐴^𝛾,𝑝 in (54). This is for two reasons. First, when 𝛾∈[𝛽,1), we are not assured that [(−𝐴)^𝜃]^∘e^𝑡𝐴𝑥∈𝑋_𝐴^𝛾,𝑝 for every 𝑥∈𝑋. For if 𝛾∈[𝛽,1), then the space 𝑋_𝐴^𝛾,𝑝 may be smaller than the domain 𝒟(𝐴) to which [(−𝐴)^𝜃]^∘e^𝑡𝐴𝑥 belongs by virtue of [19, Lemma 3.1]. The second reason is that, even limiting to 𝛾∈(0,𝛽) in order that 𝒟(𝐴)↪𝑋_𝐴^𝛾,𝑝, from (31) we only get ‖[(−𝐴)^𝜃]^∘e^𝑡𝐴𝑥‖_{(𝑋,𝒟(𝐴))𝛾,𝑝}≤2‖[(−𝐴)^𝜃]^∘e^𝑡𝐴𝑥‖_𝑋𝐴^𝛾,𝑝, 𝑥∈𝑋, and we do not know if the right-hand side can be bounded from above by some constant times 𝑡^{(𝛽−𝛾−ℜe𝜃−1)/𝛼}‖𝑥‖_𝑋. Of course, we can employ (32), but in this way all that we can reach is the estimate (−𝐴)^𝜃∘e^𝑡𝐴_{ℒ(𝑋;𝑋𝐴𝛾+𝛽−1,𝑝)}≤𝑐₁₁𝑡^{(𝛽−𝛾−ℜe𝜃−1)/𝛼},ℜe𝜃≥0,𝑡>0, (55) where 𝑐₁₁=𝑐₂𝑐₁₀, 𝛾∈(1−𝛽,1) and 𝑝∈[1,∞]. Letting 𝛿=𝛾+𝛽−1, (55) can be rewritten equivalently as (−𝐴)^𝜃∘e^𝑡𝐴_{ℒ(𝑋;𝑋𝐴𝛿,𝑝)}≤𝑐₁₁𝑡^{(2𝛽−𝛿−ℜe𝜃−2)/𝛼},ℜe𝜃≥0,𝑡>0, (56) where 𝛿∈(0,𝛽) and 𝑝∈[1,∞]. When 𝛽<1, there are good motivations to believe that estimate (56) is not the best one. In fact, for instance, when (𝜃,𝑝)=(0,∞), (56) leads us to an estimate which is rougher than the estimate e^𝑡𝐴_{ℒ(𝑋;𝑋𝐴𝛿,∞)}≤𝑐₁₂𝑡^{(𝛽−𝛿−1)/𝛼},𝛿∈(0,1),𝑡>0, (57) as shown in [2, Proposition 3.2], with 𝑐₁₂ being a positive constant depending on 𝛼, 𝛽, and 𝛿. Also, (57) ensures that e^𝑡𝐴𝑥, 𝑥∈𝑋, belongs to 𝑋_𝐴^𝛿,∞ for every 𝛿∈(0,1) and not only for 𝛿∈(0,𝛽) as (56) suggests. Furthermore, due to (31), estimate (57) yields (54) with (𝜃,𝛾,𝑝)=(0,𝛿,∞). This leads us to believe that (57) can be improved and that estimate (54) holds the same if 𝑋_𝐴^𝛾,∞ is taken in place of (𝑋,𝒟(𝐴))_𝛾,∞.

Now let 𝑌^𝑝_𝛾∈{(𝑋,𝒟(𝐴))_𝛾,𝑝,𝑋_𝐴^𝛾,𝑝}, 𝛾∈(0,1), 𝑝∈[1,∞]. As far as the estimates for the ℒ(𝑌^𝑝_𝛾;𝑋)-norm of operators [(−𝐴)^𝜃]^∘e^𝑡𝐴 are concerned, instead, at the moment only the following estimates for the case 𝜃=1 are available (cf. [24, Lemma 5.1]): (−𝐴)^1∘e^𝑡𝐴_{ℒ(𝑌𝑝𝛾;𝑋)}≤𝑐₁₃𝑡^{(𝛽+𝛾−2)/𝛼},𝑡>0,𝛾∈(0,1),𝑝∈[1,∞], (58) with 𝑐₁₃ being a positive constant depending on 𝛼, 𝛽, 𝛾, and 𝑝. Estimates (58) are successfully applied in [24, Corollary 5.4] to prove that if 𝛼+𝛽>1, then the map 𝑡→e^𝑡𝐴 is Hölder continuous from [0,∞) to ℒ(𝑌^𝑝_𝛾;𝑋), 𝛾∈(2−𝛼−𝛽,1), 𝑝∈[1,∞], with Hölder exponent 𝜎=(𝛼+𝛽+𝛾−2)/𝛼. In Section 3 we will extend (58), proving some estimates for the ℒ(𝑌^𝑝_𝛾;𝑋)-norm of [(−𝐴)^𝜃]^∘e^𝑡𝐴, ℜe𝜃≥1, which reduce to (58) in the case 𝜃=1.

Remark 8. Observe that an estimate for the norm ‖[(−𝐴)^𝜃]^∘e^𝑡𝐴‖_ℒ(𝑌^𝑝_𝛾;𝑋), ℜe𝜃≥1, 𝑡>0, 𝑌^𝑝_𝛾∈{(𝑋,𝒟(𝐴))_𝛾,𝑝,𝑋_𝐴^𝛾,𝑝}, 𝛾∈(0,1), 𝑝∈[1,∞], can be obtained combining (14), (15), and (58). Indeed, using (15), for every ℜe𝜃≥1, 𝑡>0 and 𝑥∈𝑌^𝑝_𝛾, we have (−𝐴)^𝜃∘e^𝑡𝐴𝑥_𝑋=(−𝐴)^𝜃−1∘e^(𝑡/2)𝐴(−𝐴)^1∘e^(𝑡/2)𝐴𝑥_𝑋≤(−𝐴)^𝜃−1∘e^(𝑡/2)𝐴_ℒ(𝑋)(−𝐴)^1∘e^(𝑡/2)𝐴𝑥_𝑋. (59) Therefore, due to (14) and (58), from (59) we deduce that (−𝐴)^𝜃∘e^𝑡𝐴_{ℒ(𝑌𝑝𝛾;𝑋)}≤𝑐₁₄𝑡^{(2𝛽+𝛾−ℜe𝜃−2)/𝛼},ℜe𝜃≥1,𝑡>0, (60) where 𝛾∈(0,1), 𝑝∈[1,∞] and 𝑐₁₄=2^{(2+ℜe𝜃−𝛾−2𝛽)/𝛼}𝑐̃_{𝛼,𝛽,𝜃−1}𝑐₁₃. As we will see in the next section estimate (60) is not optimal, in the sense that the negative exponent (2𝛽+𝛾−ℜe𝜃−2)/𝛼 can be refined; of course, unless 𝛽=1. The main reason to believe that (60) can be improved is that its derivation consists of two steps: the first in which [(−𝐴)^𝜃]^∘e^𝑡𝐴 is decomposed with the help of (15), and the second in which (60) is obtained combining estimates of very different nature, such as (14) and (58). It is thus to be expected that in this double step derivation some regularity goes missing and that a better result can be reached analyzing more detailedly [(−𝐴)^𝜃]^∘e^𝑡𝐴𝑥 for 𝑥∈𝑌^𝑝_𝛾.

3. Behaviour of [(−𝐴)^𝜃]^∘e^𝑡𝐴 in (𝑋,𝒟(𝐴))_𝛾,𝑝 and 𝑋_𝐴^𝛾,𝑝

According to Remark 7 we begin by improving (54), showing that the same estimate holds with (𝑋,𝒟(𝐴))_𝛾,𝑝 being replaced by 𝑋_𝐴^𝛾,∞ if 𝑝=∞ and by 𝑋_𝐴^{𝛽𝛾,𝑝} if 𝑝∈[1,∞). Throughout this and the next section, 𝐴 will be an m. l. operator in 𝑋 having nonempty domain 𝒟(𝐴) and satisfying the resolvent condition (H1) of Section 2.

Proposition 9. Let ℜe𝜃≥0, 𝛾∈(0,1) and let 𝑝∈[1,∞]. Then, there exist positive constants 𝑐_𝑗, 𝑗=15,16, depending on 𝛼, 𝛽, 𝛾, 𝜃, and 𝑝 such that (−𝐴)^𝜃∘𝑒^𝑡𝐴_{ℒ(𝑋;𝑋𝐴𝛾,∞)}≤𝑐₁₅𝑡^{(𝛽−𝛾−ℜe𝜃−1)/𝛼},)𝑡>0,𝑝=∞,(−𝐴^𝜃∘𝑒^𝑡𝐴_{ℒ(𝑋;𝑋𝐴𝛽𝛾,𝑝)}≤𝑐₁₆𝑡^{(𝛽−𝛾−ℜe𝜃−1)/𝛼},𝑡>0,𝑝∈[1,∞). (61)(62)

Proof. If 𝛽=1, then (𝑋,𝒟(𝐴))_𝛾,𝑝≅𝑋_𝐴^𝛾,𝑝 and (61) and (62) with 𝑐_𝑗=𝑐₂𝑐₁₀, 𝑗=15,16, follow by taking 𝛽=1 in (32) and (54). Therefore, without the loss of generality, we assume that 𝛽∈(0,𝛼] is such that 𝛽<𝛼 if 𝛼=1. Let 𝜃∈𝐂, ℜe𝜃≥0, 𝛾∈(0,1), and 𝑝∈[1,∞) be fixed and let 𝑥 be an arbitrary element of 𝑋. Then, for every 𝑡>0 we have (−𝐴)^𝜃∘e^𝑡𝐴𝑥_{𝑋𝐴𝛾,∞}=(−𝐴)^𝜃∘e^𝑡𝐴𝑥_𝑋+𝜉^𝛾𝐴^∘(𝜉𝐼−𝐴)⁻¹(−𝐴)^𝜃∘e^𝑡𝐴𝑥_{𝐿∗∞(𝑋)},(−𝐴)^𝜃∘e^𝑡𝐴𝑥_{𝑋𝐴𝛽𝛾,𝑝}=(−𝐴)^𝜃∘e^𝑡𝐴𝑥_𝑋+𝜉^𝛽𝛾𝐴^∘(𝜉𝐼−𝐴)⁻¹(−𝐴)^𝜃∘e^𝑡𝐴𝑥_{𝐿∗𝑝(𝑋)}. (63)(64) Of course, from estimate (54) we find (−𝐴)^𝜃∘e^𝑡𝐴𝑥_𝑋≤𝑐_𝛾,𝑝𝑐₁₀‖𝑥‖_𝑋𝑡^{(𝛽−𝛾−ℜe𝜃−1)/𝛼},𝑡>0, (65) with 𝑐_𝛾,𝑝 being such that ‖𝑦‖_𝑋≤𝑐_𝛾,𝑝‖𝑦‖_{(𝑋,𝒟(𝐴))𝛾,𝑝}, 𝑦∈(𝑋,𝒟(𝐴))_𝛾,𝑝, 𝑝∈[1,∞]. It thus suffices to investigate only the second terms on the right-hand side of (63) and (64). We begin by proving (61). First, using the second identity in (6), for every 𝜉∈(0,∞) we get 𝜉^𝛾𝐴^∘(𝜉𝐼−𝐴)⁻¹(−𝐴)^𝜃∘e^𝑡𝐴𝑥=12𝜋𝑖_Γ𝜉^𝛾(−𝜆)^𝜃e^𝑡𝜆𝐴^∘(𝜉𝐼−𝐴)⁻¹(𝜆𝐼−𝐴)⁻¹𝑥d𝜆=𝜉^𝛾12𝜋𝑖_Γ(−𝜆)^𝜃e^𝑡𝜆(𝜆−𝜉)⁻¹d𝜆𝐴^∘(𝜉𝐼−𝐴)⁻¹𝑥−12𝜋𝑖_Γ𝜉^𝛾(−𝜆)^𝜃e^𝑡𝜆(𝜆−𝜉)⁻¹𝐴^∘(𝜆𝐼−𝐴)⁻¹1𝑥d𝜆=−2𝜋𝑖_Γ𝜉^𝛾(−𝜆)^𝜃e^𝑡𝜆(𝜆−𝜉)⁻¹𝜆(𝜆𝐼−𝐴)⁻¹=1−𝐼𝑥d𝜆2𝜋𝑖_Γ𝜉^𝛾(−𝜆)^𝜃+1e^𝑡𝜆(𝜆−𝜉)⁻¹(𝜆𝐼−𝐴)⁻¹𝑥d𝜆. (66) Here we have used twice the equality ∫_Γ(−𝜆)^𝜃e^𝑡𝜆(𝜆−𝜉)⁻¹d𝜆=0, 𝜉∈(0,∞), which follows from Cauchy’s formula after having enclosed Γ on the left with an arc of the circle {𝑧∈𝐂∶|𝑧+𝑐|=𝑅}, 𝑅>0, and letting 𝑅 to infinity. From (66), using ‖(𝜆𝐼−𝐴)⁻¹‖_ℒ(𝑋)≤𝐶(|𝜆|+1)^−𝛽≤𝐶|𝜆|^−𝛽, 𝜆∈Σ_𝛼, it follows that 𝜉^𝛾𝐴^∘(𝜉𝐼−𝐴)⁻¹(−𝐴)^𝜃∘e^𝑡𝐴𝑥_𝑋≤𝐶(2𝜋)⁻¹‖𝑥‖_𝑋×_Γ𝜉^𝛾|𝜆|^{1+ℜe𝜃−𝛽}e^{−ℑm𝜃arg(−𝜆)}e^𝑡ℜe𝜆𝜆−𝜉⁻¹||()d𝜆≤𝐶2𝜋⁻¹e^{(𝜋/2)|ℑm𝜃|}‖𝑥‖_𝑋×_Γ𝜉|𝜆|^𝛾|𝜆|^{𝛾+ℜe𝜃−𝛽}e^𝑡ℜe𝜆𝜉1−𝜆⁻¹|d𝜆|. (67) Now, since ℜe𝜆≤−𝑐<0 for every 𝜆∈Γ and since 𝜉∈(0,∞), we have 𝜉1−𝜆=𝜉1−𝜆|𝜆|²=𝜉1+|𝜆|²−2𝜉ℜe𝜆|𝜆|^21/2≥𝜉1+|𝜆|^21/2. (68) Therefore, for every 𝜆∈Γ and 𝜉∈(0,∞) the following inequality holds: 𝜉|𝜆|^𝛾𝜉1−𝜆⁻¹≤𝜉|𝜆|^𝛾𝜉1+|𝜆|^2−1/2≤𝛾^𝛾/21−𝛾^(1−𝛾)/2=∶𝑐_𝛾, (69) where we have used the fact that the function 𝑓(𝑠)=𝑠^𝛾(1+𝑠²)^−1/2, 𝑠≥0, 𝛾∈(0,1), attains its maximum value 𝑐_𝛾 at the point 𝑠_𝛾=𝛾^1/2(1−𝛾)^−1/2. Coming back to (67) and setting 𝑐₁₇=𝐶(2𝜋)⁻¹e^{(𝜋/2)|ℑm𝜃|}𝑐_𝛾, we thus find (here we use also that on Γ it holds |𝜆|≥𝑐, so that ℜe𝜆=−𝑐(|ℑm𝜆|+1)^𝛼≥−𝑐(1+𝑐⁻¹)^𝛼|𝜆|^𝛼): 𝜉^𝛾𝐴^∘(𝜉𝐼−𝐴)⁻¹(−𝐴)^𝜃∘e^𝑡𝐴𝑥_𝑋≤𝑐₁₇‖𝑥‖_𝑋Γ|𝜆|^{𝛾+ℜe𝜃−𝛽}e^𝑡ℜe𝜆|d𝜆|≤𝑐₁₇‖𝑥‖_𝑋Γ|𝜆|^{𝛾+ℜe𝜃−𝛽}e^{−𝑐(1+𝑐−1)𝛼𝑡|𝜆|𝛼}|d𝜆|≤2𝑐₁₇‖𝑥‖_𝑋^∞₀𝜇^{𝛾+ℜe𝜃−𝛽}e^{−𝑐𝛼𝑡𝜇𝛼}d𝜇, (70) where 𝑐_𝛼=𝑐(1+𝑐⁻¹)^𝛼. Finally, taking the supremum with respect to 𝜉∈(0,∞) in (70) and performing the transformation 𝑐_𝛼𝑡𝜇^𝛼=𝑠 in the integral on the right, we obtain 𝜉^𝛾𝐴^∘(𝜉𝐼−𝐴)⁻¹(−𝐴)^𝜃∘e^𝑡𝐴𝑥_{𝐿∗∞(𝑋)}≤𝑐₁₈‖𝑥‖_𝑋𝑡^{(𝛽−𝛾−ℜe𝜃−1)/𝛼}, (71) where 𝑐₁₈=2𝑐₁₇𝛼⁻¹𝑐^{(𝛽−𝛾−ℜe}_𝛼^{𝜃−1)/𝛼}𝐸((𝛾+ℜe𝜃+1−𝛽)/𝛼), 𝐸(𝜒), 𝜒>0, being the Euler gamma function ∫^∞₀𝑠^𝜒−1e^−𝑠d𝑠. Then, summing up (65) and (71), from (63) it follows that (−𝐴)^𝜃∘e^𝑡𝐴𝑥_{𝑋𝐴𝛾,∞}≤𝑐_𝛾,∞𝑐₁₀+𝑐₁₈‖𝑥‖_𝑋𝑡^{(𝛽−𝛾−ℜe𝜃−1)/𝛼},ℜe𝜃≥0,𝑡>0. (72) Since 𝑥∈𝑋 was arbitrary, this completes the proof of (61) with 𝑐₁₅=𝑐_𝛾,∞𝑐₁₀+𝑐₁₈. Let us now prove (62). For every 𝑝∈[1,∞) we write 𝜉^𝛽𝛾𝐴^∘(𝜉𝐼−𝐴)⁻¹(−𝐴)^𝜃∘e^𝑡𝐴𝑥^𝑝_{𝐿∗𝑝(𝑋)}=𝐼₁+𝐼₂, (73) where 𝐼_𝑗=∫^𝑏𝑗_𝑎𝑗‖𝜉^𝛽𝛾𝐴^∘(𝜉𝐼−𝐴)⁻¹[(−𝐴)^𝜃]^∘e^𝑡𝐴𝑥‖^𝑝_𝑋(d𝜉/𝜉), 𝑗=1,2, (𝑎₁,𝑏₁,𝑎₂,𝑏₂)=(0,1,1,∞). First, (35) with 𝑌^𝑝_𝛾=(𝑋,𝒟(𝐴))_𝛾,𝑝 yields 𝐼₁≤(−𝐴)^𝜃∘e^𝑡𝐴𝑥^𝑝_{(𝑋,𝒟(𝐴))𝛾,𝑝}¹₀𝜉^{𝛽𝛾𝑝−1}𝑐₃(𝜉+1)^{1−𝛽−𝛾𝑝}d𝜉. (74) Therefore, since (𝜉+1)^{1−𝛽−𝛾}≤𝑐_𝛽,𝛾 for every 𝜉∈(0,1], where 𝑐_𝛽,𝛾=2^{1−𝛽−𝛾} or 𝑐_𝛽,𝛾=1 according that 𝛾∈(0,1−𝛽) or 𝛾∈[1−𝛽,1), from (54), we deduce that 𝐼₁≤𝑐_𝛽,𝛾𝑐₃^𝑝(−𝐴)^𝜃∘e^𝑡𝐴𝑥^𝑝_{(𝑋,𝒟(𝐴))𝛾,𝑝}¹₀𝜉^{𝛽𝛾𝑝−1}=𝑐d𝜉₁₉‖𝑥‖_𝑋𝑡^{(𝛽−𝛾−ℜe𝜃−1)/𝛼𝑝}, (75) with 𝑐₁₉=𝑐_𝛽,𝛾𝑐₃𝑐₁₀(𝛽𝛾𝑝)^−1/𝑝. As far as 𝐼₂ is concerned, exploiting (71) and recalling that we have assumed 𝛽<1, we obtain 𝐼₂=^∞₁𝜉^{(𝛽−1)𝛾𝑝}𝜉^𝛾𝐴^∘(𝜉𝐼−𝐴)⁻¹(−𝐴)^𝜃∘e^𝑡𝐴𝑥^𝑝_𝑋d𝜉𝜉≤𝑐₁₈‖𝑥‖_𝑋𝑡^{(𝛽−𝛾−ℜe𝜃−1)/𝛼𝑝∞}₁𝜉^{(𝛽−1)𝛾𝑝−1}≤𝑐d𝜉₂₀‖𝑥‖_𝑋𝑡^{(𝛽−𝛾−ℜe𝜃−1)/𝛼𝑝}, (76) where 𝑐₂₀=𝑐₁₈[(1−𝛽)𝛾𝑝]^−1/𝑝. Summing up (73)–(76), it thus follows that 𝜉^𝛽𝛾𝐴^∘(𝜉𝐼−𝐴)⁻¹(−𝐴)^𝜃∘e^𝑡𝐴𝑥_{𝐿∗𝑝(𝑋)}≤𝑐₂₁‖𝑥‖_𝑋𝑡^{(𝛽−𝛾−ℜe𝜃−1)/𝛼}, (77) where 𝑐₂₁=[(𝑐₁₉)^𝑝+(𝑐₂₀)^𝑝]^1/𝑝. Finally, (65) and (77) lead us to (−𝐴)^𝜃∘e^𝑡𝐴𝑥_{𝑋𝐴𝛽𝛾,𝑝}≤𝑐_𝛾,𝑝𝑐₁₀+𝑐₂₁‖𝑥‖_𝑋𝑡^{(𝛽−𝛾−ℜe𝜃−1)/𝛼},ℜe𝜃≥0,𝑡>0. (78) Since 𝑥∈𝑋 was arbitrary, this completes the proof of (62) with 𝑐₁₆=𝑐_𝛾,𝑝𝑐₁₀+𝑐₂₁.

Remark 10. If 𝜃=0, then (61) is precisely the estimate (57). In this sense our result improves [2] and shows that (54) holds the same with (𝑋,𝒟(𝐴))_𝛾,𝑝 being replaced with 𝑋_𝐴^𝛾,∞ if 𝑝=∞ and 𝑋_𝐴^{𝛽𝛾,𝑝} and if 𝑝∈[1,∞). Also, when 𝛽<1, (61) and (62) are in two aspects better than the estimate (55) deduced from (54) with the help of (32). First, here we do not need to restrict 𝛾 to (1−𝛽,1). Further, despite limiting 𝛾 to (1−𝛽,1), (61) and (62) show that [(−𝐴)^𝜃]^∘e^𝑡𝐴𝑥, ℜe𝜃≥0, 𝑡>0, 𝑥∈𝑋, enjoys more regularity than that predicted by (55). For, since when 𝛽<1 it holds 0<𝛾+𝛽−1<𝛽𝛾<𝛾, from (38) and (39) it follows 𝑋_𝐴^𝛾,∞↪𝑋_𝐴^{𝛽𝛾,𝑝}↪𝑋_𝐴^{𝛾+𝛽−1,𝑝}, 𝑝∈[1,∞].

Remark 11. We recall that when 𝛽<1 the spaces 𝑋_𝐴^𝜎,𝑝, 𝜎∈(0,1), 𝑝∈[1,∞], are intermediate spaces between 𝑋 and 𝒟(𝐴) for 𝜎∈(0,𝛽), but they may be contained in 𝒟(𝐴) for 𝜎∈[𝛽,1). Therefore, whereas (61) is satisfied for spaces 𝑋_𝐴^𝜎,∞ eventually smaller than 𝒟(𝐴), for (62) to hold we have to consider only spaces 𝑋_𝐴^𝜎,𝑝, 𝑝∈[1,∞), bigger than 𝒟(𝐴). In fact, letting 𝜎=𝛽𝛾, we have 𝜎∈(0,𝛽) for every 𝛾∈(0,1).

In accordance with Remark 8 we now improve estimate (58).

Proposition 12. Let ℜe𝜃≥1, 𝛾∈(0,1), 𝑝∈[1,∞] and let 𝑌^𝑝_𝛾∈{(𝑋,𝒟(𝐴))_𝛾,𝑝,𝑋_𝐴^𝛾,𝑝}. Then, there exists a positive constant 𝑐₂₂ depending on 𝛼, 𝛽, 𝛾, 𝜃, and 𝑝 such that (−𝐴)^𝜃∘𝑒^𝑡𝐴_{ℒ(𝑌𝑝𝛾;𝑋)}≤𝑐₂₂𝑡^{(𝛽+𝛾−ℜe𝜃−1)/𝛼},𝑡>0. (79)

Proof. First, using the identity 𝐴^∘(𝑧𝐼−𝐴)⁻¹=𝑧(𝑧𝐼−𝐴)⁻¹−𝐼, 𝑧∈Σ_𝛼, for every 𝑥∈𝑋, we rewrite [(−𝐴)^𝜃]^∘e^𝑡𝐴𝑥, ℜe𝜃≥0, in the following way: (−𝐴)^𝜃∘e^𝑡𝐴𝑥1=−2𝜋𝑖_Γ(−𝜆)^𝜃−1e^𝑡𝜆𝜆(𝜆𝐼−𝐴)⁻¹1𝑥d𝜆=−2𝜋𝑖_Γ(−𝜆)^𝜃−1e^𝑡𝜆𝐴^∘(𝜆𝐼−𝐴)⁻¹1𝑥+𝐼𝑥d𝜆=−2𝜋𝑖_Γ(−𝜆)^𝜃−1e^𝑡𝜆𝐴^∘(𝜆𝐼−𝐴)⁻¹𝑥d𝜆,𝑡>0. (80) Here we have used ∫_Γ(−𝜆)^𝜃−1e^𝑡𝜆d𝜆=0, which follows from the Cauchy formula applied to (−𝜆)^𝜃e^𝑡𝜆 after having enclosed Γ on the left with an arc of the circle {𝑧∈𝐂∶|𝑧+𝑐|=𝑅}, 𝑅>0, and letting 𝑅 to infinity. Let now 𝜃∈𝐂, ℜe𝜃≥1, 𝛾∈(0,1), and 𝑝∈[1,∞] be fixed and let 𝑥 be an arbitrary element of 𝑌^𝑝_𝛾. From (35) it then follows that (−𝐴)^𝜃∘e^𝑡𝐴𝑥_𝑋≤c₂₃‖𝑥‖_𝑌^𝑝_𝛾Γ|𝜆|^ℜe𝜃−1e^𝑡ℜe𝜆(|𝜆|+1)^{1−𝛽−𝛾}|d𝜆|,𝑡>0, (81) where 𝑐₂₃=(2𝜋)⁻¹e^{(𝜋/2)|ℑm𝜃|}𝑐₃. Now, recalling that |𝜆|≥𝑐>0 for every 𝜆∈Γ, we have |𝜆|≤|𝜆|+1≤(1+𝑐⁻¹)|𝜆|, 𝜆∈Γ. As a consequence, the following inequality holds: (|𝜆|+1)^{1−𝛽−𝛾}≤𝑐̃_𝛽,𝛾|𝜆|^{1−𝛽−𝛾},∀𝜆∈Γ, (82) where 𝑐̃_𝛽,𝛾=(1+𝑐⁻¹)^{1−𝛽−𝛾} or 𝑐̃_𝛽,𝛾=1 according that 𝛾∈(0,1−𝛽] or 𝛾∈(1−𝛽,1) ((0,1−𝛽]=∅ if 𝛽=1). Therefore, setting 𝑐₂₄=2𝑐̃_𝛽,𝛾𝑐₂₃, (81) and (82) yield (−𝐴)^𝜃∘e^𝑡𝐴𝑥_𝑋≤𝑐₂₄‖𝑥‖_𝑌^𝑝_𝛾^∞₀𝜇^{ℜe𝜃−𝛽−𝛾}e^{−𝑐𝛼𝑡𝜇𝛼}d𝜇,𝑡>0, (83) with 𝑐_𝛼 being as in (70). Finally, the transformation 𝑐_𝛼𝑡𝜇^𝛼=𝑠 in the last integral leads us to the following estimate: (−𝐴)^𝜃∘e^𝑡𝐴𝑥_𝑋≤𝑐₂₅‖𝑥‖_𝑌^𝑝_𝛾𝑡^{(𝛽+𝛾−ℜe𝜃−1)/𝛼},𝑡>0, (84) where 𝑐₂₅=𝑐₂₄𝛼⁻¹𝑐^{(𝛽+𝛾−ℜe}_𝛼^{𝜃−1)/𝛼}𝐸((ℜe𝜃+1−𝛽−𝛾)/𝛼), 𝐸(𝜒), 𝜒>0, is the Euler’s gamma function. Notice that here ℜe𝜃≥1 implies ℜe𝜃+1−𝛽−𝛾≥2−𝛽−𝛾>0 for every 𝛽∈(0,1] and 𝛾∈(0,1), so that 𝐸((ℜe𝜃+1−𝛽−𝛾)/𝛼) makes sense. Since (84) is satisfied for every arbitrary element 𝑥∈𝑌^𝑝_𝛾, the proof is complete with 𝑐₂₂=𝑐₂₅.

Remark 13. Estimate (79) is better than (60) obtained in Remark 8 using (14), (15), and (58). In fact, for every 𝛽∈(0,𝛼], 𝛼∈(0,1], 𝛾∈(0,1) and ℜe𝜃≥1, the following inequality holds: 𝜌₁∶=2𝛽+𝛾−ℜe𝜃−2𝛼≤𝛽+𝛾−ℜe𝜃−1𝛼∶=𝜌₂<0. (85) Then, 𝑡^𝜌2≤𝑡^𝜌1, 𝑡∈(0,1], and (79) is more accurate than (60) for small values of 𝑡.

Estimate (79) with 𝜃=1 yields the following result which we will need in Section 5 to prove the equivalence between problem (170) and the fixed-point equation (179).

Corollary 14. Let 𝛼+𝛽>1 in (H1). Then, for every 𝑥∈𝑋 the following equalities hold: 𝐴⁻¹𝑒^𝑡𝐴−𝐼𝑥=𝑒^𝑡𝐴−𝐼𝐴⁻¹𝑥=^𝑡₀𝑒^{(𝑡−𝑠)𝐴}𝑥d𝑠,𝑡≥0. (86)

Proof. The assertion is obvious for 𝑡=0. Let 𝑡>0 and let 𝑥∈𝑋. Commuting 𝐴⁻¹∈ℒ(𝑋) with the integral sign, from (9) and the resolvent equation, we have 𝐴⁻¹e^𝑡𝐴𝑥=e^𝑡𝐴𝐴⁻¹𝑥, which proves the first equality in (86). To prove the second equality, we first write e^𝑡𝐴−𝐼𝐴⁻¹𝑥=^𝑡₀𝐷_𝑟e^𝑟𝐴_{𝑟=𝑡−𝑠}𝐴⁻¹𝑥d𝑠=−^𝑡₀(−𝐴)^1∘e^{(𝑡−𝑠)𝐴}𝐴⁻¹𝑥d𝑠, (87) and we show that the latter integral is convergent. Indeed, since 𝛼+𝛽>1, we may consider 𝐴⁻¹𝑥∈𝒟(𝐴) as an element of (𝑋,𝒟(𝐴))_𝛾,𝑝, where 𝛾∈(2−𝛼−𝛽,1) and 𝑝∈[1,∞]. With this choice for 𝛾, from (79) with 𝜃=1 and (25) we obtain (here we use also ‖𝐴⁻¹𝑥‖_𝒟(𝐴)=inf_{𝑦∈𝐴(𝐴}⁻¹_𝑥)‖𝑦‖_𝑋=inf_{𝑦∈(𝐴𝐴}⁻¹_)𝑥‖𝑦‖_𝑋=‖𝑥‖_{𝒟(𝐴𝐴}⁻¹₎≤‖𝑥‖_𝑋, due to 𝐼⊂𝐴𝐴⁻¹. Then, ‖𝐴⁻¹𝑥‖_{(𝑋,𝒟(𝐴))𝛾,𝑝}≤𝑐₀‖𝐴⁻¹𝑥‖_𝑋^1−𝛾‖𝐴⁻¹𝑥‖^𝛾_𝒟(𝐴)≤𝑐₀‖𝐴⁻¹‖^1−𝛾_ℒ(𝑋)‖𝑥‖_𝑋): (−𝐴)^𝑡₀^1∘e^{(𝑡−𝑠)𝐴}𝐴⁻¹𝑥d𝑠_𝑋≤𝑐₂₂𝐴⁻¹𝑥_{(𝑋,𝒟(𝐴))𝛾,𝑝}^𝑡₀(𝑡−𝑠)^{(𝛽+𝛾−2)/𝛼}d𝑠≤𝑐₂₂𝑐_{𝛼,𝛽,𝛾}𝑐₀𝐴⁻¹_ℒ(𝑋)^1−𝛾‖𝑥‖_𝑋𝑡^{(𝛼+𝛽+𝛾−2)/𝛼}, (88) where 𝑐_{𝛼,𝛽,𝛾}=𝛼(𝛼+𝛽+𝛾−2)⁻¹. We now recall that (cf. [24, formula (3.21)]) (−𝐴)^1∘e^𝑡𝐴(−𝐴)^−𝜁=(−𝐴)^1−𝜁∘e^𝑡𝐴,ℜe𝜁∈1−𝛽,1,𝑡>0, (89) with (−𝐴)^−𝜁 being the negative fractional powers of −𝐴 defined by (cf. [24, Section 3]) (2𝜋𝑖)⁻¹∫_Γ(−𝜆)^−𝜁(𝜆𝐼−𝐴)⁻¹d𝜆, ℜe𝜁>1−𝛽. To complete the proof it thus suffices to apply (89) with 𝜁=1 to (87) and to recall that [(−𝐴)⁰]^∘e^𝑡𝐴=e^𝑡𝐴, 𝑡>0. Notice that the integral on the right-hand side of (86) is convergent, too. In fact, from (14), it follows that ‖∫^𝑡₀e^{(𝑡−𝑠)𝐴}𝑥d𝑠‖_X≤𝑐̃_𝛼,𝛽,0‖𝑥‖_𝑋∫^𝑡₀(𝑡−𝑠)^{(𝛽−1)/𝛼}d𝑠=𝛼(𝛼+𝛽−1)⁻¹𝑐̃_𝛼,𝛽,0‖𝑥‖_𝑋𝑡^{(𝛼+𝛽−1)/𝛼}.

Remark 15. In particular, from (86) it follows that if 𝛼+𝛽>1, then ∫^𝑡₀e^{(𝑡−𝑠)𝐴}𝑥ds∈𝒟(𝐴) for every 𝑥∈𝑋 and (e^𝑡𝐴−𝐼)𝑥⊆𝐴∫^𝑡₀e^{(𝑡−𝑠)𝐴}𝑥d𝑠. This extends to m. l. operators satisfying (H1) the well-known result for sectorial single-valued linear operators (see, for instance, [9, Proposition 2.1.4(ii)] and [11, Proposition 1.2(ii)]).

With the help of (54) and Proposition 12, we can now derive the following interpolation estimates (90) for the operators [(−𝐴)^𝜃]^∘e^𝑡𝐴, ℜe𝜃≥1, which are considered as operators from (𝑋,𝒟(𝐴))_𝛾,𝑝 to (𝑋,𝒟(𝐴))_𝛿,𝑝. As we will see in the proof of Proposition 16, here the fact that the spaces (𝑋,𝒟(𝐴))_𝜎,𝑝 are real interpolation spaces between 𝑋 and 𝒟(𝐴) plays a key role. For it allows us to exploit the interpolation inequality (24) in the derivation of our estimates in the case 𝛾+𝛿<1.

Proposition 16. Let ℜe𝜃≥1, 𝛾,𝛿∈(0,1), and 𝑝∈[1,∞]. Then, there exist positive constants 𝑐_𝑗, 𝑗=26,27, depending on 𝛼, 𝛽, 𝛾, 𝛿, 𝜃, and 𝑝 such that for every 𝑡>0(−𝐴)^𝜃∘𝑒^𝑡𝐴_{ℒ((𝑋,𝒟(𝐴))𝛾,𝑝;(𝑋,𝒟(𝐴))𝛿,𝑝)}≤𝑐₂₆𝑡^{(2𝛽+𝛾−𝛿−ℜe𝜃−2)/𝛼}(),𝑐,𝛾,𝛿∈0,1₂₇𝑡^{(𝛽+𝛾−𝛿−ℜe𝜃−1)/𝛼},𝑖𝑓𝛾+𝛿<1. (90)

Proof. For brevity, we will use the shortenings 𝑌^𝑝_𝜎=(𝑋,𝒟(𝐴))_𝜎,𝑝, 𝜎∈(0,1), 𝑝∈[1,∞]. We begin by proving the first estimate in (90). Let 𝜃∈𝐂, ℜe𝜃≥1, 𝛾,𝛿∈(0,1) and 𝑝∈[1,∞] be fixed and let 𝑥 be an arbitrary element of 𝑌^𝑝_𝛾. Moreover, let 𝜁 and 𝜁^′ be two arbitrary complex numbers such that 𝜃=𝜁+𝜁^′ and whose real parts satisfy ℜe𝜁≥0 and ℜe𝜁^′≥1. From the decomposition formula (15) it then follows for every 𝑡>0: (−𝐴)^𝜃∘e^𝑡𝐴𝑥_𝑌𝑝𝛿=(−𝐴)^𝜁∘e^(𝑡/2)𝐴(−𝐴)^𝜁′∘e^(𝑡/2)𝐴𝑥_𝑌𝑝𝛿≤(−𝐴)^𝜁∘e^(𝑡/2)𝐴_{ℒ(𝑋;𝑌𝑝𝛿)}(−𝐴)^𝜁′∘e^(𝑡/2)𝐴𝑥_𝑋≤(−𝐴)^𝜁∘e^(𝑡/2)𝐴_{ℒ(𝑋;𝑌𝑝𝛿)}(−𝐴)^𝜁′∘e^(𝑡/2)𝐴_{ℒ(𝑌𝑝𝛾;𝑋)}‖𝑥‖_𝑌^𝑝_𝛾. (91) Therefore, using (54) and (79) with the triplet (𝜃,𝛾,𝑡) being equal to (𝜁,𝛿,𝑡/2) and (𝜁^′,𝛾,𝑡/2), respectively, from (91) and ℜe𝜃=ℜe𝜁+ℜe𝜁^′, we deduce that (−𝐴)^𝜃∘e^𝑡𝐴𝑥_𝑌𝑝𝛿≤𝑐₁₀𝑐₂₂𝑡2^{(𝛽−𝛿−ℜe𝜁−1)/𝛼}𝑡2^{(𝛽+𝛾−ℜe𝜁′−1)/𝛼}‖𝑥‖_𝑌^𝑝_𝛾≤𝑐₂₆𝑡^{(2𝛽+𝛾−𝛿−ℜe𝜃−2)/𝛼}‖𝑥‖_𝑌^𝑝_𝛾,𝑡>0, (92) where 𝑐₂₆=2^{(2+ℜe𝜃+𝛿−𝛾−2𝛽)/𝛼}𝑐₁₀𝑐₂₂. This completes the proof of the first estimate in (90), due to the arbitrariness of 𝑥∈𝑌^𝑝_𝛾. Let us now prove the second estimate in (90). Let 𝜃∈𝐂, ℜe𝜃≥1, 𝛾,𝛿∈(0,1), 𝛾+𝛿<1, and 𝑝∈[1,∞] be fixed. Using 𝛾+𝛿<1, we fix 𝛾₂∈(𝛾/(1−𝛿),1)⊊(𝛾,1), and we let 𝛾₁=(𝛾₂𝛿)/(𝛾₂−𝛾). Clearly, since 𝛾₂∈(𝛾/(1−𝛿),1), we have 𝛾₁∈(𝛿,1). In addition, it holds: 𝛾1−𝛿>¹−𝛿𝛾₁=𝛾₂𝛿𝛾₂𝛾−𝛾−𝛿²−𝛾𝛾₂𝛿=𝛾𝛾₂>𝛾. (93) Due to (93), we now set 𝛾₀=𝛾/𝛾₂=(𝛾₁−𝛿)/𝛾₁∈(𝛾,1−𝛿), so that 𝛾=𝛾₀𝛾₂ and 𝛿=(1−𝛾₀)𝛾₁. From (24) with 𝑝₀=𝑝 it thus follows that (−𝐴)^𝜃∘e^𝑡𝐴_{ℒ(𝑌𝑝𝛾;𝑌𝑝𝛿)}≤(−𝐴)^𝜃∘e^{𝑡𝐴1−𝛾0}_{ℒ(𝑋;𝑌𝑝1𝛾1)}(−𝐴)^𝜃∘e^{𝑡𝐴𝛾0}_{ℒ(𝑌𝑝2𝛾2;𝑋)},𝑡>0, (94) where 𝑝_𝑗∈[1,∞], 𝑗=1,2. Applying (54) and (79) with the pair (𝛾,𝑝) being replaced with (𝛾₁,𝑝₁) and (𝛾₂,𝑝₂), respectively, from (94) we finally obtain (−𝐴)^𝜃∘e^𝑡𝐴_{ℒ(𝑌𝑝𝛾;𝑌𝑝𝛿)}≤𝑐₁₀𝑡^{(𝛽−𝛾1−ℜe𝜃−1)/𝛼1−𝛾0}𝑐₂₂𝑡^{(𝛽+𝛾2−ℜe𝜃−1)/𝛼𝛾0}≤𝑐₁₀^1−𝛾0𝑐₂₂^𝛾0𝑡^{[𝛽+𝛾0𝛾2−(1−𝛾0)𝛾1−ℜe𝜃−1]/𝛼}=𝑐₁₀^𝛿/𝛾1𝑐₂₂^𝛾/𝛾2𝑡^{(𝛽+𝛾−𝛿−ℜe𝜃−1)/𝛼},𝑡>0. (95) This completes the proof of the second estimate in (90) with 𝑐₂₇=(𝑐₁₀)^𝛿/𝛾1(𝑐₂₂)^𝛾/𝛾2.

Remark 17. We stress that if 𝛽<1 and 𝛾+𝛿<1, then the first estimate in (90) is rougher than the second one for small values of 𝑡, which justify our special attention to the case 𝛾+𝛿<1. Indeed, if 𝛽<1, then for every ℜe𝜃≥1 the following inequality holds: 𝜌₃∶=2𝛽+𝛾−𝛿−ℜe𝜃−2𝛼<𝛽+𝛾−𝛿−ℜe𝜃−1𝛼=∶𝜌₄<0, (96) so that 𝑡^𝜌4≤𝑡^𝜌3 for 𝑡∈(0,1]. In other words, if 𝛽 and 𝛾+𝛿 are both less than one, then the second estimate in (90) establishes that the norm ‖[(−𝐴)^𝜃]^∘e^𝑡𝐴‖_{ℒ((𝑋,𝒟(𝐴))𝛾,𝑝;(𝑋,𝒟(𝐴))𝛿,𝑝)}, ℜe𝜃≥1, may blow up as 𝑡 goes to 0, but with an order of singularity lower than that predicted by the first estimate. In this sense, though less general, the second estimate in (90) is better than the first one.

Remark 18. The reason why the second estimate in (90) yields a better exponent than the first one is the same mentioned in Remark 8. That is, while the first estimate is obtained in two steps: decomposing [(−𝐴)^𝜃]^∘e^𝑡𝐴 through (15) and then applying (54) and (79), the second estimate is essentially derived in a single step, using (24).

The following Remark 19 points out why, with the exception of the case when 𝛽=1 and 𝐴 is single-valued, to prove (90) we can not proceed as in [9, Proposition 2.2.9].

Remark 19. In the optimal case 𝛽=1, the exponents in both estimates (90) coincide equals to 𝜈=𝛾−𝛿−ℜe𝜃. Hence, in this special case, the assumption 𝛾+𝛿<1 does not give any enhancement. Also, if we further assume that 𝜃∈𝐍, then we restore the same estimates as in [9, Proposition 2.2.9(i)]. In this respect, our result extends [9] to the m. l. case, even though our proof really differs from that in [9]. For, there, the norms in the spaces (𝑋,𝒟(𝐴))_𝜎,𝑝 are replaced with the norms in the spaces 𝒟_𝐴(𝜎,𝑝), with the latter being the spaces of all 𝑥∈𝑋 such that ‖𝑥‖_{𝒟𝐴(𝜎,𝑝)}=‖𝑥‖_𝑋+[𝑥]_{𝒟𝐴(𝜎,𝑝)}<∞, where [𝑥]_{𝒟𝐴(𝜎,𝑝)}=‖𝜉^{(2−𝛽−𝜎)/𝛼}[(−𝐴)¹]^∘e^𝜉𝐴‖_𝐿^∗_𝑝(𝑋). It is well known that if 𝛽=1 and 𝐴 is single-valued, then (𝑋,𝒟(𝐴))_𝜎,𝑝≅𝒟_𝐴(𝜎,𝑝) (cf. [31, Theorem 3], [9, Proposition 2.2.2] and [27, Theorem 1.14.5]). On the contrary, if (𝛼,𝛽)≠(1,1) and/or 𝐴 is really an m. l. operator, such equivalence is no longer true and we have 𝑋_𝐴^𝜎,𝑝↪(𝑋,𝒟(𝐴))_𝜎,𝑝↪𝒟_𝐴[),𝑋𝛼𝜎,𝑝,𝑝∈1,∞_𝐴𝜎,∞↪(𝑋,𝒟(𝐴))_𝜎,∞↪𝒟_𝐴(𝜎,∞),𝑝=∞. (97) Differently from the spaces 𝑋_𝐴^𝜎,𝑝 and as a consequence of 𝐴0⊆⋂_𝑡>0𝒩([(−𝐴)¹]^∘e^𝑡𝐴), the spaces 𝒟_𝐴(𝜎,𝑝) contain 𝐴0. It can thus be shown that if 𝛼+𝛽>1, then for every 𝜎∈(2−𝛼−𝛽,1) and 𝜑∈(0,(𝛼+𝛽+𝜎−2)/𝛼) (here (𝛼+𝛽+𝜎−2)/𝛼<1, since 𝜎<1≤2−𝛽) the following embeddings hold: {0}∪𝒟_𝐴𝜎,𝑝⧵𝐴0↪𝑋_𝐴^𝜑,𝑝↪(𝑋,𝒟(𝐴))_𝜑,𝑝,[),{0}∪𝒟𝑝∈1,∞_𝐴(𝜎,∞)⧵𝐴0↪𝑋_𝐴^{(𝛼+𝛽+𝜎−2)/𝛼,∞}↪(𝑋,𝒟(𝐴))_{(𝛼+𝛽+𝜎−2)/𝛼,∞}, (98) with {0}∪[𝒟_𝐴(𝜎,𝑝)⧵𝐴0] being endowed with the norm of 𝒟_𝐴(𝜎,𝑝). Obviously, due to (29), it suffices to prove the embeddings on the right of (97) and on the left of (98). It is out of the aims of this paper to go into the details of these proofs, and for them we refer the readers to [24, Proposition 6.3]. Here we want only to make clear that, with the exception of the case when 𝛽=1 and 𝐴 is single-valued, embeddings (97) and (98) prevent us from carrying out the proof of estimates (90) simply by repeating the computations in [9]. Notice that, due to the property [𝑋_𝐴^𝜎,𝑝∩𝐴0]={0}, from the second embeddings in (97) and (98) it follows that if 𝛼+𝛽>1 and 𝜎∈(2−𝛼−𝛽,1), then 𝑋_𝐴^𝜎,∞↪{0}∪𝒟_𝐴(𝜎,∞)⧵𝐴0↪𝑋_𝐴^{(𝛼+𝛽+𝜎−2)/𝛼,∞}. (99) Since (𝛼+𝛽+𝜎−2)/𝛼≤𝜎 (indeed, 𝛼≤1≤(2−𝛽−𝜎)/(1−𝜎) implies 𝛼+𝛽+𝜎−2≤𝛼𝜎), (99) agrees with (38) for 𝑝=∞. In addition, if 2𝛼+𝛽>2 and 𝜎∈((2−𝛼−𝛽)/𝛼,1), then the first embeddings in (97) and (98) yield for every 𝜑∈(0,(𝛼+𝛽+𝛼𝜎−2)/𝛼) the following: 𝑋_𝐴^𝜎,𝑝↪{0}∪𝒟_𝐴𝛼𝜎,𝑝⧵𝐴0↪𝑋_𝐴^𝜑,𝑝,𝑝∈[1,∞). (100) Since 𝜑<(𝛼+𝛽+𝛼𝜎−2)/𝛼≤𝜎, (100) agrees with (38) for 𝑝∈[1,∞). Furthermore, if 𝛽=1, then from (29), (30), and (99) it follows that (𝑋,𝒟(𝐴))_𝜎,∞≅𝑋_𝐴^𝜎,∞≅{0}∪[𝒟_𝐴(𝜎,∞)⧵𝐴0], 𝜎∈(0,1). This confirms that in the real m. l. case the equivalence between 𝑋_𝐴^𝜎,𝑝, (𝑋,𝒟(𝐴))_𝜎,𝑝 and 𝒟_𝐴(𝜎,𝑝) does not hold even when 𝛽=1.

Using Propositions 9 and 12, we now obtain estimates for the operators [(−𝐴)^𝜃]^∘e^𝑡𝐴, ℜe𝜃≥1, considered as operators from 𝑋_𝐴^𝛾,𝑝 to  𝑋_𝐴^𝛿,𝑝. Clearly, since 𝛽<1 the spaces 𝑋_𝐴^𝜎,𝑝 may be not real interpolation spaces between 𝑋 and 𝒟(𝐴), we can not proceed as in the proof of the second estimate in (90) and a weaker result has to be expected.

Proposition 20. Let ℜe𝜃≥1, 𝛾,𝛿∈(0,1), and 𝑝∈[1,∞]. Then, there exist positive constants 𝑐_𝑗, 𝑗=28,29,30, depending on 𝛼, 𝛽, 𝛾, 𝛿, 𝜃, and 𝑝 such that (−𝐴)^𝜃∘𝑒^𝑡𝐴_{ℒ(𝑋𝐴𝛾,∞;𝑋𝐴𝛿,∞)}≤𝑐₂₈𝑡^{(2𝛽+𝛾−𝛿−ℜe𝜃−2)/𝛼},)𝑝=∞,𝑡>0,(−𝐴^𝜃∘𝑒^𝑡𝐴_{ℒ(𝑋𝐴𝛾,𝑝;𝑋𝐴𝛽𝛿,𝑝)}≤𝑐₂₉𝑡^{(2𝛽+𝛾−𝛿−ℜe𝜃−2)/𝛼},𝑝∈[1,∞),𝑡>0. (101)(102) Moreover, if 𝛾∈(0,1) and 𝛿∈(1−𝛽,1) are such that 𝛾+𝛿<1, then (−𝐴)^𝜃∘𝑒^𝑡𝐴_{ℒ(𝑋𝐴𝛾,𝑝;𝑋𝐴𝛿+𝛽−1,𝑝)}≤𝑐₃₀𝑡^{(𝛽+𝛾−𝛿−ℜe−1)/𝛼},𝑝∈[1,∞],𝑡>0. (103)

Proof. Due to (61) and (79), in order to prove (101) and (102) it suffices to repeat the same computations as in (91) and (92), with the pair ((𝑋,𝒟(𝐴))_𝛾,𝑝,(𝑋,𝒟(𝐴))_𝛿,𝑝) being replaced with (𝑋_𝐴^𝛾,∞,𝑋_𝐴^𝛿,∞) or with (𝑋_𝐴^𝛾,𝑝,𝑋_𝐴^{𝛽𝛿,𝑝}) provided that 𝑝=∞ or 𝑝∈[1,∞). In this way we derive (101) and (102) with 𝑐_𝑗+13=2^{(2+ℜe𝜃+𝛿−𝛾−2𝛽)/𝛼}𝑐_𝑗𝑐₂₂, 𝑗=15,16. As far as (103) is concerned, we recall that if 𝑋_𝑗, 𝑗=1,…,4, are four Banach spaces such that 𝑋_𝑗↪𝑋_𝑗+2, 𝑗=1,2, and 𝐿∈ℒ(𝑋₃;𝑋₂), then 𝐿∈ℒ(𝑋₁;𝑋₄) with ‖𝐿‖_{ℒ(𝑋1;𝑋4)}≤𝐶₁𝐶₂‖𝐿‖_{ℒ(𝑋3;𝑋2)}, 𝐶₁ and 𝐶₂ being the positive constants such that ‖𝑥‖_𝑋𝑗+2≤𝐶_𝑗‖𝑥‖_𝑋𝑗, 𝑥∈𝑋_𝑗, 𝑗=1,2. Applying this result to 𝐿=[(−𝐴)^𝜃]^∘e^𝑡𝐴 with (𝑋₁,𝑋₂,𝑋₃,𝑋₄)=(𝑋_𝐴^𝛾,𝑝,(𝑋,𝒟(𝐴))_𝛿,𝑝,(𝑋,𝒟(𝐴))_𝛾,𝑝,𝑋_𝐴^{𝛿+𝛽−1,𝑝}), from (29)–(32) and the second estimate in (90) we deduce (103) with 𝑐₃₀=2𝑐₂𝑐₂₇. This completes the proof.

Remark 21. The assumption 𝛾+𝛿<1 with 𝛾∈(0,1) and 𝛿∈(1−𝛽,1) implies that 𝛾∈(0,1−𝛿)⊊(0,𝛽). Therefore (cf. Remark 11), we conclude that for (103) to hold we have to consider [(−𝐴)^𝜃]^∘e^𝑡𝐴, ℜe𝜃≥1, as an operator between the intermediate spaces 𝑋_𝐴^𝛾,𝑝 and 𝑋_𝐴^𝜀,𝑝, where 𝛾,𝜀∈(0,𝛽), 𝜀=𝛿+𝛽−1, 𝛿∈(1−𝛽,1), 𝛾+𝛿<1.

4. Hölder Regularity of Some Operator Functions

Here, we study the Hölder regularity of those operator functions that we will need in Section 5. From now on, with (𝑍,‖⋅‖_𝑍) being a complex Banach space, 𝐶([𝑎,𝑏];𝑍)=𝐶⁰([𝑎,𝑏];𝑍) and 𝐶^𝛿([𝑎,𝑏];𝑍), 𝛿∈(0,1), 𝑎<𝑏, denote, respectively, the spaces of all continuous and 𝛿-Hölder continuous functions from [𝑎,𝑏] into 𝑍 endowed with the norms ‖𝑔‖_{0,𝑎,𝑏;𝑍}=sup_{𝑡∈[𝑎,𝑏]}‖𝑔(𝑡)‖_𝑍 and ‖𝑔‖_{𝛿,𝑎,𝑏;𝑍}=‖𝑔‖_{0,𝑎,𝑏;𝑍}+|𝑔|_{𝛿,𝑎,𝑏;𝑍}, where |𝑔|_{𝛿,𝑎,𝑏;𝑍} is the seminorm sup_{𝑎≤𝑡1<𝑡2≤𝑏}(𝑡₂−𝑡₁)^−𝛿‖𝑔(𝑡₂)−𝑔(𝑡₁)‖_𝑍. We endow the subspace 𝐶^𝛿₀([𝑎,𝑏];𝑍)={𝑔∈𝐶^𝛿([𝑎,𝑏];Z)∶𝑔(𝑎)=0}, 𝛿∈[0,1) with the norm ‖⋅‖_{𝛿,𝑎,𝑏;𝑍}. Further, for 𝑘∈𝐍 and 𝛿∈(0,1) we set 𝐶^𝑘([𝑎,𝑏];𝑍)={𝑔∈𝐶([𝑎,𝑏];𝑍)∶𝐷^𝑘_𝑡𝑔∈𝐶([𝑎,𝑏];𝑍)}, ‖𝑔‖_{𝑘,𝑎,𝑏;𝑍}=∑^𝑘₀‖𝐷^𝑗_𝑡𝑔‖_{0,𝑎,𝑏;𝑍} (𝐷⁰_𝑡=𝐼), and 𝐶^𝑘+𝛿([𝑎,𝑏];𝑍)={𝑔∈𝐶^𝑘([𝑎,𝑏];𝑍)∶𝐷^𝑘_𝑡𝑔∈𝐶^𝛿([𝑎,𝑏];𝑍)}, ‖𝑔‖_{𝑘+𝛿,𝑎,𝑏;𝑍}=‖𝑔‖_{𝑘,𝑎,𝑏;𝑍}+|𝐷^𝑘_𝑡𝑔|_{𝛿,𝑎,𝑏;𝑍}. Recall that if 0≤𝛿₂≤𝛿₁≤1, then 𝐶^𝛿1([𝑎,𝑏];𝑍)↪𝐶^𝛿2([𝑎,𝑏];𝑍) and ‖𝑔‖_{𝛿2,𝑎,𝑏;𝑍}≤max{1,(𝑏−𝑎)^{𝛿1−𝛿2}}‖𝑔‖_{𝛿1,𝑎,𝑏;𝑍}, 𝑔∈𝐶^𝛿1([𝑎,𝑏];𝑋). Finally, given three complex Banach spaces (𝑋_𝑘,‖⋅‖_𝑋𝑘), 𝑘=1,2,3, and a bilinear bounded operator 𝒫 from 𝑋₁×𝑋₂ to 𝑋₃ with norm 𝐶₀, that is, 𝒫∈ℬ(𝑋₁×𝑋₂;𝑋₃) and ‖𝒫‖_{ℬ(𝑋1×𝑋2;𝑋3)}=sup_{‖𝑥𝑘‖𝑋𝑘=1,𝑘=1,2}‖𝒫(𝑥₁,𝑥₂)‖_𝑋3=𝐶₀, we denote by 𝒦 the convolution operator 𝒦𝑣₁,𝑣₂(𝑡)=^𝑡₀𝒫𝑣₁(𝑡−𝑟),𝑣₂(𝑟)[]d𝑟,𝑡∈0,𝑏,𝑏>0, (104) where 𝑣_𝑘∶[0,𝑏]→𝑋_𝑘, 𝑘=1,2. Of course, if (𝑋₁,𝑋₂)=(𝐂,𝑋₃) and if 𝒫 is the scalar multiplication in 𝑋₃, that is, 𝒫(𝑧,𝑥)=𝑧𝑥, 𝑧∈𝐂, 𝑥∈𝑋₃, then 𝐶₀=1 and 𝒦 reduces to the usual convolution operator 𝒦(𝑣₁,𝑣₂)(𝑡)=∫^𝑡₀𝑣₁(𝑡−𝑟)𝑣₂(𝑟)d𝑟. As usual, for every 𝑞∈[1,∞], we will denote by 𝑞^′ the conjugate exponent of 𝑞.

Now let 𝑋₃=𝑋 and introduce the following linear operators 𝑄_𝑗, 𝑗=1,…,6, where 𝑔_𝑗∈𝐶^𝛿𝑗([0,𝑇];𝑋), 𝑗=1,2,5, 𝑔_𝑙𝑘∈𝐶^𝛿𝑙𝑘([0,𝑇],𝑋_𝑘), 𝑙=3,6, 𝑘=1,2, 𝑔₄∈𝐶^𝛿4([0,𝑇];𝐂), 𝑦∈𝑌^𝑝_𝛾, 𝑌^𝑝_𝛾∈{(𝑋,𝒟(𝐴))_𝛾,𝑝,𝑋_𝐴^𝛾,𝑝}, 𝑝∈[1,∞], and 𝑡∈[0,𝑇], 𝑇>0 as follows: 𝑄₁𝑔₁(𝑡)∶=^𝑡₀e^{(𝑡−𝑠)𝐴}𝑔₁(𝑠)𝑄d𝑠,₂𝑔₂(𝑡)∶=^𝑡₀(−𝐴)^1∘e^{(𝑡−𝑠)𝐴}𝑔₂(𝑠)−𝑔₂(𝑡𝑄)d𝑠,₃𝑔₃₁,𝑔₃₂(𝑡)∶=𝑄₂𝒦𝑔₃₁,𝑔₃₂𝑡),𝑄₄(𝑔₄,𝑦(𝑡)∶=𝑄₂𝑔₄𝑦𝑡),𝑄₅(𝑔₅(𝑡)∶=e^𝑡𝐴−𝐼𝑔₅(𝑡),𝑄₆𝑔₆₁,𝑔₆₂(𝑡)∶=𝑄₅𝒦𝑔₆₁,𝑔₆₂(𝑡), (105)(106)(107)(108)(109)(110) with 𝑔₄𝑦 being the function from [0,𝑇] to 𝑌^𝑝_𝛾 defined by (𝑔₄𝑦)(𝑡)=𝑔₄(𝑡)𝑦. We will find conditions on 𝛿_𝑗,𝛿_𝑙𝑘,𝛿₄,𝛾∈(0,1), 𝑗=1,2,5, 𝑙=3,6, 𝑘=1,2, in order that 𝑄_𝑗𝑔_𝑗∈𝐶^𝜏𝑗([0,𝑇];𝑋), 𝑄_𝑙(𝑔_𝑙1,𝑔_𝑙2)∈𝐶^𝜏𝑙([0,𝑇];𝑋) and 𝑄₄(𝑔₄,𝑦)∈𝐶^𝜏4([0,𝑇];𝑋) for opportunely chosen 𝜏_𝑗,𝜏_𝑙,𝜏₄∈(0,1). We emphasize of the presence of the increment 𝑔₂(𝑠)−𝑔₂(𝑡) inside the integral defining 𝑄₂𝑔₂. As we will see, and differently from 𝑄₁, it is just this presence which makes 𝑄₂𝑔₂ well-defined for smooth enough functions 𝑔₂. This is the reason why the operator 𝑄₂ as it was defined in [20, formula (4.12)] can make no sense and has to be replaced with that defined by the present (106) (cf. the appendix below). We begin our analysis on the 𝑄_𝑗’s with the following result proven in [20, Lemma 4.1]. Since we will need it later, here, removing some misprints in [20], we report its short proof for the reader’s convenience.

Lemma 22. Let 𝛼+𝛽>1 in (H1). Then, for every 𝛿₁∈(0,(𝛼+𝛽−1)/𝛼), the operator 𝑄₁ defined by (105) maps 𝐶^𝛿1([0,𝑇];𝑋) into 𝐶^𝛿1₀([0,𝑇];𝑋), and for every 𝑡∈[0,𝑇] satisfies the following estimate, where 𝑝∈(𝛼/(𝛼+𝛽−1−𝛼𝛿₁),∞) as follows: 𝑄₁𝑔_{1𝛿1,0,𝑡;𝑋}≤𝐶₁(𝑡)^𝑡₀𝑔₁^𝑝_{𝛿1,0,𝑠;𝑋}d𝑠^1/𝑝. (111) Here 𝐶₁(𝑡) is a nondecreasing function of 𝑡 depending also on 𝛼, 𝛽, 𝛿₁, and 𝑝^′.

Proof. Let 𝑔₁∈𝐶^𝛿1([0,𝑇];𝑋), 𝛿₁∈(0,(𝛼+𝛽−1)/𝛼), and 𝑡∈[0,𝑇]. From (14) and the Hölder inequality with 𝑝∈(𝛼/(𝛼+𝛽−1−𝛼𝛿₁),∞)⊊(1,∞), for any 𝜏∈[0,𝑡], we deduce that 𝑄₁𝑔₁(𝜏)_𝑋≤𝑐̃_𝛼,𝛽,0^𝜏₀(𝜏−𝑠)^{(𝛽−1)/𝛼}𝑔_{10,0,𝑠;𝑋}d𝑠≤𝑐₃₁𝜏^{[𝛼−(1−𝛽)𝑝′]/(𝛼𝑝′)𝜏}₀𝑔₁^𝑝_{𝛿1,0,𝑠;𝑋}d𝑠^1/𝑝≤𝑐₃₁𝜏^{[𝛼−(1+𝛼𝛿1−𝛽)𝑝′]/(𝛼𝑝′)}𝜏^𝛿1𝜏₀𝑔₁^𝑝_{𝛿1,0,𝑠;𝑋}d𝑠^1/𝑝, (112) where 𝑐₃₁=𝑐̃_𝛼,𝛽,0𝛼^1/𝑝′[𝛼−(1−𝛽)𝑝^′]^−1/𝑝′. Here 𝛼−(1+𝛼𝛿₁−𝛽)𝑝^′>0, since 𝑝^′∈(1,𝛼/(1+𝛼𝛿₁−𝛽)). For 1−1/𝑝>1−(𝛼+𝛽−1−𝛼𝛿₁)/𝛼=(1+𝛼𝛿₁−𝛽)/𝛼. passing to the supremum with respect to 𝜏∈[0,𝑡] in (112) we thus find 𝑄₁𝑔_{10,0,𝑡;𝑋}≤𝑐₃₁𝑡^{[𝛼−(1+𝛼𝛿1−𝛽)𝑝′]/(𝛼𝑝′)}𝑡^𝛿1𝑡₀𝑔₁^𝑝_{𝛿1,0,𝑠;𝑋}d𝑠^1/𝑝. (113) Now let (since [𝑄₁𝑔₁](0)=0, the case 𝑡₁=0 follows from (112) with 𝜏=𝑡₂) 0<𝑡₁<𝑡₂≤𝑡. The change of variable 𝑡−𝑠=𝑟 in (105) leads us to [𝑄₁𝑔₁](𝑡₂)−[𝑄₁𝑔₁](𝑡₁)=∑²_𝑘=1𝐼_{𝑘;𝑡1,𝑡2,𝑔1}, where 𝐼_{1;𝑡1,𝑡2,𝑔1}∶=∫^𝑡2_𝑡1e^𝑟𝐴𝑔₁(𝑡₂−𝑟)d𝑟 and 𝐼_{2;𝑡1,𝑡2,𝑔1}∶=∫^𝑡1₀e^𝑟𝐴[𝑔₁(𝑡₂−𝑟)−𝑔₁(𝑡₁−𝑟)]d𝑟. Reasoning as in (112) and using the inequality 𝑡^𝜇₂−𝑡^𝜇₁≤(𝑡₂−𝑡₁)^𝜇, 𝜇∈(0,1], we get 𝐼_{1;𝑡1,𝑡2,𝑔1𝑋}≤𝑐₃₁𝑡₂−𝑡₁^{[𝛼−(1−𝛽)𝑝′]/(𝛼𝑝′)𝑡2}_𝑡1𝑔₁^𝑝_{𝛿1,0,𝑡2−𝑟;𝑋}d𝑟^1/𝑝≤𝑐₃₁𝑡^{[𝛼−(1+𝛼𝛿1−𝛽)𝑝′]/(𝛼𝑝′)}₂𝑡₂−𝑡₁^𝛿1𝑡₀𝑔₁^𝑝_{𝛿1,0,𝑡−𝑟;𝑋}d𝑟^1/𝑝. (114) Similarly, but taking advantage from 𝑔₁∈𝐶^𝛿1([0,𝑇];𝑋), we obtain 𝐼_{2;𝑡1,𝑡2,𝑔1𝑋}≤𝑐̃_𝛼,𝛽,0𝑡₂−𝑡₁^𝛿1𝑡1₀𝑟^{(𝛽−1)/𝛼}𝑔_{1𝛿1,0,𝑡2−𝑟;𝑋}d𝑟≤𝑐₃₁𝑡^{[𝛼−(1+𝛼𝛿1−𝛽)𝑝′]/(𝛼𝑝′)}₁𝑡^𝛿1₁𝑡₂−𝑡₁^𝛿1𝑡₀𝑔₁^𝑝_{𝛿1,0,𝑡−𝑟;𝑋}d𝑟^1/𝑝. (115) Thus, letting 𝑐₁̃(𝑡)=𝑐₃₁𝑡^{[𝛼−(1+𝛼𝛿1−𝛽)𝑝′]/(𝛼𝑝′)} from (114) and (115) it follows that 𝑄₁𝑔₁𝑡₂−𝑄₁𝑔₁𝑡_1𝑋≤𝑐₁̃(𝑡)𝑡^𝛿1+1𝑡₂−𝑡₁^𝛿1𝑡₀𝑔₁^𝑝_{𝛿1,0,𝑡−𝑟;𝑋}d𝑟^1/𝑝. (116) Finally, summing up (113) and (116) and using ∫^𝑡₀‖𝑔₁‖^𝑝_{𝛿1,0,𝑡−𝑟;𝑋}d𝑟=∫^𝑡₀‖𝑔₁‖^𝑝_{𝛿1,0,𝑠;𝑋}d𝑠, we derive (111) with 𝐶₁(𝑡)=𝑐₁̃(𝑡)(2𝑡^𝛿1+1). This completes the proof.

Remark 23. We stress that if we renounce to its Hölder regularity, then for 𝑄₁𝑔₁ to be well-defined it suffices that 𝛼 and 𝛽 are as in Lemma 22 and that 𝑔₁ is merely in 𝐶([0,𝑇];𝑋). In fact (see the last part of the proof of Corollary 14, replacing there 𝑥 with 𝑔₁(𝑠)), ‖[𝑄₁𝑔₁](𝑡)‖_𝑋≤𝛼(𝛼+𝛽−1)⁻¹𝑐̃_𝛼,𝛽,0‖𝑔₁‖_{0,0,𝑡;𝑋}𝑡^{(𝛼+𝛽−1)/𝛼}, 𝑡∈[0,𝑇].

Lemma 24. Let 3𝛼+𝛽>3 in (H1). Then, for every 𝛿₂∈((3−2𝛼−𝛽)/𝛼,1), the operator 𝑄₂ defined by (106) maps 𝐶^𝛿2([0,𝑇];𝑋) into 𝐶^𝜈2₀([0,𝑇];𝑋), 𝜈₂=(𝛼𝛿₂+2𝛼+𝛽−3)/𝛼∈(0,𝛿₂], and for every 𝑡∈[0,𝑇] it satisfies the following estimate: 𝑄₂𝑔_{2𝜈2,0,𝑡;𝑋}≤𝐶₂(𝑡)𝑔_{2𝛿2,0,𝑡;𝑋}. (117) Here 𝐶₂(𝑡) is a nondecreasing function of 𝑡 depending also on 𝛼, 𝛽, and 𝛿₂.

Proof. Denote by 𝛼 the number (1−𝛼)/𝛼. In particular, since 3𝛼+𝛽>3 implies 𝛼∈(2/3,1], we have 𝛼∈[0,1/2). Let 𝑡∈[0,𝑇], 𝑔₂∈𝐶^𝛿2([0,𝑇];𝑋), 𝛿₂∈((3−2𝛼−𝛽)/𝛼,1), and 𝜈₂=(𝛼𝛿₂+2𝛼+𝛽−3)/𝛼∈(0,𝛿₂]. We notice that (𝛼𝛿₂+𝛽−2)/𝛼=𝜈₂+𝛼−1 and (𝛼𝛿₂+𝛽−3)/𝛼=𝜈₂−2. Then, using (14) with 𝜃=1, for every 𝜏∈[0,𝑡] we obtain 𝑄₂𝑔₂(𝜏)_𝑋≤𝑐̃_𝛼,𝛽,1𝑔_{2𝛿2,0,𝜏;𝑋}^𝜏₀(𝜏−𝑠)^{(𝛼𝛿2+𝛽−2)/𝛼}d𝑠=𝑐₃₂𝑔_{2𝛿2,0,𝜏;𝑋}𝜏^𝜈2+𝛼, (118) where 𝑐₃₂=𝑐̃_𝛼,𝛽,1(𝜈₂+𝛼)⁻¹. Hence 𝑄₂𝑔_{20,0,𝑡;𝑋}≤𝑐₃₂𝑔_{2𝛿2,0,𝑡;𝑋}𝑡^𝜈2+𝛼. (119) Now let (since [𝑄₂𝑔₂](0)=0, the case 𝑡₁=0 follows from (118) with 𝜏=𝑡₂) 0<𝑡₁<𝑡₂≤𝑡. We have [𝑄₂𝑔₂](𝑡₂)−[𝑄₂𝑔₂](𝑡₁)=∑³_𝑘=1𝐽_{𝑘;𝑡1,𝑡2,𝑔2}, where for a function 𝑔∶[0,𝑇]→𝑋 we set 𝐽_{1;𝑡1,𝑡2,𝑔}∶=^𝑡1₀(−𝐴)^1∘e^{(𝑡2−𝑠)𝐴}−(−𝐴)^1∘e^{(𝑡1−𝑠)𝐴}×𝑔(𝑠)−𝑔𝑡₁𝐽d𝑠,_{2;𝑡1,𝑡2,𝑔}∶=^𝑡1₀(−𝐴)^1∘e^{(𝑡2−𝑠)𝐴}𝑔𝑡₁−𝑔𝑡₂𝐽d𝑠,_{3;𝑡1,𝑡2,𝑔}∶=^𝑡2_𝑡1(−𝐴)^1∘e^{(𝑡2−𝑠)𝐴}𝑔(𝑠)−𝑔𝑡₂d𝑠. (120) First, using (13) with (𝑠,𝑡,𝜃)=(𝑡₁−𝑠,𝑡₂−𝑠,1), 𝑠∈(0,𝑡₁), and (14) with 𝜃=2, and letting (𝑐₃₃,𝑐₃₄)=(𝑐̃_𝛼,𝛽,2(1−𝜈₂)⁻¹,𝑐₃₃𝜈₂⁻¹), we get 𝐽_{1;𝑡1,𝑡2,𝑔2𝑋}≤𝑐̃_𝛼,𝛽,2𝑔_{2𝛿2,0,𝑡1;𝑋}^𝑡1₀^𝑡2_𝑡^−𝑠_1−𝑠𝜉^{(𝛽−3)/𝛼}d𝜉𝑡₁−𝑠^𝛿2≤𝑐̃d𝑠_𝛼,𝛽,2𝑔_{2𝛿2,0,𝑡1;𝑋}^𝑡1₀^𝑡2_𝑡^−𝑠_1−𝑠𝜉^{(𝛼𝛿2+𝛽−3)/𝛼}d𝜉d𝑠=𝑐₃₃𝑔_{2𝛿2,0,𝑡1;𝑋}^𝑡1₀𝑡₁−𝑠^𝜈2−1−𝑡₂−𝑠^𝜈2−1d𝑠=𝑐₃₄𝑔_{2𝛿2,0,𝑡1;𝑋}𝑡^𝜈2₁+𝑡₂−𝑡₁^𝜈2−𝑡^𝜈2₂≤𝑐₃₄𝑔_{2𝛿2,0,𝑡2;𝑋}𝑡₂−𝑡₁^𝜈2. (121) Let us turn to 𝐽_{2;𝑡1,𝑡2,𝑔2}. We first observe that the integral ∫^𝑡1₀[(−𝐴)¹]^∘e^{(𝑡2−𝑠)𝐴}d𝑠 is convergent. For, ‖∫^𝑡1₀[(−𝐴)¹]^∘e^{(𝑡2−𝑠)𝐴}d𝑠‖_𝑋≤𝑐̃_𝛼,𝛽,1∫^𝑡1₀(𝑡₂−𝑠)^{(𝛽−2)/𝛼}d𝑠≤𝐶_{𝛼,𝛽,𝑡1,𝑡2}, where 𝐶_{𝛼,𝛽,𝑡1,𝑡2} is equal to 𝑐̃_𝛼,𝛽,1ln[𝑡₂(𝑡₂−𝑡₁)⁻¹] if 𝛽=1 and to 𝛼(2−𝛼−𝛽)⁻¹𝑐̃_𝛼,𝛽,1[(𝑡₂−𝑡₁)^{(𝛼+𝛽−2)/𝛼}−𝑡₂^{(𝛼+𝛽−2)/𝛼}] if 𝛽∈(0,1). Thus, we may rewrite it as −∫^{𝑡2−𝑡1}_𝑡2[(−𝐴)¹]^∘e^𝑟𝐴d𝑟=∫^{𝑡2−𝑡1}_𝑡2𝐷_𝑟e^𝑟𝐴d𝑟=e^{(𝑡2−𝑡1)𝐴}−e^𝑡2𝐴. Consequently, 𝐽_{2;𝑡1,𝑡2,𝑔2𝑋}≤𝑐̃_𝛼,𝛽,0𝑡₂−𝑡₁^{(𝛽−1)/𝛼}+𝑡₂^{(𝛽−1)/𝛼}𝑔_{2𝛿2,0,𝑡2;𝑋}𝑡₂−𝑡₁^𝛿2≤𝑐̃_𝛼,𝛽,01+𝑡₂𝑡₂−𝑡₁^{−1(𝛽−1)/𝛼}×𝑔_{2𝛿2,0,𝑡2;𝑋}𝑡₂−𝑡₁^{(𝛼𝛿2+𝛽−1)/𝛼}≤2𝑐̃_𝛼,𝛽,0𝑔_{2𝛿2,0,𝑡2;𝑋}𝑡₂−𝑡₁^𝜈2+2𝛼, (122) where we have used [𝑡₂(𝑡₂−𝑡₁)⁻¹]^{(𝛽−1)/𝛼}≤1 and (𝛼𝛿₂+𝛽−1)/𝛼=𝜈₂+2𝛼. As far as 𝐽_{3;𝑡1,𝑡2,𝑔2} is concerned, instead, reasoning as in the derivation of (118) we find 𝐽_{3;𝑡1,𝑡2,𝑔2𝑋}≤𝑐̃_𝛼,𝛽,1𝑔_{2𝛿2,0,𝑡2;𝑋}^𝑡2_𝑡1𝑡₂−𝑠^{𝜈2+𝛼−1}d𝑠=𝑐₃₂𝑔_{2𝛿2,0,𝑡2;𝑋}𝑡₂−𝑡₁^𝜈2+𝛼. (123) Then, summing up (121)–(123) and letting 𝑐₂̃(𝑡)=𝑐₃₄+2𝑐̃_𝛼,𝛽,0𝑡^2𝛼+𝑐₃₂𝑡^𝛼, we obtain 𝑄₂𝑔₂𝑡₂−𝑄₂𝑔₂𝑡_1𝑋≤³_𝑘=1𝐽_{𝑘;𝑡1,𝑡2,𝑔2𝑋}≤𝑐₂̃(𝑡)𝑔_{2𝛿2,0,𝑡;𝑋}𝑡₂−𝑡₁^𝜈2. (124) Finally, (119) and (124) yield (117) with 𝐶₂(𝑡)=𝑐₃₂𝑡^𝜈2+𝛼+𝑐₂̃(𝑡).

Remark 25. In particular, Lemma 24 establishes that, with the exception of the case 𝛽=1 in which 𝜈₂=𝛿₂, 𝑄₂ produces a loss of regularity equal to 𝛿₂−𝜈₂=(3−2𝛼−𝛽)/𝛼.

As Corollary 14, the next result will be needed to prove the equivalence between problem (170) and the fixed-point equation (179). From now on, if 𝐴⁻¹∈ℒ(𝑋) and 𝑔∈𝐶^𝛿([0,𝑇];𝑋), 𝛿∈[0,1), with 𝐴⁻¹𝑔 we will always mean the function in 𝐶^𝛿([0,𝑇];𝒟(𝐴)) defined by (𝐴⁻¹𝑔)(𝑡)=𝐴⁻¹(𝑔(𝑡)). Notice that ‖𝐴⁻¹𝑔‖_{𝛿,0,𝑡;𝒟(𝐴)}≤‖𝑔‖_{𝛿,0,𝑡;𝑋}, 𝑡∈[0,𝑇].

Corollary 26. (i)Let 2𝛼+𝛽>2 in (H1). Then, for every 𝑔∈𝐶^𝛿([0,𝑇];𝑋), 𝛿∈((2−𝛼−𝛽)/𝛼,1), 𝐴⁻¹𝑄₂𝑔(𝑡)=−^𝑡₀𝑒^{(𝑡−𝑠)𝐴}𝑔(𝑠)−𝑔(𝑡)d𝑠,𝑡∈[0,𝑇]. (125)(ii)Let 𝛼+𝛽>1 in (H1). Then, for every 𝑔∈𝐶([0,𝑇];𝑋)𝑄₂𝐴⁻¹𝑔(𝑡)=−^𝑡₀𝑒^{(𝑡−𝑠)𝐴}𝑔(𝑠)−𝑔(𝑡)d𝑠,𝑡∈[0,𝑇]. (126)

Proof. Of course, it suffices to assume that 𝑡∈(0,𝑇]. Let us first prove (i). So, let 2𝛼+𝛽>2, 𝑔∈𝐶^𝛿([0,𝑇];𝑋), 𝛿∈((2−𝛼−𝛽)/𝛼,1), and 𝑡∈(0,𝑇], and we observe that both sides of (125) are well defined. Indeed, replacing the pair (𝑔₂,𝛿₂) with (𝑔,𝛿), from (118) we get 𝑄₂𝑔(𝑡)_𝑋≤𝑐̃_𝛼,𝛽,1𝛼𝛼𝛿+𝛼+𝛽−2⁻¹𝑔_{𝛿,0,𝑡;𝑋}𝑡^{(𝛼𝛿+𝛼+𝛽−2)/𝛼}. (127) On the other side, 𝐼_𝑡,𝑔=∫^𝑡₀e^{(𝑡−𝑠)𝐴}[𝑔(𝑠)−𝑔(𝑡)]d𝑠 satisfies 𝐼_{𝑡,𝑔𝑋}≤𝑐̃_𝛼,𝛽,0𝑔_{𝛿,0,𝑡;𝑋}^𝑡₀(𝑡−𝑠)^{(𝛼𝛿+𝛽−1)/𝛼}d𝑠≤𝑐₃₅𝑔_{𝛿,0,𝑡;𝑋}𝑡^{(𝛼+𝛼𝛿+𝛽−1)/𝛼}, (128) where 𝑐₃₅=𝛼(𝛼𝛿+𝛼+𝛽−1)⁻¹𝑐̃_𝛼,𝛽,0. Then, commuting 𝐴⁻¹∈ℒ(𝑋) with the integral signs, using (80) with 𝜃=1, and taking into account (7), we find 𝐴⁻¹𝑄₂𝑔₂(𝑡)=𝐴^−1𝑡₀−12𝜋𝑖_Γe^{(𝑡−𝑠)𝜆}𝐴^∘(𝜆𝐼−𝐴)⁻¹d𝜆𝑔₂(𝑠)−𝑔₂(𝑡)d𝑠=−^𝑡₀12𝜋𝑖_Γe^{(𝑡−𝑠)𝜆}𝐴⁻¹𝐴^∘(𝜆𝐼−𝐴)⁻¹d𝜆𝑔₂(𝑠)−𝑔₂(𝑡)d𝑠=−^𝑡₀12𝜋𝑖_Γe^{(𝑡−𝑠)𝜆}(𝜆𝐼−𝐴)⁻¹d𝜆𝑔₂(𝑠)−𝑔₂(𝑡)d𝑠. (129) Since (2𝜋𝑖)⁻¹∫_Γe^{(𝑡−𝑠)𝜆}(𝜆𝐼−𝐴)⁻¹d𝜆=e^{(𝑡−𝑠)𝐴}, the proof of (125) is complete. We now prove (ii). Let 𝛼+𝛽>1, 𝑔∈𝐶([0,𝑇];𝑋) and 𝑡∈(0,𝑇]. Then, for every 𝛾∈(2−𝛼−𝛽,1), the same reasonings made to derive (88), except for replacing 𝑥 with g(𝑠)−𝑔(𝑡), yield 𝑄₂𝐴⁻¹𝑔(𝑡)_𝑋≤2𝑐₂₂𝑐_{𝛼,𝛽,𝛾}𝑐₀𝐴^−11−𝛾_ℒ(𝑋)𝑔_{0,0,𝑡;𝑋}𝑡^{(𝛼+𝛽+𝛾−2)/𝛼}. (130) Hence, [𝑄₂(𝐴⁻¹𝑔)](𝑡) being meaningful, we obtain (126) simply applying to it formula (89) with 𝜁=1 and then using [(−𝐴)⁰]^∘e^{(𝑡−𝑠)𝐴}=e^{(𝑡−𝑠)𝐴}, 𝑠∈(0,𝑡). In particular, a better estimate than (130) holds. For, [𝑄₂(𝐴⁻¹𝑔)](𝑡)=−∫^𝑡₀e^{(𝑡−𝑠)𝐴}[𝑔(𝑠)−𝑔(𝑡)]d𝑠 satisfies 𝑄₂𝐴⁻¹𝑔(𝑡)_𝑋≤2𝑐̃_𝛼,𝛽,0𝑔_{0,0,𝑡;𝑋}^𝑡₀(𝑡−𝑠)^{(𝛽−1)/𝛼}d𝑠≤2𝑐₃₆𝑔_{0,0,𝑡;𝑋}𝑡^{(𝛼+𝛽−1)/𝛼}, (131) where 𝑐₃₆=𝛼(𝛼+𝛽−1)⁻¹𝑐̃_𝛼,𝛽,0. The proof is complete.

Let us now examine the operator 𝑄₃ defined by (107). To this purpose we need the following result which is proved in [20, Corollary 3.2].

Lemma 27. Let 𝛿_3𝑘∈(0,1), 𝑘=1,2, be such that 𝜎₃=𝛿₃₁+𝛿₃₂∈(0,1/𝑝^′), 𝑝∈(1/(1−𝛿₃₁),∞). Then the convolution operator 𝒦 defined by (104) maps 𝐶^𝛿31([0,𝑇];𝑋₁)×𝐶^𝛿32([0,𝑇];𝑋₂) into 𝐶^𝜎3₀([0,𝑇];𝑋), and for every 𝑡∈[0,𝑇] satisfies the following estimate: 𝒦(𝑔₃₁,𝑔₃₂)_{𝜎3,0,𝑡;𝑋}≤𝑡^{−𝜎3+1/𝑝′}𝑐₃̃(𝑡)𝑔_{31𝛿31,0,𝑡;𝑋1}^𝑡₀𝑔₃₂^𝑝_{𝛿32,0,𝑠;𝑋2}d𝑠^1/𝑝. (132) Here 𝑐₃̃(𝑡) is a nondecreasing function of 𝑡 depending also on 𝛿₃₁ and 𝛿₃₂. Further, in the cases 𝛿₃₁∈(0,1), 𝛿₃₂=0, and 𝛿₃₁=𝛿₃₂=0, the following estimates hold, respectively, as follows: 𝒦𝑔₃₁,𝑔_{32𝛿31,0,𝑡;𝑋}≤𝐶₀𝑡^1−𝛿311+𝑡^𝛿31𝑔_{31𝛿31,0,𝑡;𝑋1}𝑔_{320,0,𝑡;𝑋2},𝒦𝑔₃₁,𝑔_{320,0,𝑡;𝑋}≤𝐶₀𝑡𝑔_{310,0,𝑡;𝑋1}𝑔_{320,0,𝑡;𝑋2}. (133)

From Lemmas 24 and 27 we obtain the following Lemma 28.

Lemma 28. Let 𝛼 and 𝛽 be as in Lemma 24. Then, for every 𝛿₃₁∈((3−2𝛼−𝛽)/𝛼,1) and 𝛿₃₂∈(0,1) such that 𝜎₃=𝛿₃₁+𝛿₃₂∈((3−2𝛼−𝛽)/𝛼,1/𝑝^′), 𝑝∈(1/(1−𝛿₃₁),∞), the operator 𝑄₃ defined by (107) maps 𝐶^𝛿31([0,𝑇];𝑋₁)×𝐶^𝛿32([0,𝑇];𝑋₂) into 𝐶^𝜈3₀([0,𝑇];𝑋), 𝜈₃=(𝛼𝜎₃+2𝛼+𝛽−3)/𝛼, and for every 𝑡∈[0,𝑇] satisfies the following estimate: 𝑄₃𝑔₃₁,𝑔_{32𝜈3,0,𝑡;𝑋}≤t^{−𝜎3+1/𝑝′}𝐶₂(𝑡)𝑐₃̃(𝑡)𝑔_{31𝛿31,0,𝑡;𝑋1}^𝑡₀𝑔₃₂^𝑝_{𝛿32,0,𝑠;𝑋2}d𝑠^1/𝑝. (134)

Proof. First, if 𝛿₃₁∈((3−2𝛼−𝛽)/𝛼,1) and 𝑝∈(1/(1−𝛿₃₁),∞), then 1/𝑝^′∈(𝛿₃₁,1)⊊((3−2𝛼−𝛽)/𝛼,1). Consequently, the assumption 𝜎₃=𝛿₃₁+𝛿₃₂∈((3−2𝛼−𝛽)/𝛼,1/𝑝^′), 𝛿₃₂∈(0,1), makes sense. Now, Lemma 27 yields 𝒦(𝑔₃₁,𝑔₃₂)∈𝐶^𝜎3₀([0,𝑇];𝑋) for any pair (𝑔₃₁,𝑔₃₂)∈𝐶^𝛿31([0,𝑇];𝑋₁)×𝐶^𝛿32([0,𝑇];𝑋₂). Then, recalling that 𝑄₃(𝑔₃₁,𝑔₃₂)=𝑄₂𝒦(𝑔₃₁,𝑔₃₂), the assertion follows from Lemma 24, with 𝛿₂ and 𝑔₂ being replaced by 𝜎₃ and 𝒦(𝑔₃₁,𝑔₃₂), respectively. Finally, (134) follows from (117) and (132).

We can now restore the loss of regularity produced by 𝑄₂.

Proposition 29. Let 5𝛼+2𝛽>6 in (H1). Then, for every 𝛿₃∈((3−2𝛼−𝛽)/𝛼,1/2), the operator 𝑄₃ defined by (107) maps 𝐶^𝛿3([0,𝑇];𝑋₁)×𝐶^𝛿3([0,𝑇];𝑋₂) into 𝐶^𝛿3₀([0,𝑇];𝑋), and for every 𝑡∈[0,𝑇] satisfies the following estimate, where 𝑝∈(1/(1−2𝛿₃),∞) and 𝐶₃(𝑡)=𝐶₂(𝑡)𝑐₃̃(𝑡)max{1,𝑡^{(𝛼𝛿3+2𝛼+𝛽−3)/𝛼}}: 𝑄₃𝑔₃₁,𝑔_{32𝛿3,0,𝑡;𝑋}≤𝑡^{1−2𝛿3−1/𝑝}𝐶₃(𝑡)𝑔_{31𝛿3,0,𝑡;𝑋1}^𝑡₀𝑔₃₂^𝑝_{𝛿3,0,𝑠;𝑋2}d𝑠^1/𝑝. (135)

Proof. Let 𝛿₃∈((3−2𝛼−𝛽)/𝛼,1/2) and let 𝑝∈(1/(1−2𝛿₃),∞)⊊(1/(1−𝛿₃),∞). Then, 2𝛿₃∈((6−4𝛼−2𝛽)/𝛼,1/𝑝^′)⊆((3−2𝛼−𝛽)/𝛼,1/𝑝^′). We are thus in position to apply Lemma 28 with 𝛿₃₁=𝛿₃₂=𝛿₃ from which we deduce that 𝑄₃ maps 𝐶^𝛿3([0,𝑇];𝑋₁)×𝐶^𝛿3([0,𝑇];𝑋₂) into 𝐶^𝜈3₀([0,𝑇];𝑋), 𝜈₃=(2𝛼𝛿₃+2𝛼+𝛽−3)/𝛼. But, since our choice for 𝛿₃ implies 𝜈₃>𝛿₃, we a fortiori have the fact that 𝑄₃ maps 𝐶^𝛿3([0,𝑇];𝑋₁)×𝐶^𝛿3([0,𝑇];𝑋₂) into 𝐶^𝛿3₀([0,𝑇];𝑋). Finally, (135) follows from (134) and the estimate ‖𝑔‖_{𝛾,0,𝑡;𝑋}≤max{1,𝑡^𝛿−𝛾}‖𝑔‖_{𝛿,0,𝑡;𝑋}, 𝑔∈𝐶^𝛿([0,𝑇];𝑋), 𝛿≥𝛾.

The next Lemma 30 concerns the operator 𝑄₄. Its proof is similar to that of Lemma 24, but with the essential difference that the presence of 𝑦∈𝑌^𝑝_𝛾 allows us to use estimate (79) in place of (14). As a consequence and provided to choose 𝛾 large enough, we will achieve a better result in which any loss of regularity is observed.

Lemma 30. Let 2𝛼+𝛽>2 in (H1) and 𝑟∈[1,∞]. Then, for every 𝛿₄∈(0,1) and 𝛾∈(3−2𝛼−𝛽,1) the operator 𝑄₄ defined by (108) maps 𝐶^𝛿4([0,𝑇];𝐂)×𝑌^𝑟_𝛾, 𝑌^𝑟_𝛾∈{(𝑋,𝒟(𝐴))_𝛾,𝑟,𝑋_𝐴^𝛾,𝑟}, into 𝐶^𝛿4₀([0,𝑇];𝑋), and for every 𝑡∈[0,𝑇] satisfies the following estimate: 𝑄₄𝑔₄,𝑦_{𝛿4,0,𝑡;𝑋}≤𝐶₄(𝑡)𝑡^{(2𝛼+𝛽+𝛾−3)/𝛼}𝑔_{4𝛿4,0,𝑡;𝐂}𝑦_𝑌𝑟𝛾. (136) Here 𝐶₄(𝑡) is a nondecreasing function of 𝑡 depending on 𝛼, 𝛽, 𝛿₄, 𝛾 and 𝑟.

Proof. Let 𝑡∈[0,𝑇], 𝑔₄∈𝐶^𝛿4([0,𝑇];𝐂), 𝛿₄∈(0,1), and 𝑦∈𝑌^𝑟_𝛾, 𝛾∈(3−2𝛼−𝛽,1), 𝑟∈[1,∞]. As in the proof of Lemma 24 we set 𝛼=(1−𝛼)/𝛼 and we observe that, since 2𝛼+𝛽>2 implies 𝛼∈(1/2,1], here 𝛼∈[0,1). Furthermore, we denote by 𝜎_{𝛼,𝛽,𝛾} the number (2𝛼+𝛽+𝛾−3)/𝛼∈(0,1), so that the exponents (𝛽+𝛾−2)/𝛼 and (𝛽+𝛾−3)/𝛼 appearing in (79) with 𝜃=1 and 𝜃=2 may be rewritten, as 𝜎_{𝛼,𝛽,𝛾}+𝛼−1 and 𝜎_{𝛼,𝛽,𝛾}−2, respectively. Then, using (79) with 𝜃=1, we obtain 𝑄₄𝑔₄,𝑦(𝜏)_𝑋≤𝑐₂₂𝑔_{4𝛿4,0,𝜏;𝐂}𝑦_𝑌𝑟𝛾^𝜏₀(𝜏−𝑠)^{𝛿4+𝜎𝛼,𝛽,𝛾+𝛼−1}d𝑠=𝑐₃₇𝑔_{4𝛿4,0,𝜏;𝐂}𝑦_𝑌𝑟𝛾𝜏^{𝛿4+𝜎𝛼,𝛽,𝛾+𝛼},∀𝜏∈[0,𝑡], (137) where 𝑐₃₇=𝑐₂₂(𝛿₄+𝜎_{𝛼,𝛽,𝛾}+𝛼)⁻¹. Hence, taking the supremum with respect to 𝜏∈[0,𝑡], one has 𝑄₄𝑔₄,𝑦_{0,0,𝑡;𝑋}≤𝑐₃₇𝑔_{4𝛿4,0,𝑡;𝐂}𝑦_𝑌𝑟𝛾𝑡^{𝛿4+𝜎𝛼,𝛽,𝛾+𝛼}. (138) Now, let (since [𝑄₄(𝑔₄,𝑦)](0)=0, the case 𝑡₁=0 follows from (137) with 𝜏=𝑡₂) 0<𝑡₁<𝑡₂≤𝑡. We have [𝑄₄(𝑔₄,𝑦)](𝑡₂)−[𝑄₄(𝑔₄,𝑦)](𝑡₁)=∑³_𝑘=1𝐽_{𝑘;𝑡1,𝑡2,𝑔4𝑦}, the 𝐽_{𝑘;𝑡1,𝑡2,𝑔}’s, 𝑔∶[0,𝑇]→𝑋, being as in (120). Using (13) with (𝑠,𝑡,𝜃)=(𝑡₁−𝑠,𝑡₂−𝑠,1), 𝑠∈(0,𝑡₁), and (79) with 𝜃=2, and letting (𝑐₃₈,𝑐₃₉)=(𝑐₂₂(1−𝛿₄)⁻¹,𝑐₃₈𝛿₄⁻¹), we get 𝐽_{1;𝑡1,𝑡2,𝑔4𝑦𝑋}≤𝑐₂₂𝑔_{4𝛿4,0,𝑡1;𝐂}𝑦_𝑌𝑟𝛾×^𝑡1₀^𝑡2_𝑡^−𝑠_1−𝑠𝜉^{𝜎𝛼,𝛽,𝛾−2}d𝜉𝑡₁−𝑠^𝛿4d𝑠≤𝑐₂₂𝑔_{4𝛿4,0,𝑡1;𝐂}𝑦_𝑌𝑟𝛾^𝑡1₀^𝑡2_𝑡^−𝑠_1−𝑠𝜉^{𝛿4+𝜎𝛼,𝛽,𝛾−2}d𝜉d𝑠≤𝑐₂₂𝑔_{4𝛿4,0,𝑡1;𝐂}𝑦_𝑌𝑟𝛾×^𝑡1₀𝑡₂−𝑠^{𝜎𝛼,𝛽,𝛾𝑡2}_𝑡^−𝑠_1−𝑠𝜉^𝛿4−2d𝜉d𝑠≤𝑐₃₈𝑔_{4𝛿4,0,𝑡2;𝐂}𝑦_𝑌𝑟𝛾𝑡^{𝜎𝛼,𝛽,𝛾}₂×^𝑡1₀𝑡₁−𝑠^𝛿4−1−𝑡₂−𝑠^𝛿4−1d𝑠=𝑐₃₉𝑔_{4𝛿4,0,𝑡2;𝐂}𝑦_𝑌𝑟𝛾𝑡^{𝜎𝛼,𝛽,𝛾}₂𝑡^𝛿4₁+𝑡₂−𝑡₁^𝛿4−𝑡^𝛿4₂≤𝑐₃₉𝑔_{4𝛿4,0,𝑡2;𝐂}𝑦_𝑌𝑟𝛾𝑡^{𝜎𝛼,𝛽,𝛾}₂𝑡₂−𝑡₁^𝛿4. (139) Now, let us examine 𝐽_{𝑘;𝑡1,𝑡2,𝑔4𝑦}, 𝑘=2,3. First, using (79) with 𝜃=1, we find 𝐽_{2;𝑡1,𝑡2,𝑔4𝑦𝑋}≤𝑐₂₂𝑔_{4𝛿4,0,𝑡2;𝐂}𝑦_𝑌𝑟𝛾^𝑡1₀𝑡₂−𝑠^{𝜎𝛼,𝛽,𝛾+𝛼−1}d𝑠𝑡₂−𝑡₁^𝛿4=𝑐₄₀𝑔_{4𝛿4,0,𝑡2;𝐂}𝑦_𝑌𝑟𝛾𝑡^{𝜎𝛼,𝛽,𝛾+𝛼}₂−𝑡₂−𝑡₁^{𝜎𝛼,𝛽,𝛾+𝛼}𝑡₂−𝑡₁^𝛿4≤𝑐₄₀𝑔_{4𝛿4,0,𝑡2;𝐂}𝑦_𝑌𝑟𝛾𝑡^{𝜎𝛼,𝛽,𝛾+𝛼}₂𝑡₂−𝑡₁^𝛿4. (140) Instead, the same computations made to derive (137) yield 𝐽_{3;𝑡1,𝑡2,𝑔4𝑦𝑋}≤𝑐₂₂𝑔_{4𝛿4,0,𝑡2;𝐂}𝑦_𝑌𝑟𝛾^𝑡2_𝑡1𝑡₂−𝑠^{𝛿4+𝜎𝛼,𝛽,𝛾+𝛼−1}d𝑠=𝑐₃₇𝑔_{4𝛿4,0,𝑡2;𝐂}𝑦_𝑌𝑟𝛾𝑡₂−𝑡₁^{𝛿4+𝜎𝛼,𝛽,𝛾+𝛼}. (141) From (139)–(141) and ‖[𝑄₄(𝑔₄,𝑦)](𝑡₂)−[𝑄₄(𝑔₄,𝑦)](𝑡₁)‖_𝑋≤∑³_𝑘=1‖𝐽_{𝑘;𝑡1,𝑡2,𝑔4𝑦}‖_𝑋, it follows that 𝑄₄𝑔₄,𝑦𝑡₂−𝑄₄𝑔₄,𝑦𝑡_1𝑋≤𝑐₄̃(𝑡)𝑡^{𝜎𝛼,𝛽,𝛾}𝑔_{4𝛿4,0,𝑡;𝐂}𝑦_𝑌𝑟𝛾𝑡₂−𝑡₁^𝛿4, (142) where 𝑐₄̃(𝑡)=𝑐₃₉+(𝑐₃₇+𝑐₄₀)𝑡^𝛼. Finally, summing up (138) and (142) we get (136) with 𝐶₄(𝑡)=𝑐₃₇𝑡^𝛿4+𝛼+𝑐₄̃(𝑡). The proof is complete.

Remark 31. Notice that if 𝑌^𝑟_𝛾=𝑋_𝐴^𝛾,𝑟, then in order to be sure that the conclusions of Lemma 30 hold with 𝑦 which really belongs to some intermediate space between 𝑋 and 𝒟(𝐴) we have to choose 𝛾∈(3−2𝛼−𝛽,𝛽). This is possible, provided that the stronger assumption 2𝛼+𝛽>3−𝛽≥2 is satisfied. Otherwise, if 2𝛼+𝛽∈(2,3−𝛽], 𝛽<1, then 𝛾∈(3−2𝛼−𝛽,1)⊊[𝛽,1) and 𝑦 may be contained in 𝒟(𝐴).

Finally, for the operator 𝑄₅ we have the following result. Again a loss of regularity is exhibited, even though of an amount smaller than that in Lemma 24 (cf. Remark 33).

Lemma 32. Let 2𝛼+𝛽>2 in (H1). Then, for every 𝛿₅∈((2−𝛼−𝛽)/𝛼,1), the operator 𝑄₅ defined by (109) maps 𝐶^𝛿5₀([0,𝑇];𝑋) into 𝐶^𝜈5₀([0,𝑇];𝑋), 𝜈₅=(𝛼𝛿₅+𝛼+𝛽−2)/𝛼∈(0,𝛿₅], and for every 𝑡∈[0,𝑇] satisfies the following estimate: 𝑄₅𝑔_{5𝜈5,0,𝑡;𝑋}≤𝐶₅(𝑡)𝑔_{5𝛿5,0,𝑡;𝑋}. (143) Here 𝐶₅(𝑡) is a nondecreasing function of 𝑡 depending also on 𝛼, 𝛽, and 𝛿₅.

Proof. Let 𝑔₅∈𝐶^𝛿5₀([0,𝑇];𝑋), 𝛿₅∈((2−𝛼−𝛽)/𝛼,1), and 𝜈₅=(𝛼𝛿₅+𝛼+𝛽−2)/𝛼∈(0,𝛿₅]. We still let 𝛼=(1−𝛼)/𝛼 and as in Lemma 30 we have 𝛼∈[0,1). Further, observe that 𝛿₅+(𝛽−1)/𝛼=𝜈₅+𝛼∈(0,𝛿₅]. Let 𝑡∈[0,𝑇]. Then, using (14) and 𝑔₅(0)=0, we get 𝑄₅𝑔_{50,0,𝑡;𝑋}≤sup_{[]𝜏∈0,𝑡}𝑐̃_𝛼,𝛽,0𝜏^{(𝛽−1)/𝛼}+1𝑔_{5𝛿5,0,𝜏;𝑋}𝜏^𝛿5≤𝑐̃_𝛼,𝛽,0+𝑡^{(1−𝛽)/𝛼}𝑔_{5𝛿5,0,𝑡;𝑋}𝑡^𝜈5+𝛼. (144) Now, let (since [𝑄₅𝑔₅](0)=0, the case 𝑡₁=0 follows from (144) and ‖[𝑄₅𝑔₅](𝑡₂)‖_𝑋≤‖𝑄₅𝑔₅‖_{0,0,𝑡2;𝑋}) 0<𝑡₁<𝑡₂≤𝑡. We have [𝑄₅𝑔₅](𝑡₂)−[𝑄₅𝑔₅](𝑡₁)=∑³_𝑘=1𝑈_{𝑘;𝑡1,𝑡2,𝑔5}, where for a function 𝑔∶[0,𝑇]→𝑋 we let 𝑈_{1;𝑡1,𝑡2,𝑔}∶=e^𝑡2𝐴𝑔𝑡₂−𝑔𝑡₁,𝑈_{2;𝑡1,𝑡2,𝑔}∶=e^𝑡2𝐴−e^𝑡1𝐴𝑔𝑡₁,𝑈_{3;𝑡1,𝑡2,𝑔}∶=𝑔𝑡₁−𝑔𝑡₂. (145) First, since 𝑡₂^{(𝛽−1)/𝛼}≤(𝑡₂−𝑡₁)^{(𝛽−1)/𝛼} for every 𝛽∈(0,1], we deduce that 𝑈_{1;𝑡1,𝑡2,𝑔5𝑋}≤𝑐̃_𝛼,𝛽,0𝑡₂^{(𝛽−1)/𝛼}𝑔_{5𝛿5,0,𝑡2;𝑋}𝑡₂−𝑡₁^𝛿5≤𝑐̃_𝛼,𝛽,0𝑔_{5𝛿5,0,𝑡2;𝑋}𝑡₂−𝑡₁^𝜈5+𝛼. (146) As far as 𝑈_{2;𝑡1,𝑡2,𝑔5} is concerned, instead, rewriting e^𝑡2𝐴−e^𝑡1𝐴 as −∫^𝑡2_𝑡1[(−𝐴)¹]^∘e^𝑟𝐴d𝑟 and using both 𝑔₅(0)=0 and (𝛼𝛿₅+𝛽−2)/𝛼=𝜈₅−1, it follows that 𝑈_{2;𝑡1,𝑡2,𝑔5𝑋}≤𝑐̃_𝛼,𝛽,1𝑔_{5𝛿5,0,𝑡1;𝑋}𝑡^𝛿5₁^𝑡2_𝑡1𝑟^{(𝛽−2)/𝛼}≤𝑐̃d𝑟_𝛼,𝛽,1𝑔_{5𝛿5,0,𝑡1;𝑋}^𝑡2_𝑡1𝑟^𝜈5−1≤𝑐̃d𝑟_𝛼,𝛽,1𝜈₅⁻¹𝑔_{5𝛿5,0,𝑡1;𝑋}𝑡^𝜈5₂−𝑡^𝜈5₁≤𝑐̃_𝛼,𝛽,1𝜈₅⁻¹𝑔_{5𝛿5,0,𝑡1;𝑋}𝑡₂−𝑡₁^𝜈5. (147) Then, since ‖𝑈_{3;𝑡1,𝑡2,𝑔5}‖_𝑋≤|𝑔₅|_{𝛿5,0,𝑡2;𝑋}(𝑡₂−𝑡₁)^𝛿5, from (146) and (147) we find 𝑄₅𝑔₅𝑡₂−𝑄₅𝑔₅𝑡_1𝑋≤³_𝑘=1𝑈_{𝑘;𝑡1,𝑡2,𝑔5𝑋}≤𝑐₅̃(𝑡)𝑔_{5𝛿5,0,𝑡;𝑋}𝑡₂−𝑡₁^𝜈5, (148) where 𝑐₅̃(𝑡)=𝑐̃_𝛼,𝛽,0𝑡^𝛼+𝑐̃_𝛼,𝛽,1𝜈₅⁻¹+𝑡^{𝛿5−𝜈5}. Summing up (144) and (148) we obtain (143) with 𝐶₅(𝑡)=[𝑐̃_𝛼,𝛽,0+𝑡^{(1−𝛽)/𝛼}]𝑡^𝜈5+𝛼+𝑐₅̃(𝑡). This completes the proof.

Remark 33. Thus, with the exception of 𝛽=1, 𝑄₅ produces a loss of regularity equal to 𝛿₅−𝜈₅=(2−𝛼−𝛽)/𝛼≤(3−2𝛼−𝛽)/𝛼. In this sense 𝑄₅ behaves better than 𝑄₂.

Remark 34. Notice that, under the weaker assumptions 𝛼+𝛽>1 and 𝑔₅∈𝐶([0,𝑇];𝑋), (86) with 𝑥=𝑔₅(𝑡), 𝑡∈[0,𝑇], yields 𝐴⁻¹[𝑄₅𝑔₅](𝑡)=[𝑄₅(𝐴⁻¹𝑔₅)](𝑡)=∫^𝑡₀e^{(𝑡−𝑠)𝐴}𝑔₅(𝑡)d𝑠.

Similarly as we have done in Proposition 29 for restoring the loss of regularity produced by 𝑄₂, we now show how Lemma 27 allows to restore that produced by 𝑄₅. We begin with the following version of Lemma 28 relative to 𝑄₆, and which is obtained combining Lemma 27 with Lemma 32 instead of Lemma 24.

Lemma 35. Let 𝛼 and 𝛽 be as in Lemma 32. Then, for every 𝛿₆₁∈((2−𝛼−𝛽)/𝛼,1) and 𝛿₆₂∈(0,1) such that 𝜎₆=𝛿₆₁+𝛿₆₂∈((2−𝛼−𝛽)/𝛼,1/𝑝^′), 𝑝∈(1/(1−𝛿₆₁),∞), the operator 𝑄₆ defined by (110) maps 𝐶^𝛿61([0,𝑇];𝑋₁)×𝐶^𝛿62([0,𝑇];𝑋₂) into 𝐶^𝜈6₀([0,𝑇];𝑋), 𝜈₆=(𝛼𝜎₆+𝛼+𝛽−2)/𝛼, and for every 𝑡∈[0,𝑇] satisfies the following estimate: 𝑄₆𝑔₆₁,𝑔_{62𝜈6,0,𝑡;𝑋}≤𝑡^{−𝜎6+1/𝑝′}𝐶₅(𝑡)𝑐₃̃(𝑡)𝑔_{61𝛿61,0,𝑡;𝑋1}^𝑡₀𝑔₆₂^𝑝_{𝛿62,0,𝑠;𝑋2}d𝑠^1/𝑝. (149)

Proof. First, if 𝛿₆₁∈((2−𝛼−𝛽)/𝛼,1) and 𝑝∈(1/(1−𝛿₆₁),∞), then 1/𝑝^′∈(𝛿₆₁,1)⊊((2−𝛼−𝛽)/𝛼,1). Consequently, the assumption 𝜎₆=𝛿₆₁+𝛿₆₂∈((2−𝛼−𝛽)/𝛼,1/𝑝^′) makes sense, provided to choose 𝛿₆₂∈(0,1) small enough. Lemma 27 then yields 𝒦(𝑔₆₁,𝑔₆₂)∈𝐶^𝜎6₀([0,𝑇];𝑋) for any pair (𝑔₆₁,𝑔₆₂)∈𝐶^𝛿61([0,𝑇];𝑋₁)×𝐶^𝛿62([0,𝑇];𝑋₂). Then, since 𝑄₆(𝑔₆₁,𝑔₆₂)=𝑄₅𝒦(𝑔₆₁,𝑔₆₂), the assertion follows from Lemma 32, with the pair (𝛿₅,𝑔₅) being replaced by (𝜎₆,𝒦(𝑔₆₁,𝑔₆₂)). Finally, (149) follows from (143) and (132).

From Lemma 35 we obtain the analogous of Proposition 29 for 𝑄₆.

Proposition 36. Let 3𝛼+2𝛽>4 in (H1). Then, for every 𝛿₆∈((2−𝛼−𝛽)/𝛼,1/2), the operator 𝑄₆ defined by (110) maps 𝐶^𝛿6([0,𝑇];𝑋₁)×𝐶^𝛿6([0,𝑇];𝑋₂) into 𝐶^𝛿6₀([0,𝑇];𝑋), and for every 𝑡∈[0,𝑇] satisfies the following estimate, where 𝑝∈(1/(1−2𝛿₆),∞) and 𝐶₆(𝑡)=𝐶₅(𝑡)𝑐₃̃(𝑡)max{1,𝑡^{(𝛼𝛿6+𝛼+𝛽−2)/𝛼}}: 𝑄₆𝑔₆₁,𝑔_{62𝛿6,0,𝑡;𝑋}≤𝑡^{1−2𝛿6−1/𝑝}𝐶₆(𝑡)𝑔_{61𝛿6,0,𝑡;𝑋1}^𝑡₀𝑔₆₂^𝑝_{𝛿6,0,𝑠;𝑋2}d𝑠^1/𝑝. (150)

Proof. Let 𝛿₆∈((2−𝛼−𝛽)/𝛼,1/2) and 𝑝∈(1/(1−2𝛿₆),∞)⊊(1/(1−𝛿₆),∞). Then, 2𝛿₆∈((4−2𝛼−2𝛽)/𝛼,1/𝑝^′)⊆((2−𝛼−𝛽)/𝛼,1/𝑝^′) and we can apply Lemma 35 with 𝛿_6𝑘=𝛿₆, 𝑘=1,2. We thus deduce that 𝑄₆ maps 𝐶^𝛿6([0,𝑇];𝑋₁)×𝐶^𝛿6([0,𝑇];𝑋₂) into 𝐶^𝜈6₀([0,𝑇];𝑋), 𝜈₆=(2𝛼𝛿₆+𝛼+𝛽−2)/𝛼. But, since 𝛿₆>(2−𝛼−𝛽)/𝛼 implies 𝜈₆>𝛿₆, we a fortiori have the fact that 𝑄₆ maps 𝐶^𝛿6([0,𝑇];𝑋₁)×𝐶^𝛿6([0,𝑇];𝑋₂) into 𝐶^𝛿6₀([0,𝑇];𝑋). Finally, (150) follows from (149) and ‖𝑄₆(𝑔₆₁,𝑔₆₂)‖_{𝛿6,0,𝑡;𝑋}≤max{1,𝑡^{𝜈6−𝛿6}}‖𝑄₆(𝑔₆₁,𝑔₆₂)‖_{𝜈6,0,𝑡;𝑋}.

In Section 6 we will also encounter 𝑄₅ acting on functions which enjoy some space regularity, that is, functions 𝑔₅ which are Hölder continuous in time with values on 𝑌^𝑟_𝛾∈{(𝑋,𝒟(𝐴))_𝛾,𝑟,𝑋_𝐴^𝛾,𝑟}. In this case Lemma 32 can be refined, and the loss of regularity produced by 𝑄₅ is naturally restored by the additional condition of space regularity on 𝑔₅. In some sense, the forthcoming Corollary 38 is the analogous of Lemma 30, where the function 𝑔₄𝑦 involved in the definition of 𝑄₄(𝑔₄,𝑦) (cf. (108)) was of class 𝐶^𝛿4([0,𝑇];𝑌^𝑟_𝛾).

Lemma 37. Let 𝛼+𝛽>1 in (H1) and 𝑌^𝑟_𝛾∈{(𝑋,𝒟(𝐴))_𝛾,𝑟,𝑋_𝐴^𝛾,𝑟}, 𝛾∈(2−𝛼−𝛽,1), 𝑟∈[1,∞]. Then, for every 𝛿₅∈(0,(𝛼+𝛽+𝛾−2)/𝛼], the operator 𝑄₅ defined by (109) maps 𝐶^𝛿5([0,𝑇];𝑌^𝑟_𝛾) into 𝐶^𝛿5₀([0,𝑇],𝑋), and for every 𝑡∈[0,𝑇] satisfies the following estimate: 𝑄₅𝑔_{5𝛿5,0,𝑡;𝑋}≤𝑐₄₁𝑡^{(𝛼+𝛽+𝛾−2−𝛼𝛿5)/𝛼}2𝑡^𝛿5+1𝑔_{5𝛿5,0,𝑡;𝑌𝑟𝛾}. (151) Here 𝑐₄₁ is a positive constant depending on 𝛼, 𝛽, 𝛾, and 𝑟.

Proof. Let 𝛾∈(2−𝛼−𝛽,1)⊆(1−𝛽,1) and let 𝜒_{𝛼,𝛽,𝛾} be the number (𝛼+𝛽+𝛾−2)/𝛼∈(0,1), so that the exponent (𝛽+𝛾−2)/𝛼 in (79) with 𝜃=1 is equal to 𝜒_{𝛼,𝛽,𝛾}−1. Let 𝑔₅∈𝐶^𝛿5([0,𝑇];𝑌^𝑟_𝛾), 𝛿₅∈(0,𝜒_{𝛼,𝛽,𝛾}], 𝑟∈[1,∞]. Since [𝑄₅𝑔₅](0)=0, we assume that 𝑡∈(0,𝑇] and we observe that, due to Propositions 6 and 12, [𝑄₅𝑔₅](𝑡) is rewritten as follows: 𝑄₅𝑔₅(𝑡)=e^𝑡𝐴−𝐼𝑔₅(𝑡)=lim_𝜀→0+e^𝑡𝐴−e^𝜀𝐴𝑔₅(𝑡)=lim_𝜀→0+^𝑡_𝜀𝐷_𝑠e^𝑠𝐴𝑔₅(𝑡)d𝑠=−lim_𝜀→0+^𝑡_𝜀(−𝐴)^1∘e^𝑠𝐴𝑔₅(𝑡)d𝑠=−^𝑡₀(−𝐴)^1∘e^𝑠𝐴𝑔₅(𝑡)d𝑠. (152) Indeed, for every 𝜀∈[0,𝑡) and 𝑥∈𝑌^𝑟_𝛾, (79) with 𝜃=1 yields (−𝐴)^𝑡_𝜀^1∘e^𝑠𝐴𝑥d𝑠_𝑋≤𝑐₂₂‖𝑥‖_𝑌^𝑟_𝛾^𝑡_𝜀𝑠^{𝜒𝛼,𝛽,𝛾−1}d𝑠≤𝑐₄₁‖𝑥‖_𝑌^𝑟_𝛾(𝑡−𝜀)^{𝜒𝛼,𝛽,𝛾}, (153) where 𝑐₄₁=𝑐₂₂𝜒⁻¹_{𝛼,𝛽,𝛾}. From (152) and (153) with (𝜀,𝑡,𝑥)=(0,𝜏,𝑔₅(𝜏)) we thus get 𝑄₅𝑔_{50,0,𝑡;𝑋}=sup_{𝜏∈[0,𝑡]}𝑄₅𝑔₅(𝜏)_𝑋≤𝑐₄₁𝑔_{50,0,𝑡;𝑌𝑟𝛾}𝑡^{𝜒𝛼,𝛽,𝛾}. (154) Now, let 0≤𝑡₁<𝑡₂≤𝑡. From (152) it follows that [𝑄₅𝑔₅](𝑡₂)−[𝑄₅𝑔₅](𝑡₁)=−∑²_𝑘=1𝑉_{𝑘;𝑡1,𝑡2,𝑔5}, where for every function 𝑔∶[0,𝑇]→𝑌^𝑝_𝛾 we have set 𝑉_{1;𝑡1,𝑡2,𝑔}∶=^𝑡1₀(−𝐴)^1∘e^𝑠𝐴𝑔𝑡₂−𝑔𝑡₁𝑉d𝑠,_{2;𝑡1,𝑡2,𝑔}∶=^𝑡2_𝑡1(−𝐴)^1∘e^𝑠𝐴𝑔𝑡₂d𝑠. (155) Hence, using (153) with the triplet (𝜀,𝑡,𝑥) being replaced by (0,𝑡₁,𝑔₅(𝑡₂)−𝑔₅(𝑡₁)) and (𝑡₁,𝑡₂,𝑔₅(𝑡₂)), respectively, we deduce that 𝑉_{1;𝑡1,𝑡2,𝑔5𝑋}≤𝑐₄₁𝑔_{5𝛿5,0,𝑡2;𝑌𝑟𝛾}𝑡^{𝜒𝛼,𝛽,𝛾}₁𝑡₂−𝑡₁^𝛿5,𝑉_{2;𝑡1,𝑡2,𝑔5𝑋}≤𝑐₄₁𝑔_{50,0,𝑡2;𝑌𝑟𝛾}𝑡₂−𝑡₁^{𝜒𝛼,𝛽,𝛾}. (156) As a consequence, since 𝛿₅∈(0,𝜒_{𝛼,𝛽,𝛾}], 𝑄₅𝑔₅𝑡₂−𝑄₅𝑔₅𝑡_1𝑋≤𝑐₄₁𝑡^{𝜒𝛼,𝛽,𝛾−𝛿5}𝑡^𝛿5+1𝑔_{5𝛿5,0,𝑡;𝑌𝑟𝛾}𝑡₂−𝑡₁^𝛿5. (157) Summing up (154) and (157), we obtain (151). The proof is complete.

Since in Lemma 37 it is not required that 𝑔₅(0)=0, the special case of the constant function 𝑔₅(𝑡)=𝑥∈𝑌^𝑝_𝛾, 𝑡∈[0,𝑇], is admissible, and we obtain the following result.

Corollary 38. Let 𝛼, 𝛽, and 𝑌^𝑟_𝛾 be as in Lemma 37, and let 𝑥∈𝑌^𝑟_𝛾, 𝛾∈(2−𝛼−𝛽,1), and 𝑟∈[1,∞]. Then, for every 𝛿₇∈(0,(𝛼+𝛽+𝛾−2)/𝛼], the function [𝑄₇𝑥](⋅)∶=(𝑒^⋅𝐴−𝐼)𝑥 belongs to 𝐶^𝛿7₀([0,𝑇];𝑋), and for every 𝑡∈[0,𝑇] satisfies the estimate 𝑄₇𝑥_{𝛿7,0,𝑡;𝑋}≤𝑐₄₁𝑡^{(𝛼+𝛽+𝛾−2−𝛼𝛿7)/𝛼}𝑡^𝛿7+1𝑐₄₁‖𝑥‖_𝑌^𝑟_𝛾. (158)

Proof. Let 𝑔₅(𝑡)=𝑥 in the proof of Lemma 37, and observe that 𝑉_{1,𝑡1,𝑡2,𝑔5} reduces to the zero element of 𝑋. Estimate (158) then follows from (154) and the second estimate in (156).

For later purposes, we conclude the section with the following remark.

Remark 39. The condition 5𝛼+2𝛽>6 in (H1) required in Proposition 29 is the strongest among the conditions for the pair (𝛼,𝛽) required in Corollary 14 and the other results of this section. Indeed, 5𝛼+2𝛽>6⟹3𝛼+2𝛽>6−2𝛼≥4⟹3𝛼+𝛽>4−𝛽≥3⟹2𝛼+𝛽>3−𝛼≥2⟹𝛼+𝛽>2−𝛼≥1. (159) Hence, if 5𝛼+2𝛽>6, then Corollary 14 and all the results from Lemma 22 to Corollary 38 are applicable. Next we will make large usage of this fact, but we warn the reader that, for brevity and regarding it as acquired, we will not mention it anymore.

5. Application to Maximal Time Regularity

The results of the previous sections are here applied to correct, refine, and extend the results in [20] concerning the maximal time regularity of the solutions to a class of degenerate abstract evolution equations. Let (𝑋,‖⋅‖_𝑋) and (𝑍,‖⋅‖_𝑍) be two complex Banach spaces, and consider the following degenerate first-order integrodifferential Cauchy problem for 𝑣∶𝐼_𝑇→𝑋, where 𝐼_𝑇=[0,𝑇], 𝑇>0, and 𝑛₁,𝑛₂∈𝐍: 𝐷_𝑡(𝑀𝑣(𝑡))=𝜆₀𝑀+𝐿𝑣(𝑡)+^𝑛1_𝑖1=1𝒦𝑘_𝑖1,𝐿_𝑖1𝑣(𝑡)+^𝑛2_𝑖2=1ℎ_𝑖2(𝑡)𝑦_𝑖2+𝑓(𝑡),𝑡∈𝐼_𝑇,𝑀𝑣(0)=𝑀𝑣₀. (160) Here 𝒦 is the convolution operator (104) in which (𝑋₁,𝑋₂,𝑋₃)=(𝑍,𝑋,𝑋), whereas 𝑀, 𝐿, and 𝐿_𝑖1, 𝑖₁=1,…,𝑛₁, are closed single-valued linear operators from 𝑋 to itself, whose domains fulfill the relation 𝒟(𝐿)⊆⋂^𝑛1_𝑖1=1[𝒟(𝑀)∩𝒟(𝐿_𝑖1)]. Further, we assume that 𝐿admitsacontinuousinverseoperator𝐿⁻¹(𝑋),(𝐿),∈ℒi.e.,0∈𝜌 (161) whereas we allow 𝑀 to have no bounded inverse. Hence, in general, 𝐴∶=𝐿𝑀⁻¹ is only the m. l. operator defined by 𝒟(𝐴)=𝑥∈𝒟𝑀⁻¹∶𝐿𝑀⁻¹𝑥=(𝑀)≠∅𝑥∈ℛ∶𝑀⁻¹(𝐿)={(𝑀)∶(𝐿)𝑥∩𝒟≠∅𝑥∈ℛthereexists𝑦∈𝒟suchthat𝑦∈𝑀⁻¹𝑥=(𝑀)(𝐿(𝒟(𝐿,𝑥∈ℛ∶𝑥=𝑀𝑦forsome𝑦∈𝒟)=𝑀))𝐴𝑥=_{𝑦∈𝑀−1(𝐿)𝑥∩𝒟}=(𝐿),(𝐴).𝐿𝑦𝐿𝑦∶𝑦∈𝒟suchthat𝑥=𝑀𝑦𝑥∈𝒟 (162) Therefore, problem (160) can not be reduced, via the change of unknown 𝑢=𝑀𝑣, to an integrodifferential problem related to single-valued linear operators. On the contrary, due to (161) and the closed graph theorem, 𝑀𝐿⁻¹,𝐿_𝑖1𝐿⁻¹∈ℒ(𝑋), 𝑖₁=1,…,𝑛₁. As far as the data vector (𝜆₀,𝑣₀,𝑘₁,…,𝑘_𝑛1,ℎ₁,…,ℎ_𝑛2,𝑦₁,…,𝑦_𝑛2,𝑓) is concerned, at the moment, we only assume 𝜆₀∈𝐂, 𝑣₀∈𝒟(𝑀), 𝑘_𝑖1∶𝐼_𝑇→𝑍, ℎ_𝑖2∶𝐼_𝑇→𝐂, 𝑦_𝑖2∈𝑋, 𝑖_𝑙=1,…,𝑛_𝑙, 𝑙=1,2, and 𝑓∶𝐼_𝑇→𝑋, in order that (160) makes sense in 𝑋. This minimal assumptions will be refined later. In general, only strict solutions 𝑣 to (160) shall be investigated, where (cf. [22, 23]) by a strict solution 𝑣 to (160) we mean that, 𝒟(𝐿) being endowed with the graph norm ‖⋅‖_𝒟(𝐿)=‖⋅‖_𝑋+‖𝐿⋅‖_𝑋, 𝑣∈𝐶(𝐼_𝑇;𝒟(𝐿)), 𝑀𝑣∈𝐶¹(𝐼_𝑇;𝑋), and (160) holds. Clearly, if 𝑀⁻¹ is really a m. l. operator, then 𝑀𝑣(0)=𝑀𝑣₀ does not necessarily mean 𝑣(0)=𝑣₀, but only 𝑣(0)−𝑣₀∈𝑀⁻¹0. As we will see below, if 𝑣₀∈𝒟(𝐿) and the data 𝑘_𝑖1, ℎ_𝑖2, 𝑦_𝑖2 and 𝑓, 𝑖_𝑙=1,…,𝑛_𝑙, 𝑙=1,2, satisfy suitable assumptions, then for a strict solution 𝑣 to (160) it just holds 𝑣(0)=𝑣₀. Throughout the section, 𝑌^𝑞_𝜓, 𝜓∈(0,1), 𝑞∈[1,∞], will always denote one between the spaces (𝑋,𝒟(𝐴))_𝜓,𝑞 and 𝑋_𝐴^𝜓,𝑞, 𝐴 being defined by (162). That is, 𝑌^𝑞_𝜓∈{(𝑋,𝒟(𝐴))_𝜓,𝑞,𝑋_𝐴^𝜓,𝑞}. To avoid confusion, if more than a single 𝑌^𝑞_𝜓 is involved in some statement, that is, if we write 𝑥_𝑗∈𝑌^𝑞_𝜓𝑗, 𝑗=1,…,𝑛, 𝑛∈𝐍, then it is understood that the same choice has been made for all the 𝑌^𝑞_𝜓𝑗 in the sense that 𝑌^𝑞_𝜓𝑗=(𝑋,𝒟(𝐴))_{𝜓𝑗,𝑞} or 𝑌^𝑞_𝜓𝑗=𝑋^𝜓𝑗_𝐴^,𝑞 for every 𝑗=1,…,𝑛.

According to [2, Section 1.6], we recall that the 𝑀-modified resolvent set 𝜌_𝑀(𝐿) of 𝐿 is defined to be the set {𝑧∈𝐂∶(𝑧𝑀−𝐿)⁻¹∈ℒ(𝑋)}. The bounded operator (𝑧𝑀−𝐿)⁻¹ is called the 𝑀 modified resolvent of 𝐿. It is easy to prove that 𝜌_𝑀(𝐿)⊆𝜌(𝐴) and that 𝑀(𝑧𝑀−𝐿)⁻¹=(𝑧𝐼−𝐴)⁻¹, 𝑧∈𝜌_𝑀(𝐿) (cf. [2, Theorem 1.14]). With the notion of 𝑀-modified resolvent of 𝐿 at hand, we assume that(H2)𝜌_𝑀(𝐿) contains a region Σ_𝛼={𝑧∈𝐂∶ℜe𝑧≥−𝑐(|ℑm𝑧|+1)^𝛼,ℑm𝑧∈𝐑}, 𝛼∈(0,1], 𝑐>0, and for some exponent 𝛽∈(0,𝛼] and constant 𝐶>0 the estimate ‖𝑀(𝜆𝑀−𝐿)⁻¹‖_ℒ(𝑋)≤𝐶(|𝜆|+1)^−𝛽 holds for every 𝜆∈Σ_𝛼.

Before we proceed with our analysis we remark that, due to the wide range of choices for the data vector, problem (160) contains many subcases at its interior. So, in spite of the case when at least one between the 𝑘_𝑖’s is different from zero and problem (160) is really an integrodifferential one, the choice 𝑘_𝑖1=0, 𝑖₁=1,…,𝑛₁, yields to consider also various nonintegrodifferential degenerate problems. For instance, those corresponding to 𝜆₀=𝑘_𝑖1=ℎ_𝑖2=0 and 𝜆₀=𝑘_𝑖1=𝑓=0, 𝑖_𝑙=1,…,𝑛_𝑙, 𝑙=1,2, respectively: 𝐷_𝑡(𝑀𝑣(𝑡))=𝐿𝑣(𝑡)+𝑓(𝑡),𝑡∈𝐼_𝑇,𝑀𝑣(0)=𝑀𝑣₀,𝐷_𝑡(𝑀𝑣(𝑡))=𝐿𝑣(𝑡)+^𝑛2_𝑖2=1ℎ_𝑖2(𝑡)𝑦_𝑖2,𝑡∈𝐼_𝑇,𝑀𝑣(0)=𝑀𝑣₀. (163)(164) Although (164) differs from (163) only in the fact that 𝑓 is replaced with ∑^𝑛2_𝑖2=1ℎ_𝑖2(𝑡)𝑦_𝑖2; nevertheless a very different result is achieved when the 𝑦_𝑖2’s are assumed to belong to 𝑌^𝑟_𝛾𝑖2, at least for opportunely chosen 𝛾_𝑖2∈(0,1), 𝑖₂=1,…,𝑛₂. As we will see (cf. Remark 51 and Theorem 56), in this situation the loss of time regularity for the pair (𝐿𝑣,𝐷_𝑡𝑀𝑣) with respect to that of 𝑓, typical of the case 𝛽<1 in (H2) (see [21, Theorem 9], [2, Theorem 3.26], and [22, Theorem 7.2]), can be restored in order that (𝐿𝑣,𝐷_𝑡𝑀𝑣) possesses the maximal time regularity which is the minimal between the time regularities of the ℎ_𝑖2’s. The same phenomenon is carried over into the integrodifferential case for the following problems, corresponding to 𝜆₀=ℎ_𝑖2=0, 𝑖₂=1,…,𝑛₂, and 𝜆₀=𝑓=0: 𝐷_𝑡(𝑀𝑣(𝑡))=𝐿𝑣(𝑡)+^𝑛1_𝑖1=1𝒦𝑘_𝑖1,𝐿_𝑖1𝑣(𝑡)(𝑡),(0)+𝑓𝑀𝑣=𝑀𝑣₀,𝐷_𝑡(𝑀𝑣(𝑡))=𝐿𝑣(𝑡)+^𝑛1_𝑖1=1𝒦𝑘_𝑖1,𝐿_𝑖1𝑣(𝑡)+^𝑛2_𝑖2=1ℎ_𝑖2(𝑡)𝑦_𝑖2,𝑀𝑣(0)=𝑀𝑣₀, (165)(166)𝑡∈𝐼_𝑇. When 𝛽<1, the loss of time regularity for the pair (𝐿𝑣,𝐷_𝑡𝑀𝑣) with respect to that of the vector (𝑘₁,…,𝑘_𝑛1,𝑓) in problem (165) (cf. [22, Theorem 7.1] and [23, Theorem 2.1] for 𝑛₁=1) can be restored in problem (166) assuming that 𝑦_𝑖2∈𝑌^𝑟_𝛾𝑖2, 𝑖₂=1,…,𝑛₂. In this context (cf. Remark 51 and Theorem 53) the pair (𝐿𝑣,𝐷_𝑡𝑀𝑣) has the maximal time regularity which is the minimal between the time regularities of the 𝑘_𝑖1’s and ℎ_𝑖2’s.

We stress that, if 𝛽=1, then no loss of time regularity is observed and all the quoted results agree with the well-known theory of maximal regularity in spaces of continuous functions for the nondegenerate version of (160), corresponding to the case when 𝑀=𝐼 and 𝐿 generates an analytic semigroup. Hence, roughly speaking, one can verify the consistency of any result on problem (160) with condition (H2) simply by letting 𝛽=1 on its statement, and then checking if it is compatible with those for the nondegenerate case. To this purpose, we recall that the question of maximal regularity for the nondegenerate (possibly nonautonomous) version of (160) has been deeply investigated by several authors. See, for instance, [4, 6–8, 10, 32] for problem (165) with (𝑀,𝛽,𝑛₁)=(𝐼,1,1) and [9, 11] for problem (163) with (𝑀,𝛽)=(𝐼,1).

Finally, assumption (161) excludes the case of 𝐿=0 in (160), so that our results cannot be compared with those in [5, 33, 34]. There, assuming that the bilinear bounded operator 𝒫 underlying the definition of 𝒦 is the scalar multiplication in 𝑋, the problem 𝐷_𝑡𝑣(𝑡)=𝒦𝑘₁,𝐿₁𝑣(𝑡)+𝑓(𝑡),𝑡∈𝐼_𝑇,𝑣(0)=𝑣₀ (167) is treated under the following assumptions: (i) 𝐿₁ is a closed densely defined linear operator generating an analytic semigroup; (ii) 𝑘₁∶[0,∞)→𝐑 is absolutely Laplace transformable. Observe that, if (𝑘₁,𝑓)=(1,0), then problem (167) reduces to the abstact wave equation 𝐷²_𝑡𝑣(𝑡)=𝐿₁𝑣(𝑡), 𝐷_𝑡𝑣(0)=0, 𝑣(0)=𝑣₀, whereas when 𝑀=𝐼 and 𝜆₀=𝑘_𝑖1=ℎ_𝑖2=𝑓=0, 𝑖_𝑙=1,…,𝑛_𝑙, 𝑙=1,2, problem (160) reduces to the abstract heat equation 𝐷_𝑡𝑣(𝑡)=𝐿𝑣(𝑡), 𝑣(0)=𝑣₀. In other words, whereas [5, 33, 34] are concerned with the hyperbolic case, here we are concerned with the parabolic one.

Let us now come back to problem (160). Of course, assumption (H2) implies that the operator 𝐴 defined by (162) satisfies (H1), so that it generates a semigroup {e^𝑡𝐴}_𝑡≥0 defined by e^0𝐴=𝐼 and (9) and satisfying (14). Assuming that 𝑣₀∈𝒟(𝐿), we let 𝑤=𝐿𝑣−𝑣₀⟺𝑣=𝐿⁻¹𝑤+𝑣₀. (168) Then, by setting 𝐹_𝑤(𝑡)=𝜆₀𝐴⁻¹𝑤(𝑡)+^𝑛1_𝑖1=1𝒦𝑘_𝑖1,𝑆_𝑖1𝑤(𝑡)+𝒦𝑘_𝑖1,𝐿_𝑖1𝑣₀(𝑡+_𝑛2𝑖2=1)ℎ_𝑖2(𝑡)𝑦_𝑖2+𝑣₁+𝑓(𝑡),𝑡∈𝐼_𝑇, (169) where 𝐴⁻¹=𝑀𝐿⁻¹∈ℒ(𝑋), 𝑆_𝑖1=𝐿_𝑖1𝐿⁻¹∈ℒ(𝑋), 𝑖₁=1,…,𝑛₁, and 𝑣₁=(𝜆₀𝑀+𝐿)𝑣₀, we see that 𝑣 is a strict solution to (160) if and only if 𝑤 satisfies (indeed, if 𝑣∈𝐶(𝐼_𝑇;𝒟(𝐿)), then ‖𝑤(𝑡)−𝑤(𝑠)‖_𝑋=‖𝐿[𝑣(𝑡)−𝑣(𝑠)]‖_𝑋≤‖𝑣(𝑡)−𝑣(𝑠)‖_𝒟(𝐿)→0 as 𝑠→𝑡, 𝑡,𝑠∈𝐼_𝑇, that is, 𝑤∈𝐶(𝐼_𝑇;𝑋). Conversely, if 𝑤∈𝐶(𝐼_𝑇;𝑋), then 𝑣=𝐿⁻¹𝑤+𝑣₀∈𝒟(𝐿) and ‖𝑣(𝑡)−𝑣(𝑠)‖_𝒟(𝐿)≤(‖𝐿⁻¹‖_ℒ(𝑋)+1)‖𝑤(𝑡)−𝑤(𝑠)‖_𝑋→0 as 𝑠→𝑡, 𝑡,𝑠∈𝐼_𝑇, that is, 𝑣∈𝐶(𝐼_𝑇;𝒟(𝐿)). Finally, since 𝑀𝑣=𝐴⁻¹𝑤+𝑀𝑣₀, we have 𝑀𝑣∈𝐶¹(𝐼_𝑇;𝑋) if and only if 𝐴⁻¹𝑤∈𝐶¹(𝐼_𝑇;𝑋)) 𝑤∈𝐶(𝐼_𝑇;𝑋), 𝐴⁻¹𝑤∈𝐶¹(𝐼_𝑇;𝑋), and solves to the following problem: 𝐷_𝑡𝐴⁻¹𝑤(𝑡)=𝑤(𝑡)+𝐹_𝑤(𝑡)∈𝐴𝐴⁻¹𝑤(𝑡)+𝐹_𝑤(𝑡),𝑡∈𝐼_𝑇,𝐴⁻¹𝑤(0)=0(i.e.,𝑤(0)∈𝐴0). (170) Now let 2𝛼+𝛽>2, and assume that 𝑘_𝑖1∈𝐶^𝜂𝑖1(𝐼_𝑇;𝑍), ℎ_𝑖2∈𝐶^𝜎𝑖2(𝐼_𝑇;𝐂), and 𝑓∈𝐶^𝜇(𝐼_𝑇;𝑋), where 𝜂_𝑖1,𝜎_𝑖2,𝜇∈(2−𝛼−𝛽/𝛼,1), 𝑖_𝑙=1,…,𝑛_𝑙, 𝑙=1,2. Then, if 𝑤∈𝐶(𝐼_𝑇;𝑋) is a solution to (170) such that 𝐴⁻¹𝑤∈𝐶¹(𝐼_𝑇;𝑋), the function 𝐹_𝑤 satisfies 𝐹_𝑤∈𝐶^𝛿𝐼_𝑇;𝑋,𝛿=min_{𝑖𝑘=1,…,𝑛𝑘,𝑘=1,2}𝜂_𝑖1,𝜎_𝑖2,𝜇∈2−𝛼−𝛽𝛼,1. (171) Indeed, 𝛿 being the smallest Hölder exponent, for every 𝑖_𝑙=1,…,𝑛_𝑙, 𝑙=1,2, we have 𝐴⁻¹𝑤,ℎ_𝑖2𝑦_𝑖2,𝑓∈𝐶^𝛿(𝐼_𝑇;𝑋) and 𝒦(𝑘_𝑖1,𝑆_𝑖1𝑤), 𝒦(𝑘_𝑖1,𝐿_𝑖1𝑣₀)∈𝐶^𝜂𝑖1₀(𝐼_𝑇;𝑋)↪𝐶^𝛿₀(𝐼_𝑇;𝑋) (cf. Lemma 27 for the case (𝛿₃₁,𝛿₃₂,𝑋₁,𝑋₂)=(𝜂_𝑖1,0,𝑍,𝑋) with the pair (𝑔₃₁,𝑔₃₂) being replaced by (in fact, since 𝑆_𝑖1=𝐿_𝑖1𝐿⁻¹∈ℒ(𝑋), 𝑖₁=1,…,𝑛₁, if 𝑤∈𝐶(𝐼_𝑇;𝑋), then 𝑆_𝑖1𝑤∈𝐶(𝐼_𝑇;𝑋), whereas the constant functions 𝜅_𝑖1(𝑡)=𝐿_𝑖1𝑣₀, 𝑡∈𝐼_𝑇, 𝑖₁=1,…,𝑛₁, obviously belong to 𝐶(𝐼_𝑇;𝑋)) (𝑘_𝑖1,𝑆_𝑖1𝑤) and (𝑘_𝑖1,𝐿_𝑖1𝑣₀), resp.). Consequently (cf. [2, Theorem 3.7 and Remark p. 54] with 𝑢₀=0), the solution 𝐴⁻¹𝑤 to the multivalued evolution problem 𝐷_𝑡(𝐴⁻¹𝑤)∈𝐴(𝐴⁻¹𝑤)+𝐹_𝑤, 𝐴⁻¹𝑤(0)=0 is necessarily of the form 𝐴⁻¹𝑤(𝑡)=𝑄₁𝐹_𝑤(𝑡),𝑡∈𝐼_𝑇, (172) with 𝑄₁ being the operator defined by (105). Further (cf. [2, Remark p. 55] with 𝑢₀=0, and where 𝐴^∘e^𝑡𝐴 stands for 𝐷_𝑡e^𝑡𝐴=−[(−𝐴)¹]^∘e^𝑡𝐴) the derivative of 𝐴⁻¹𝑤 is given by 𝐷_𝑡𝐴⁻¹𝑤(𝑡)=e^𝑡𝐴𝐹_𝑤(𝑡)−𝑄₂𝐹_𝑤(𝑡),𝑡∈𝐼_𝑇⧵{0}, (173) with 𝑄₂ being the operator in (106). Notice that 𝑄₂𝐹_𝑤 is well defined by virtue of (127) with 𝑔=𝐹_𝑤. Now let 𝑦_𝑖2∈𝑌^𝑟_𝛾𝑖2 and 𝑣₁+𝑓(0)∈𝑌^𝑟_𝜑 where 𝛾_𝑖2,𝜑∈(1−𝛽,1), 𝑖₂=1,…,𝑛₂, and 𝑟∈[1,∞]. Since 𝐴⁻¹𝑤(0)=𝒦(𝑘_𝑖1,𝑆_𝑖1𝑤)(0)=𝒦(𝑘_𝑖1,𝐿_𝑖1𝑣₀)(0)=0, 𝑖₁=1,…,𝑛₁, from (169) it thus follows that 𝐹_𝑤(0)∶=𝑥₀ is independent on 𝑤 and 𝑥₀=^𝑛2_𝑖2=1ℎ_𝑖2(0)𝑦_𝑖2+𝑣₁+𝑓(0)∈𝑌^𝑟_𝛾,𝛾=min_{𝑖2=1,…,𝑛2}𝛾_𝑖2,𝜑∈1−𝛽,1. (174) Indeed (cf. (20) or (38)), we have 𝑌^𝑟_𝛾𝑖2↪𝑌^𝑟_𝛾, 𝑖₂=1,…,𝑛₂, and 𝑌^𝑟_𝜑↪𝑌^𝑟_𝛾, the embeddings being equalities for those between the numbers 𝛾₁,…,𝛾_𝑛2 and 𝜑 which are equal to 𝛾. Then, under these assumptions on the data, formula (173) for 𝐷_𝑡(𝐴⁻¹𝑤(𝑡)) can be extended until 𝑡=0. For, we have lim_𝑡→0⁺𝐷_𝑡(𝐴⁻¹𝑤(𝑡))=𝑥₀∈𝐴0+𝑥₀ and the differential equation in (170) is satisfied even at 𝑡=0. To see this, we observe that 𝐷_𝑡𝐴⁻¹𝑤(𝑡)−𝑥_0𝑋≤𝐼₁(𝑡)+𝐼_2,𝑤(𝑡)+𝐼_3,𝑤(𝑡),𝑡∈𝐼_𝑇⧵{0}, (175) where 𝐼₁(𝑡)=‖(e^𝑡𝐴−𝐼)𝑥₀‖_𝑋, 𝐼_2,𝑤(𝑡)=‖e^𝑡𝐴[𝐹_𝑤(𝑡)−𝑥₀]‖_𝑋, and 𝐼_3,𝑤(𝑡)=‖[𝑄₂𝐹_𝑤](𝑡)‖_𝑋. First, from Proposition 6 we get lim_𝑡→0⁺𝐼₁(𝑡)=0. On the other side, using 𝐹_𝑤∈𝐶^𝛿(𝐼_𝑇;𝑋), 𝛿∈((2−𝛼−𝛽)/𝛼,1)⊆((1−𝛽)/𝛼,1), we obtain 𝐼_2,𝑤(𝑡)≤𝑐̃_𝛼,𝛽,0𝐹_{𝑤𝛿,0,𝑡;𝑋}𝑡^{(𝛼𝛿+𝛽−1)/𝛼},𝑡∈𝐼_𝑇⧵{0}, (176) so that lim_𝑡→0⁺𝐼_2,𝑤(𝑡)=0. Finally, (127) with 𝑔=𝐹_𝑤 yields lim_𝑡→0⁺𝐼_3,𝑤(𝑡)=0, too. Formula (173) thus holds at 𝑡=0 with 𝐷_𝑡(𝐴⁻¹𝑤(0))=lim_𝑡→0⁺𝐷_𝑡(𝐴⁻¹𝑤(𝑡))=𝑥₀.

Remark 40. In [2, Remark p. 55], formula (173) was extended until 𝑡=0 only under the more restrictive assumption 𝑥₀∈𝑋_𝐴^𝛾,∞, 𝛾∈(1−𝛽,1). Indeed [24, Proposition 5.2] was not available at the time of [2] and only the strong continuity of {e^𝑡𝐴}_𝑡≥0 in the 𝑋-norm on the spaces 𝑋_𝐴^𝛾,∞, 𝛾∈(1−𝛽,1), was known (cf. [2, Theorem 3.3]). Notice that in the case of problem (163) the element 𝑥₀ reduces to 𝐿𝑣₀+𝑓(0), so that in the nondegenerate case (𝑀,𝛽)=(𝐼,1) we get back the classical assumption 𝐿𝑣₀+𝑓(0)∈(𝑋,𝒟(𝐿))_𝛾,𝑟, 𝛾∈(0,1), 𝑟∈[1,∞] (see, for instance, [9, Theorem 4.3.1(iii)] and [11, Theorem 4.5]).

Since (170) implies that 𝑤(𝑡)=𝐷_𝑡(𝐴⁻¹𝑤(𝑡))−𝐹_𝑤(𝑡), from (173) we thus find that 𝑤(𝑡)=𝑄₇𝑥₀(𝑡)+e^𝑡𝐴−𝐼𝐹_𝑤(𝑡)−𝑥₀−𝑄₂𝐹_𝑤(𝑡),𝑡∈𝐼_𝑇, (177) where, according to the notation in Corollary 38, we have set [𝑄₇𝑥₀](𝑡)=(e^𝑡𝐴−𝐼)𝑥₀. In particular, 𝑤(0)=0. We conclude that, under the previous assumptions on the pair (𝛼,𝛽) and on the data vector (𝑘₁,…,𝑘_𝑛1,ℎ₁,…,ℎ_𝑛2,𝑦₁,…,𝑦_𝑛2,𝑓,𝑣₁), if 𝑤∈𝐶(𝐼_𝑇;𝑋) solves (170), then necessarily 𝑤∈𝐶₀(𝐼_𝑇;𝑋). As a consequence (cf. (168)), the strict solution 𝑣 to (160) satisfies the initial condition just in the sense 𝑣(0)=𝑣₀.

Introduce the functions 𝑓∶𝐼_𝑇→𝑋 and ℎ_𝑖2∶𝐼_𝑇→𝑌^𝑟_𝛾𝑖2, 𝑖₂=1,…,𝑛₂, defined by 𝑓(𝑡)=𝑓(𝑡)−𝑓(0),ℎ_𝑖2(𝑡)=ℎ_𝑖2(𝑡)−ℎ_𝑖2(0)𝑦_𝑖2,𝑡∈𝐼_𝑇. (178) Then, replacing 𝐹_𝑤 with the right-hand side of (169), using (174), and recalling the definitions of the operators 𝑄_𝑗, 𝑗=2,…,6, in (106)–(110), from (177) we deduce that 𝑤∈𝐶₀(𝐼_𝑇;𝑋) solves the fixed-point equation 𝑤=𝑤₀+𝑤₁+𝑅𝑤, (179) the functions 𝑤_𝑙, 𝑙=0,1, and the operator 𝑅𝑤 being defined by 𝑤₀∶=𝑄₇𝑥₀+^𝑛1_𝑖1=1𝑄₆𝑘_𝑖1,𝐿_𝑖1𝑣₀+^𝑛2_𝑖2=1𝑄₅ℎ_𝑖2+𝑄₅𝑤𝑓,₁∶=−^𝑛1_𝑖1=1𝑄₃𝑘_𝑖1,𝐿_𝑖1𝑣₀−^𝑛2_𝑖2=1𝑄₄ℎ_𝑖2,𝑦_𝑖2−𝑄₂𝑓,𝑅𝑤∶=𝜆₀𝑄₅𝐴⁻¹𝑤−𝑄₂𝐴⁻¹𝑤+^𝑛1_𝑖1=1𝑄₆𝑘_𝑖1,𝑆_𝑖1𝑤−𝑄₃𝑘_𝑖1,𝑆_𝑖1𝑤. (180)(181)(182) Conversely, let 𝑤∈𝐶₀(𝐼_𝑇;𝑋) be a solution to the fixed-point equation (179), and assume that the pair (𝛼,𝛽) and the data vector (𝑘₁,…,𝑘_𝑛1,ℎ₁,…,ℎ_𝑛2,𝑦₁,…,𝑦_𝑛2,𝑓,𝑣₁) satisfy the assumptions below (170) and (173). Then, as before, 𝒦(𝑘_𝑖1,𝑆_𝑖1𝑤),𝒦(𝑘_𝑖1,𝐿_𝑖1𝑣₀)∈𝐶^𝛿₀(𝐼_𝑇;𝑋) and ℎ_𝑖2𝑦_𝑖2,𝑓∈𝐶^𝛿(𝐼_𝑇;𝑋), 𝑖_𝑙=1,…,𝑛_𝑙, 𝑙=1,2, 𝛿∈((2−𝛼−𝛽)/𝛼,1) being as in (171). We apply 𝐴⁻¹∈ℒ(𝑋) to both sides of (179), and we show that 𝐴⁻¹𝑤 satisfies (172) with 𝐹_𝑤∈𝐶(𝐼_𝑇;𝑋) as in (169), so that 𝐴⁻¹𝑤 is a solution to problem (170). To this purpose, we take into account Corollaries 14 and 26. Let 𝑡∈𝐼_𝑇. First (cf. Remark 34 and recall that 𝑄₆(⋅,⋅)=𝑄₅𝒦(⋅,⋅)), using (86), (174), and (178), we get 𝐴⁻¹𝑤₀(𝑡)=^𝑡₀e^{(𝑡−𝑠)𝐴}𝑥₀+^𝑛1_𝑖1=1𝒦𝑘_𝑖1,𝐿_𝑖1𝑣₀(𝑡)+^𝑛2_𝑖2=1ℎ_𝑖2(𝑡)+𝑓(𝑡)=d𝑠_𝑡0e^{(𝑡−𝑠)𝐴}𝑣₁+^𝑛1_𝑖1=1𝒦𝑘_𝑖1,𝐿_𝑖1𝑣₀(𝑡)+^𝑛2_𝑖2=1ℎ_𝑖2(𝑡)𝑦_𝑖2+𝑓(𝑡)d𝑠. (183)

Instead, due to the definition of 𝑄₃ and 𝑄₄, using (125) we obtain 𝐴⁻¹𝑤₁(𝑡)=^𝑡₀e^{(𝑡−𝑠)𝐴𝑛1}_𝑖1=1𝒦𝑘_𝑖1,𝐿_𝑖1𝑣₀(𝑠)−𝒦𝑘_𝑖1,𝐿_𝑖1𝑣₀𝑡+()_𝑛2𝑖2=1ℎ_𝑖2(𝑠)−ℎ_𝑖2(𝑡)𝑦_𝑖2+𝑓(𝑠)−𝑓(𝑡)d𝑠. (184) Therefore, from (183), (184), and the definition (105) of 𝑄₁ it follows that 𝐴⁻¹𝑤₀+𝑤₁𝑡)=𝑄₁(𝑣₁+^𝑛1_𝑖1=1𝒦𝑘_𝑖1,𝐿_𝑖1𝑣₀+^𝑛2_𝑖2=1ℎ_𝑖2𝑦_𝑖2+𝑓(𝑡), (185) the left-hand side being well-defined due to Remark 23. As far as 𝐴⁻¹[𝑅𝑤](𝑡) is concerned, we first observe that, 𝑤 being in 𝐶(𝐼_𝑇;𝑋), from formula (126) and Remark 34 it follows that [𝑄₂(𝐴⁻¹𝑤)](𝑡) and [𝑄₅(𝐴⁻¹𝑤)](𝑡) are both well defined and equal to −∫^𝑡₀e^{(𝑡−𝑠)𝐴}[𝑤(𝑠)−𝑤(𝑡)]d𝑠 and ∫^𝑡₀e^{(𝑡−𝑠)𝐴}𝑤(𝑡)d𝑠, respectively. Consequently 𝑄₅𝐴⁻¹𝑤−𝑄₂𝐴⁻¹𝑤𝑡)=_𝑡0(e^{(𝑡−𝑠)𝐴}𝑤(𝑠)d𝑠=𝑄₁𝑤(𝑡). (186) Hence, commuting 𝐴⁻¹∈ℒ(𝑋) with both the integral sign and the semigroup, one has 𝐴⁻¹𝑄₅𝐴⁻¹𝑤−𝑄₂𝐴⁻¹𝑤(𝑡)=𝑄₁𝐴⁻¹𝑤(𝑡). (187) Similarly, since Remark 34 and formula (125) yield 𝐴⁻¹𝑄₆𝑘_𝑖1,𝑆_𝑖1𝑤(𝑡)=^𝑡₀e^{(𝑡−𝑠)𝐴}𝒦𝑘_𝑖1,𝑆_𝑖1𝑤𝑡)𝐴(d𝑠,₋₁𝑄₃𝑘_𝑖1,𝑆_𝑖1𝑤(𝑡)=−^𝑡₀e^{(𝑡−𝑠)𝐴}𝒦𝑘_𝑖1,𝑆_𝑖1𝑤(𝑠)−𝒦𝑘_𝑖1,𝑆_𝑖1𝑤(𝑡)d𝑠, (188) we find that 𝐴⁻¹𝑄₆𝑘_𝑖1,𝑆_𝑖1𝑤−𝑄₃𝑘_𝑖1,𝑆_𝑖1𝑤𝑡)=𝑄₁(𝒦𝑘_𝑖1,𝑆_𝑖1𝑤(𝑡), (189)𝑖₁=1,…,𝑛₁. In conclusion, from (187) and (189) it follows that 𝐴⁻¹[𝑅𝑤](𝑡)=𝑄₁𝜆₀𝐴⁻¹𝑤+^𝑛1_𝑖1=1𝒦𝑘_𝑖1,𝑆_𝑖1𝑤(𝑡). (190) Summing up (185) and (190), we finally obtain 𝐴⁻¹𝑤(𝑡)=[𝑄₁𝐹_𝑤](𝑡), 𝐹_𝑤 being as in (169). This completes the proof of the equivalence between problem (170) and the fixed point equation (179), provided that the data satisfy the mentioned assumptions.

Remark 41. We can summarize the previous reasonings as follows: problem (160) has been reduced to the fixed-point equation (179) for the new unknow 𝑤=𝐿(𝑣−𝑣₀), 𝑣₀∈𝒟(𝐿). This fixed-point argument is similar to that first successfully applied in [4, 7, 8, 32] to problem (165) with (𝑀,𝛽,𝑛₁)=(𝐼,1,1) and then generalized in [23] to the degenerate case. A different approach has been followed in [6, 10] for the nondegenerate case and in [22] for the degenerate one. There, assuming that 𝑘₁ is absolutely Laplace transformable (cf. [6, 22]) or of bounded variation (cf. [10]), problem (165) with 𝑛₁=1 is solved by constructing its relative resolvent operator. We quote also [35] where the method of constructing the fundamental solution for the equation without the integral term is applied to a class of concrete degenerate integrodifferential equations.

From now on, for 5𝛼+2𝛽>6, 𝛽∈(0,𝛼], 𝛼∈(0,1], and 𝜈∈((3−2𝛼−𝛽)/𝛼,1), 𝐼_{𝛼,𝛽,𝜈}⊆((3−2𝛼−𝛽)/𝛼,1/2)⊆(0,1/2) will denote the interval defined by 𝐼_{𝛼,𝛽,𝜈}=3−2𝛼−𝛽𝛼,𝜈,if𝜈∈3−2𝛼−𝛽𝛼,12,3−2𝛼−𝛽𝛼,121,if𝜈∈2,1. (191) Clearly, if 𝜈,𝜌∈((3−2𝛼−𝛽)/𝛼,1), 𝜈≤𝜌, then 𝐼_{𝛼,𝛽,𝜈}⊆𝐼_{𝛼,𝛽,𝜌}.

Lemma 42. Assume (161), and let 5𝛼+2𝛽>6 in (H2). Assume that 𝑘_𝑖1∈𝐶^𝜂𝑖1(𝐼_𝑇;𝑍), 𝜂_𝑖1∈((3−2𝛼−𝛽)/𝛼,1), 𝑖₁=1,…,𝑛₁, and let 𝜂=min_{𝑖1=1,…,𝑛1}𝜂_𝑖1. Then, for every fixed 𝛿∈𝐼_{𝛼,𝛽,𝜂}, the operator 𝑅 defined by (182) maps continuously 𝐶^𝛿(𝐼_𝑇;𝑋) into 𝐶^𝛿₀(𝐼_𝑇;𝑋), and for every 𝑡∈𝐼_𝑇 satisfies the following estimate, where 𝑝∈(1/(1−2𝛿),∞): ‖𝑅𝑤‖_{𝛿,0,𝑡;𝑋}≤𝑐₄₂(𝑇)^𝑡₀‖𝑤‖^𝑝_{𝛿,0,𝑠;𝑋}d𝑠^1/𝑝,𝑤∈𝐶^𝛿𝐼_𝑇;𝑋. (192) Here 𝑐₄₂(𝑇) is a positive constant depending only on 𝑇, 𝜆₀, 𝛼, 𝛽, 𝜂_𝑖1, 𝛿, 𝑝, ‖𝑘_𝑖1‖_{𝜂𝑖1,0,𝑇;𝑍} and ‖𝑆_𝑖1‖_ℒ(𝑋), 𝑖₁=1,…,𝑛₁.

Proof. Let 𝑘_𝑖1∈𝐶^𝜂𝑖(𝐼_𝑇;𝑍), 𝜂_𝑖1∈((3−2𝛼−𝛽)/𝛼,1), 𝑖₁=1,…,𝑛₁, and let us fix an arbitrary number 𝛿∈𝐼_{𝛼,𝛽,𝜂}, where 𝜂=min_{𝑖1=1,…,𝑛1}𝜂_𝑖. In particular, since 𝛿≤𝜂≤𝜂_𝑖, we have 𝑘_𝑖1∈𝐶^𝛿(𝐼_𝑇;𝑍) with ‖𝑘_𝑖1‖_{𝛿,0,𝑡;𝑍}≤max{1,𝑡^{𝜂𝑖1−𝛿}}‖𝑘_𝑖1‖_{𝜂𝑖1,0,𝑡;𝑍}, 𝑖₁=1,…,𝑛₁. Now let 𝑤∈𝐶^𝛿(𝐼_𝑇;𝑋) and 𝑡∈𝐼_𝑇. First, formula (186) being applicable, we rewrite (182) as 𝑅𝑤=𝜆₀𝑄₁𝑤+^𝑛1_𝑖1=1𝑄₆𝑘_𝑖1,𝑆_𝑖1𝑤−𝑄₃𝑘_𝑖,𝑆_𝑖𝑤. (193) Now, we notice that 5𝛼+2𝛽>6 implies that 𝛼+𝛽−1𝛼=5𝛼+2𝛽−3𝛼−2>2𝛼4−3𝛼≥12𝛼2. (194) Since (1−𝛽)/𝛼≤(2−𝛼−𝛽)/𝛼≤(3−2𝛼−𝛽)/𝛼, from (194) it follows that 𝛿∈𝐼_{𝛼,𝛽,𝜂}⊆((3−2𝛼−𝛽)/𝛼,1/2)⊆((2−𝛼−𝛽)/𝛼,1/2)⊊((1−𝛽)/𝛼,(𝛼+𝛽−1)/𝛼), and, consequently, 𝛼<1𝛼+𝛽−1−𝛼𝛿.1−2𝛿 (195) We conclude (cf. Remark 39) that Lemma 22 and Propositions 29 and 36 are applicable with 𝛿∈𝐼_{𝛼,𝛽,𝜂} and 𝑝∈(1/(1−2𝛿),∞). Then, using estimates (111), (135), and (150) with the pair (𝑔₁,𝛿₁) and the quintuplets (𝑔_𝑙1,𝑔_𝑙2,𝛿_𝑙,𝑋₁,𝑋₂), 𝑙=3,6, being replaced, respectively, by (𝑤,𝛿) and (indeed, since 𝑆_𝑖1=𝐿_𝑖1𝐿⁻¹∈ℒ(𝑋), if 𝑤∈𝐶^𝛿(𝐼_𝑇;𝑋), then 𝑆_𝑖1𝑤∈𝐶^𝛿(𝐼_𝑇;𝑋) with ‖𝑆_𝑖1𝑤‖_{𝛿,0,𝑡;𝑋}≤‖𝑆_𝑖1‖_ℒ(𝑋)‖𝑤‖_{𝛿,0,𝑡;𝑋}, 𝑖₁=1,…,𝑛₁) (𝑘_𝑖1,𝑆_𝑖1𝑤,𝛿,𝑍,𝑋),   𝑖₁=1,…,𝑛₁, from (193) we finally obtain ‖𝑅𝑤‖_{𝛿,0,𝑡;𝑋}≤𝜆₀𝑄₁𝑤_{𝛿,0,𝑡;𝑋}+_{𝑙=3,6,𝑖1=1,…,𝑛1}𝑄_𝑙𝑘_𝑖1,𝑆_𝑖1𝑤_{𝛿,0,𝑡;𝑋}≤𝑐₄₂(𝑇)^𝑡₀‖𝑤‖^𝑝_{𝛿,0,𝑠;𝑋}d𝑠^1/𝑝. (196) Here we have set 𝑐₄₂(𝑇)=|𝜆₀|𝐶₁(𝑇)+𝑇^{1−2𝛿−1/𝑝}∑_{𝑙=3,6,𝑖1=1,…,𝑛1}𝐶_𝑙(𝑇)‖𝑘_𝑖1‖_{𝛿,0,𝑇;𝑍}‖𝑆_𝑖1‖_ℒ(𝑋), where 𝐶_𝑙(𝑇), 𝑙=1,3,6, are the values at 𝑡=𝑇 of the functions 𝐶_𝑙(𝑡) in Lemma 22 and Propositions 29 and 36. This completes the proof.

Remark 43. Assume that in Lemma 42 the Hölder exponents 𝜂_𝑖1∈((3−2𝛼−𝛽)/𝛼,1) are such that 𝜂=min_{𝑖1=1,…,𝑛1}𝜂_𝑖 belongs to ((3−2𝛼−𝛽)/𝛼,1/2). In this case (cf. (191)), the choice 𝛿=𝜂 is admissible, and the meaning of Lemma 42 is that the operator 𝑅 defined by (182) preserves the minimal of the time regularities of 𝑘₁,…,𝑘_𝑛1.

Corollary 44. Let the assumptions of Lemma 42 be satisfied, and let 𝜂 and 𝑅 be as there. Then, for every fixed 𝛿∈𝐼_{𝛼,𝛽,𝜂}, the sequence {𝑅^𝑛}^∞_𝑛=0 (𝑅⁰=𝐼, 𝑅^𝑛=𝑅𝑅^𝑛−1, 𝑛∈𝐍) satisfies the following estimates, where 𝑤∈𝐶^𝛿(𝐼_𝑇;𝑋) and 𝑝∈(1/(1−2𝛿),∞): 𝑅^𝑛𝑤_{𝛿,0,𝑡;𝑋}≤𝑐₄₂(𝑇)^𝑛𝑡^𝑛𝑛!^1/𝑝‖𝑤‖_{𝛿,0,𝑇;𝑋},𝑡∈𝐼_𝑇,𝑛∈𝐍∪{0}. (197)

Proof. Reasoning as in [23, p. 468], we prove (197) by induction. Since for every fixed 𝛿∈𝐼_{𝛼,𝛽,𝜂} the operator 𝑅 maps 𝐶^𝛿(𝐼_𝑇;𝑋) in 𝐶^𝛿₀(𝐼_𝑇;𝑋), replacing 𝑤 with 𝑅^𝑛𝑤 in (192) and introducing the sequence of scalar nonnegative nondecreasing functions {𝜑_𝑛}^∞_𝑛=0 defined by 𝜑_𝑛(𝑡)=‖𝑅^𝑛𝑤‖_{𝛿,0,𝑡;𝑋}, 𝑡∈𝐼_𝑇, from (192) we obtain 𝜑_𝑛+1(𝑡)≤𝑐₄₂(𝑇)^𝑡₀𝜑_𝑛(𝑠)^𝑝d𝑠^1/𝑝,𝑡∈𝐼_𝑇,𝑛∈𝐍∪{0}. (198) Then, applying to (198) an induction argument in which the first step of the induction follows from (192), we immediately deduce the following estimates: 𝜑_𝑛(𝑡)≤𝑐₄₂(𝑇)^𝑛𝑡^𝑛𝑛!^1/𝑝‖𝑤‖_{𝛿,0,𝑇;𝑋},𝑡∈𝐼_𝑇,𝑛∈𝐍∪{0}. (199) The proof is complete.

Lemma 45. Let 5𝛼+2𝛽>6 in (H2) and 𝑣₀∈⋂^𝑛1_𝑖1=1𝒟(𝐿_𝑖1). Assume that 𝑘_𝑖1∈𝐶^𝜂𝑖1(𝐼_𝑇;𝑍), ℎ_𝑖2∈𝐶^𝜎𝑖2(𝐼_𝑇;𝐂), and 𝑦_𝑖2∈𝑌^𝑟_𝛾𝑖2, where 𝜂_𝑖1,𝜎_𝑖2∈((3−2𝛼−𝛽)/𝛼,1), 𝛾_𝑖2∈(3−2𝛼−𝛽,1), 𝑖_𝑙=1,…,𝑛_𝑙, 𝑙=1,2, and 𝑟∈[1,∞]. Let 𝜏₁=min_{𝑖𝑙=1,…,𝑛𝑙,𝑙=1,2}{𝜂_𝑖1,𝜎_𝑖2}. Then, for every fixed 𝛿∈𝐼_{𝛼,𝛽,𝜏1}, the function 𝑤₁ defined by (181) belongs to 𝐶^𝛿₀(𝐼_𝑇;𝑋), provided that 𝑓∈𝐶^𝜇(𝐼_𝑇;𝑋), 𝜇∈[𝛿+𝜇_𝛼,𝛽,1), 𝜇_𝛼,𝛽=(3−2𝛼−𝛽)/𝛼.

Proof. Let us fix 𝛿∈𝐼_{𝛼,𝛽,𝜏1}, 𝜏₁=min_{𝑖𝑙=1,…,𝑛𝑙,𝑙=1,2}{𝜂_𝑖1,𝜎_𝑖2}. Of course, 𝑘_𝑖1∈𝐶^𝛿(𝐼_𝑇;𝑍) and ℎ_𝑖2∈𝐶^𝛿(𝐼_𝑇;𝐂), 𝑖_𝑙=1,…,𝑛_𝑙, 𝑙=1,2. Then, Proposition 29 and Lemma 30 applied with the quintuplets (𝑔₃₁,𝑔₃₂,𝛿₃,𝑋₁,𝑋₂) and the quadruplet (𝑔₄,𝑦,𝛿₄,𝛾) being replaced, respectively, by (the constant functions 𝜅_𝑖1(𝑡)=𝐿_𝑖1𝑣₀, 𝑡∈𝐼_𝑇, 𝑖=1,…,𝑛₁, being obviously of class 𝐶^𝛿(𝐼_𝑇;𝑋)) (𝑘_𝑖1,𝐿_𝑖1𝑣₀,𝛿,𝑍,𝑋) and (ℎ_𝑖2,𝑦_𝑖2,𝛿,𝛾_𝑖2), imply that 𝑄₃(𝑘_𝑖1,𝐿_𝑖1𝑣₀),𝑄₄(ℎ_𝑖2,𝑦_𝑖2)∈𝐶^𝛿₀(𝐼_𝑇;𝑋), 𝑖_𝑙=1,…,𝑛_𝑙, 𝑙=1,2. Now, since 𝛿∈𝐼_{𝛼,𝛽,𝜏1}⊆((3−2𝛼−𝛽)/𝛼,1/2)⊆(0,1/2), the number 𝛿+𝜇_𝛼,𝛽 satisfies 3−2𝛼−𝛽𝛼<𝛿≤𝛿+𝜇_𝛼,𝛽<6−3𝛼−2𝛽2𝛼<1, (200) and assumption 𝑓∈𝐶^𝜇(𝐼_𝑇;𝑋), 𝜇∈[𝛿+𝜇_𝛼,𝛽,1), is meaningful. Lemma 24 with (𝑔₂,𝛿₂)=(𝑓,𝜇) then yields 𝑄₂𝑓∈𝐶^{𝜈𝛼,𝛽,𝜇}₀(𝐼_𝑇;𝑋), 𝜈_{𝛼,𝛽,𝜇}=(𝛼𝜇+2𝛼+𝛽−3)/𝛼. Since 𝜈_{𝛼,𝛽,𝜇}≥𝜈_{𝛼,𝛽,𝛿+𝜇𝛼,𝛽}=𝛿, we get 𝑄₂𝑓∈𝐶^𝛿₀(𝐼_𝑇;𝑋), too. Summing up, we get the assertion.

Before considering the function 𝑤₀ in (180), we introduce the following notation. In the sequel, for 3𝛼+2𝛽>4, 𝛽∈(0,𝛼], 𝛼∈(0,1], and 𝜈∈((2−𝛼−𝛽)/𝛼,1), 𝐽_{𝛼,𝛽,𝜈}⊆((2−𝛼−𝛽)/𝛼,1/2)⊆(0,1/2) will denote the interval 𝐽_{𝛼,𝛽,𝜈}=2−𝛼−𝛽𝛼,𝜈,if𝜈∈2−𝛼−𝛽𝛼,12,2−𝛼−𝛽𝛼,121,if𝜈∈2,1. (201) Notice that, since (2−𝛼−𝛽)/𝛼≤(3−2𝛼−𝛽)/𝛼, if the stronger condition 5𝛼+2𝛽>6 is satisfied, then (191) and (201) yield 𝐼_{𝛼,𝛽,𝜈}⊆𝐽_{𝛼,𝛽,𝜈} for every fixed 𝜈∈((3−2𝛼−𝛽)/𝛼,1). The introduction of the intervals 𝐽_{𝛼,𝛽,𝜈} is justified by Lemma 46, which requires a weaker condition on the pair (𝛼,𝛽) than the one in Lemmas 42 and 45.

Lemma 46. Let 3𝛼+2𝛽>4 in (H2), and let 𝑣₀∈𝒟(𝐿). Assume that 𝑘_𝑖1∈𝐶^𝜂𝑖1(𝐼_𝑇;𝑋), ℎ_𝑖2∈𝐶^𝜎𝑖2(𝐼_𝑇;𝐂), 𝑦_𝑖2∈𝑌^𝑟_𝛾𝑖2, and 𝑣₁+𝑓(0)∈𝑌^𝑟_𝜑, where 𝜂_𝑖1,𝜎_𝑖2∈((2−𝛼−𝛽)/𝛼,1), 𝛾_𝑖2,𝜑∈(4−2𝛼−2𝛽,1), 𝑖_𝑙=1,…,𝑛_𝑙, 𝑙=1,2, 𝑟∈[1,∞], and 𝑣₁=(𝜆₀𝑀+𝐿)𝑣₀. Let 𝛾=min_{𝑖2=1,…,𝑛2}{𝛾_𝑖2,𝜑} and 𝜏₀=min_{𝑖𝑙=1,…,𝑛𝑙,𝑙=1,2}{𝜂_𝑖1,𝜎_𝑖2,𝜒_{𝛼,𝛽,𝛾}}, where 𝜒_{𝛼,𝛽,𝛾}=(𝛼+𝛽+𝛾−2)/𝛼. Then, for every fixed 𝛿∈𝐽_{𝛼,𝛽,𝜏0}, the function 𝑤₀ defined by (180) belongs to 𝐶^𝛿₀(𝐼_𝑇;𝑋), provided that 𝑓∈𝐶^𝜇(𝐼_𝑇;𝑋), 𝜇∈[𝛿+𝜚_𝛼,𝛽,1),𝜚_𝛼,𝛽=(2−𝛼−𝛽)/𝛼.

Proof. Observe that (cf. (159)) all the results from Lemma 32 to Corollary 38 will be applicable. First, since 2𝛼+2𝛽>4−𝛼≥3, the choice 𝛾_𝑖2,𝜑∈(4−2𝛼−2𝛽,1), 𝑖₂=1,…,𝑛₂, is meaningful. Moreover, since 𝛾=min_{𝑖2=1,…,𝑛2}{𝛾_𝑖2,𝜑}∈(4−2𝛼−2𝛽,1), the number 𝜒_{𝛼,𝛽,𝛾}=(𝛼+𝛽+𝛾−2)/𝛼 satisfies 𝜒_{𝛼,𝛽,𝛾}∈((2−𝛼−𝛽)/𝛼,1). Hence, 𝜏₀=min_{𝑖𝑙=1,…,𝑛𝑙,𝑙=1,2}{𝜂_𝑖1,𝜎_𝑖2,𝜒_{𝛼,𝛽,𝛾}}∈((2−𝛼−𝛽)/𝛼,1), too, and 𝐽_{𝛼,𝛽,𝜏0} is well defined. Now, let 𝛿∈𝐽_{𝛼,𝛽,𝜏0} be fixed. Due to (20) or (38), the element 𝑥₀ defined by (174) belongs to 𝑌^𝑟_𝛾, whereas the functions ℎ_𝑖2 defined by (178) are of class 𝐶^𝛿₀(𝐼_𝑇;𝑌^𝑟_𝛾𝑖2)↪𝐶^𝛿₀(𝐼_𝑇;𝑌^𝑟_𝛾). Then, since 𝛾∈(4−2𝛼−2𝛽,1)⊆(2−𝛼−𝛽,1), from Lemma 37 and Corollary 38 applied with the pairs (𝑔₅,𝛿₅) and (𝑥,𝛿₇) being replaced by (ℎ_𝑖2,𝛿) and (𝑥₀,𝛿), respectively, we deduce that 𝑄₅ℎ_𝑖2,𝑄₇𝑥₀∈𝐶^𝛿₀(𝐼_𝑇;𝑋), 𝑖₂=1,…,𝑛₂. In addition, since the 𝑘_𝑖1’s and the constant functions 𝜅_𝑖1(𝑡)=𝐿_𝑖1𝑣₀ belong to 𝐶^𝛿(𝐼_𝑇;𝑋), from Proposition 36 applied with (𝑔₆₁,𝑔₆₂,𝑋₁,𝑋₂)=(𝑘_𝑖1,𝐿_𝑖1𝑣₀,𝑍,𝑋), it follows that 𝑄₆(𝑘_𝑖1,𝐿_𝑖1𝑣₀)∈𝐶^𝛿₀(𝐼_𝑇;𝑋), 𝑖₁=1,…,𝑛₁. Finally, since 𝛿∈𝐽_{𝛼,𝛽,𝜏0}⊆((2−𝛼−𝛽)/𝛼,1/2), the number 𝛿+𝜚_𝛼,𝛽 satisfies 2−𝛼−𝛽𝛼<𝛿≤𝛿+𝜚_𝛼,𝛽<4−𝛼−2𝛽2𝛼<1, (202) and the assumption 𝑓∈𝐶^𝜇(𝐼_𝑇;𝑋), 𝜇∈[𝛿+𝜚_𝛼,𝛽,1), makes sense. Then, the function 𝑓=𝑓−𝑓(0) being of class 𝐶^𝜇₀(𝐼_𝑇;𝑋), Lemma 32 applied with (𝑔₅,𝛿₅)=(𝑓,𝜇) yields 𝑄₅𝜈𝑓∈𝐶^{𝛼,𝛽,𝜇}₀(𝐼_𝑇;𝑋), 𝜈_{𝛼,𝛽,𝜇}=(𝛼𝜇+𝛼+𝛽−2)/𝛼. Since 𝜈_{𝛼,𝛽,𝜇}≥𝜈_{𝛼,𝛽,𝛿+𝜚𝛼,𝛽}=𝛿, we conclude that 𝑄₅𝑓∈𝐶^𝛿₀(𝐼_𝑇;𝑋), too. Summing up, we get the assertion.

Remark 47. We stress that, if 𝛽∈(0,1) in (H2), then 0<𝜌_𝛼,𝛽≤𝜇_𝛼,𝛽, so that in both Lemmas 45 and 46 we have to assume that 𝑓∈𝐶^𝜇(𝐼_𝑇;𝑋) with 𝜇>𝛿. This is necessary in order to restore the loss of regularity produced by the operators 𝑄₂ and 𝑄₅.

We can now prove the main results of the section.

Theorem 48. Assume (161) and 𝑣₀∈𝒟(𝐿), and let 5𝛼+2𝛽>6 in (H2). Assume that 𝑘_𝑖1∈𝐶^𝜂𝑖1(𝐼_𝑇;𝑍), ℎ_𝑖2∈𝐶^𝜎𝑖2(𝐼_𝑇;𝐂), 𝑦_𝑖2∈𝑌^𝑟_𝛾𝑖2, and 𝑣₁+𝑓(0)∈𝑌^𝑟_𝜑, where 𝜂_𝑖1,𝜎_𝑖2∈((3−2𝛼−𝛽)/𝛼,1), 𝛾_𝑖2,𝜑∈(5−3𝛼−2𝛽,1), 𝑖_𝑙=1,…,𝑛_𝑙, 𝑙=1,2, 𝑟∈[1,∞], and 𝑣₁=(𝜆₀𝑀+𝐿)𝑣₀. Let 𝛾=min_{𝑖2=1,…,𝑛2}{𝛾_𝑖2,𝜑} and 𝜏=min_{𝑖𝑙=1,…,𝑛𝑙,𝑙=1,2}{𝜂_𝑖1,𝜎_𝑖2,𝜒_{𝛼,𝛽,𝛾}}, where 𝜒_{𝛼,𝛽,𝛾}=(𝛼+𝛽+𝛾−2)/𝛼. Then, for every fixed 𝛿∈𝐼_{𝛼,𝛽,𝜏} problem (160) admits a unique strict solution 𝑣∈𝐶^𝛿(𝐼_𝑇;𝒟(𝐿)) satisfying 𝑣(0)=𝑣₀ and such that 𝐿𝑣,𝐷_𝑡𝑀𝑣∈𝐶^𝛿(𝐼_𝑇;𝑋), provided that 𝑓∈𝐶^𝜇(𝐼_𝑇;𝑋), 𝜇∈[𝛿+𝜇_𝛼,𝛽,1), 𝜇_𝛼,𝛽=(3−2𝛼−𝛽)/𝛼.

Proof. Of course, due to (159), the assumption 𝛾_𝑖2,𝜑∈(5−3𝛼−2𝛽,1), 𝑖₂=1,…,𝑛₂, makes sense. In addition, since 𝛾=min_{𝑖2=1,…,𝑛2}{𝛾_𝑖2,𝜑}∈(5−3𝛼−2𝛽,1), we have 𝜒_{𝛼,𝛽,𝛾}=(𝛼+𝛽+𝛾−2)/𝛼∈((3−2𝛼−𝛽)/𝛼,1). Therefore, by virtue of the choice of the Hölder exponents 𝜂_𝑖1 and 𝜎_𝑖2, the number 𝜏=min_{𝑖𝑙=1,…,𝑛𝑙,𝑙=1,2}{𝜂_𝑖1,𝜎_𝑖2,𝜒_{𝛼,𝛽,𝛾}} belongs to ((3−2𝛼−𝛽)/𝛼,1) too, and the interval 𝐼_{𝛼,𝛽,𝜏} is well defined. Further, the numbers 𝜂, 𝜏₁, and 𝜏₀ being as in the statements of Lemmas 42, 45, and 46, respectively, we have 𝜏=𝜏₀≤𝜏₁≤𝜂. As a consequence, since 𝐼_{𝛼,𝛽,𝜏}⊆𝐼_{𝛼,𝛽,𝜏1}⊆𝐼_{𝛼,𝛽,𝜂} and 𝐼_{𝛼,𝛽,𝜏}⊆𝐽_{𝛼,𝛽,𝜏}, all the mentioned lemmas are applicable with 𝛿∈𝐼_{𝛼,𝛽,𝜏}. To this purpose, we stress that since ((3−2𝛼−𝛽)/𝛼,1)⊆((2−𝛼−𝛽)/𝛼,1) and (5−3𝛼−2𝛽,1)⊆(4−2𝛼−2𝛽,1)⊆(3−2𝛼−𝛽,1), the conditions for the applicability of both Lemmas 45 and 46 are fulfilled. Hence, now let 𝛿∈𝐼_{𝛼,𝛽,𝜏} being fixed. First, due to Lemma 42, the operator 𝑅=𝑅|_𝐶^𝛿_{0(𝐼𝑇;𝑋)}, 𝑅𝑔=𝑅𝑔, 𝑔∈𝐶^𝛿₀(𝐼_𝑇;𝑋), a fortiori maps 𝐶^𝛿₀(𝐼_𝑇;𝑋) into itself. Then, 𝐶^𝛿₀(𝐼_𝑇;𝑋) being endowed with the same norm ‖⋅‖_{𝛿,0,𝑇;𝑋} of 𝐶^𝛿(𝐼_𝑇;𝑋), from (197) we obtain the estimates 𝑅^𝑛_{ℒ(𝐶𝛿0(𝐼𝑇;𝑋))}≤𝑐₄₂(𝑇)^𝑛𝑇^𝑛𝑛!^1/𝑝{0},1,𝑛∈𝐍∪𝑝∈.1−2𝛿,∞ (203) In particular, (203) yields that ∑^∞_𝑛=0𝑅^𝑛 converges in ℒ(𝐶^𝛿₀(𝐼_𝑇;𝑋)). From generalized Neumann’s Theorem it thus follows that 1∈𝜌(𝑅), the inverse (𝐼−𝑅)⁻¹∈ℒ(𝐶^𝛿₀(𝐼_𝑇;𝑋)) being precisely ∑^∞_𝑛=0𝑅^𝑛. Since Lemmas 45 and 46 (both applied with (observe here that if 𝜇∈[𝛿+𝜇_𝛼,𝛽,1), then the exponent 𝜈_{𝛼,𝛽,𝜇} in the last part of the proof of Lemma 46 satisfies 𝜈_{𝛼,𝛽,𝜇}≥𝜈_{𝛼,𝛽,𝛿+𝜇𝛼,𝛽}≥𝜈_{𝛼,𝛽,𝛿+𝜚𝛼,𝛽}=𝛿. For, 𝜈_{𝛼,𝛽,𝛿+𝜇𝛼,𝛽}=(𝛼𝛿+1−𝛼)/𝛼=𝛿+(1−𝛼)/𝛼)𝑓∈𝐶^𝜇(𝐼_𝑇;𝑋), 𝜇∈[𝛿+𝜇_𝛼,𝛽,1)⊆[𝛿+𝜌_𝛼,𝛽,1)) imply that 𝑤₀,𝑤₁∈𝐶^𝛿₀(𝐼_𝑇;𝑋), we conclude that the fixed-point equation (179) admits the unique solution 𝑤=^∞_𝑛=0𝑅^𝑛𝑤₀+𝑤₁∈𝐶^𝛿₀𝐼_𝑇;𝑋. (204) Observe now that the data vector (𝑘₁,…,𝑘_𝑛1,ℎ₁,…,ℎ_𝑛2,𝑓,𝑦₁,…,𝑦_𝑛2,𝑣₁+𝑓(0)) satisfies all the assumptions which were needed to show the equivalence between the fixed-point equation (179) and problem (170). Indeed, 𝛿≤𝜏 and 𝛿≤𝛿+𝜇_𝛼,𝛽≤𝜇 imply, respectively, that 𝑘_𝑖1∈𝐶^𝛿(𝐼_𝑇;𝑍),ℎ_𝑖2∈𝐶^𝛿(𝐼_𝑇;𝐂) and 𝑓∈𝐶^𝛿(𝐼_𝑇;𝑋),𝑖_𝑙=1,…,𝑛_𝑙,𝑙=1,2, whereas, as in Lemma 46, 𝛾=min_{𝑖2=1,…,𝑛2}{𝛾_𝑖2,𝜑} implies that 𝑦_𝑖2,𝑣₁+𝑓(0)∈𝑌^𝑟_𝛾. Therefore, since 𝐴⁻¹∈ℒ(𝑋), if 𝑤∈𝐶^𝛿₀(𝐼_𝑇;𝑋) is the solution to the fixed-point equation (179), then 𝐴⁻¹𝑤∈𝐶^𝛿₀(𝐼_𝑇;𝑋), too, and the function 𝐹_𝑤 defined by (169) satisfies 𝐹_𝑤∈𝐶^𝛿𝐼_𝑇,𝑥;𝑋₀=𝐹_𝑤(0)=^𝑛2_𝑖2=1ℎ_𝑖2(0)𝑦_𝑖2+𝑣₁+𝑓(0)∈𝑌^𝑟_𝛾, (205) where 𝛿∈𝐼_{𝛼,𝛽,𝜏}⊊(2−𝛼−𝛽)/𝛼,1), 𝛾∈(5−3𝛼−2𝛽,1)⊊(1−𝛽,1), and 𝑟∈[1,∞]. Consequently, recalling (168), we have proved that problem (160) has a unique strict global solution 𝑣=𝐿⁻¹𝑤+𝑣₀∈𝐶^𝛿(𝐼_𝑇;𝒟(𝐿)) satisfying 𝑣(0)=𝐿⁻¹𝑤(0)+𝑣₀=𝑣₀ and such that 𝐿𝑣=𝑤+𝐿𝑣₀∈𝐶^𝛿(𝐼_𝑇;𝑋). As far as the regularity of 𝐷_𝑡𝑀𝑣 is concerned, instead, it suffices to observe that (168), (170), 𝑤∈𝐶^𝛿₀(𝐼_𝑇;𝑋), and 𝐹_𝑤∈𝐶^𝛿(𝐼_𝑇;𝑋) yield 𝐷_𝑡𝑀𝑣=𝐷_𝑡𝐴⁻¹𝑤=𝑤+𝐹_𝑤∈𝐶^𝛿𝐼_𝑇;𝑋. (206) The proof is complete.

Remark 49. Theorem 48 improves the faulty Thereoms 5.6 and 5.7 in [20] in two aspects. First, the assumption 3𝛼+8𝛽>10 is weakened to 5𝛼+2𝛽>6. In fact, 3𝛼+8𝛽>10 implies that 5𝛼+2𝛽=3𝛼+8𝛽+2𝛼−6𝛽>10−4𝛼≥6. Hence, in the special case 𝛼=1, the constraint 𝛽>7/8 in [20] reduces to the definitely weaker 𝛽>1/2. Second, in [20], only for 𝑛₁=𝑛₂=1 and opportunely chosen 𝛾<𝛽, the data 𝑦₁ and 𝑣₁+𝑓(0) were assumed to belong to the intermediate spaces 𝑋_𝐴^𝛾,𝑟, whereas here, removing the assumption 𝛾<𝛽 and considering the general case 𝑛₁,𝑛₂∈𝐍, we allow 𝑦₁,…,𝑦_𝑛2 and 𝑣₁+𝑓(0) to belong also to the interpolation spaces (𝑋,𝒟(𝐴))_𝛾,𝑟. To emphasize how much these aspects are decisive, let 𝛼=1 in Theorem 48. Then, if 𝛽∈(1/2,2/3] and the choice 𝑋_𝐴^𝜓,𝑟 is understood for 𝑌^𝑟_𝜓, we have 𝛾_𝑖2,𝜑∈(2−2𝛽,1)⊊[𝛽,1), and the spaces 𝑋^𝛾𝑖2_𝐴^,𝑟 and 𝑋_𝐴^𝜑,𝑟, 𝑖₂=1,…,𝑛₂, may be smaller than 𝒟(𝐴). However, the choice 𝑌^𝑟_𝜓=(𝑋,𝒟(𝐴))_𝜓,𝑟 being admissible, in this situation too we can solve problem (160) with the data in spaces larger than 𝒟(𝐴). Further, since 2/3<7/8, in this case the results in [20] would not be applicable. These observations lead us to conclude that the more delicate approach followed in this paper with respect to that in [20, Sections 4 and 5], and especially the sharper results of the present Sections 3 and 4, yield a valuable refinement in the treatment of questions of maximal time regularity for the strict solutions to (160); of course, unless that the not too much significant case 𝛽=1 is assumed in (H2).

Remark 50. The assumption 5𝛼+2𝛽>6 in (H2) implies that 𝛽∈((6−5𝛼)/2,𝛼]⊆(1/2,1] and 𝛼∈(6/7,1]. In particular, if 𝛼=1, then Theorem 48 holds with 𝛽∈(1/2,1], 𝜂_𝑖1,𝜎_𝑖2∈(1−𝛽,1), 𝛾_𝑖2,𝜑∈(2−2𝛽,1), 𝑖_𝑙=1,…,𝑛_𝑙, 𝑙=1,2, and 𝜇_1,𝛽=1−𝛽. Hence, 𝛾∈(2−2𝛽,1), 𝜒_1,𝛽,𝛾=𝛽+𝛾−1∈(1−𝛽,𝛽), and 𝛿∈𝐼_1,𝛽,𝜏 with 𝜏∈(1−𝛽,𝛽), where 𝐼_1,𝛽,𝜏=11−𝛽,𝜏,if𝜏∈1−𝛽,2,𝐼_1,𝛽,𝜏=11−𝛽,21,if𝜏∈2,𝛽. (207) Clearly, if 𝛽=1, then 5𝛼+2𝛽>6 is redundant, and Theorem 48 holds with 𝜂_𝑖1,𝜎_𝑖2,𝛾_𝑖2,𝜑∈(0,1),𝑖_𝑙=1,…,𝑛_𝑙,𝑙=1,2,𝜇_1,1=0,𝛾=𝜒_1,1,𝛾∈(0,1), and 𝛿∈𝐼_1,1,𝜏, 𝜏∈(0,1), where 𝐼_1,1,𝜏=(0,𝜏] if 𝜏∈(0,1/2) and 𝐼_1,1,𝜏=(0,1/2) if 𝜏∈[1/2,1).

Remark 51. Observe that, if the 𝜂_𝑖1’s and 𝜎_𝑖2’s are assumed to vary in the smaller interval 𝑈_𝛼,𝛽∶=((3−2𝛼−𝛽)/𝛼,(𝛼+𝛽−1)/𝛼), then 𝜑 and the 𝛾_𝑖2’s can be chosen such that 𝜏=min_{𝑖𝑙=1,…,𝑛𝑙,𝑙=1,2}{𝜂_𝑖1,𝜎_𝑖2}. To this purpose, letting 𝜌=max_{𝑖𝑙=1,…,𝑛𝑙,𝑙=1,2}{𝜂_𝑖1,𝜎_𝑖2}∈𝑈_𝛼,𝛽, it suffices to take 𝛾_𝑖2,𝜑∈𝑉_{𝛼,𝛽,𝜌}, 𝑖₂=1,…,𝑛₂, where 𝑉_{𝛼,𝛽,𝜌}∶=[2+𝛼𝜌−𝛼−𝛽,1)⊊(5−3𝛼−2𝛽,1). Then 𝛾=min_{𝑖2=1,…,𝑛2}{𝛾_𝑖2,𝜑}∈𝑉_{𝛼,𝛽,𝜌} and 𝜒_{𝛼,𝛽,𝛾}=(𝛼+𝛽+𝛾−2)/𝛼≥𝜌. In other words, provided that the data vector (𝑦₁,…,𝑦_𝑛2,𝑣₁+𝑓(0)) is smooth enough, the pair (𝐿𝑣,𝐷_𝑡𝑀𝑣) has the maximal time regularities which is the minimal between the time regularities of the 𝑘_𝑖1’s and ℎ_𝑖2’s.

We conclude with the results which follow from Theorem 48 for problems (163)–(166).

Theorem 52. Assume (161) and 𝑣₀∈𝒟(𝐿), and let 5𝛼+2𝛽>6 in (H2). Assume that 𝑘_𝑖1∈𝐶^𝜂𝑖1(𝐼_𝑇;𝑍) and 𝐿𝑣₀+𝑓(0)∈𝑌^𝑟_𝛾, where 𝜂_𝑖1∈((3−2𝛼−𝛽)/𝛼,1), 𝑖₁=1,…,𝑛₁, 𝛾∈(5−3𝛼−2𝛽,1), and 𝑟∈[1,∞]. Let 𝜏=min_{𝑖1=1,…,𝑛1}{𝜂_𝑖1,𝜒_{𝛼,𝛽,𝛾}}, where 𝜒_{𝛼,𝛽,𝛾}=(𝛼+𝛽+𝛾−2)/𝛼. Then, for every fixed 𝛿∈𝐼_{𝛼,𝛽,𝜏} problem (165) admits a unique strict solution 𝑣∈𝐶^𝛿(𝐼_𝑇;𝒟(𝐿)) satisfying 𝑣(0)=𝑣₀ and such that 𝐿𝑣,𝐷_𝑡𝑀𝑣∈𝐶^𝛿(𝐼_𝑇;𝑋), provided that 𝑓∈𝐶^𝜇(𝐼_𝑇;𝑋), 𝜇∈[𝛿+𝜇_𝛼,𝛽,1), 𝜇_𝛼,𝛽=(3−2𝛼−𝛽)/𝛼.

Proof. Repeat the proofs of Lemmas 42, 45, and 46, Corollary 44, and Theorem 48, letting there 𝜆₀=ℎ_𝑖2=0, 𝑖₂=1,…,𝑛₂. To this purpose, observe that (169) and (174) reduce to 𝐹_𝑤(𝑡)=∑^𝑛1_𝑖1=1[𝒦(𝑘_𝑖1,𝑆_𝑖1𝑤)(𝑡)+𝒦(𝑘_𝑖1,𝐿_𝑖1𝑣₀)(𝑡)]+𝐿𝑣₀+𝑓(𝑡) and 𝑥₀=𝐿𝑣₀+𝑓(0). Consequently, (180)–(182) change to 𝑤₀=𝑄₇𝑥₀+∑^𝑛1_𝑖1=1𝑄₆(𝑘_𝑖1,𝐿_𝑖1𝑣₀)+𝑄₅𝑓, 𝑤₁=−∑^𝑛1_𝑖1=1𝑄₃(𝑘_𝑖1,𝐿_𝑖1𝑣₀)−𝑄₂𝑓, and 𝑅𝑤=∑^𝑛1_𝑖1=1[𝑄₆(𝑘_𝑖1,𝑆_𝑖1𝑤)−𝑄₃(𝑘_𝑖1,𝑆_𝑖1𝑤)].

Theorem 53. Assume (161) and 𝑣₀∈𝒟(𝐿), and let 5𝛼+2𝛽>6 in (H2). Assume that 𝑘_𝑖1∈𝐶^𝜂𝑖1(𝐼_𝑇;𝑍), ℎ_𝑖2∈𝐶^𝜎𝑖2(𝐼_𝑇;𝐂), 𝑦_𝑖2∈𝑌^𝑟_𝛾𝑖2, and 𝐿𝑣₀∈𝑌^𝑟_𝜑, where 𝜂_𝑖1,𝜎_𝑖2∈((3−2𝛼−𝛽)/𝛼,1), 𝛾_𝑖2,𝜑∈(5−3𝛼−2𝛽,1), 𝑖_𝑙=1,…,𝑛_𝑙, 𝑙=1,2, and 𝑟∈[1,∞]. Let 𝛾=min_{𝑖2=1,…,𝑛2}{𝛾_𝑖2,𝜑} and 𝜏=min_{𝑖𝑙=1,…,𝑛𝑙,𝑙=1,2}{𝜂_𝑖1,𝜎_𝑖2,𝜒_{𝛼,𝛽,𝛾}}, where 𝜒_{𝛼,𝛽,𝛾}=(𝛼+𝛽+𝛾−2)/𝛼. Then, for every fixed 𝛿∈𝐼_{𝛼,𝛽,𝜏} problem (166) admits a unique strict solution 𝑣∈𝐶^𝛿(𝐼_𝑇;𝒟(𝐿)) satisfying 𝑣(0)=𝑣₀ and such that 𝐿𝑣,𝐷_𝑡𝑀𝑣∈𝐶^𝛿(𝐼_𝑇;𝑋).

Proof. Let 𝜆₀=𝑓=0 in the proofs of Lemmas 42, 45, and 46, Corollary 44, and Theorem 48. In this case, (169) and (174) reduce to 𝐹_𝑤(𝑡)=∑^𝑛1_𝑖1=1[𝒦(𝑘_𝑖1,𝑆_𝑖1𝑤)(𝑡)+𝒦(𝑘_𝑖1,𝐿_𝑖1𝑣₀)(𝑡)]+∑^𝑛2_𝑖2=1ℎ_𝑖2(𝑡)𝑦_𝑖2+𝐿𝑣₀ and 𝑥₀=∑^𝑛2_𝑖2=1ℎ_𝑖2(0)𝑦_𝑖2+𝐿𝑣₀. Hence, (180)–(182) change to 𝑤₀=𝑄₇𝑥₀+∑^𝑛1_𝑖1=1𝑄₆(𝑘_𝑖1,𝐿_𝑖1𝑣₀)+∑^𝑛2_𝑖2=1𝑄₅ℎ_𝑖2,𝑤₁=−∑^𝑛1_𝑖1=1𝑄₃(𝑘_𝑖1,𝐿_𝑖1𝑣₀)−∑^𝑛2_𝑖2=1𝑄₄(ℎ_𝑖2,𝑦_𝑖2), and 𝑅𝑤=∑^𝑛1_𝑖1=1[𝑄₆(𝑘_𝑖1,𝑆_𝑖1𝑤)−𝑄₃(𝑘_𝑖1,𝑆_𝑖1𝑤)].

Let us now turn to the degenerate differential problems (163) and (164).

Theorem 54. Assume (161) and 𝑣₀∈𝒟(𝐿), and let 5𝛼+2𝛽>6 in (H2). Assume that 𝐿𝑣₀+𝑓(0)∈𝑌^𝑟_𝛾, 𝛾∈(5−3𝛼−2𝛽,1), 𝑟∈[1,∞], and let 𝜒_{𝛼,𝛽,𝛾}=(𝛼+𝛽+𝛾−2)/𝛼. Then, for every fixed 𝛿∈𝐼_{𝛼,𝛽,𝜒𝛼,𝛽,𝛾} problem (163) admits a unique strict global solution 𝑣∈𝐶^𝛿(𝐼_𝑇;𝒟(𝐿)) satisfying 𝑣(0)=𝑣₀ and such that 𝐿𝑣,𝐷_𝑡𝑀𝑣∈𝐶^𝛿(𝐼_𝑇;𝑋), provided that 𝑓∈𝐶^𝜇(𝐼_𝑇;𝑋), 𝜇∈[𝛿+𝜇_𝛼,𝛽,1), 𝜇_𝛼,𝛽=(3−2𝛼−𝛽)/𝛼.

Proof. Let 𝜆₀=𝑘_𝑖1=ℎ_𝑖2=0,𝑖_𝑙=1,…,𝑛_𝑙,𝑙=1,2, in problem (160) and formulae (169), (174) and, (179)–(182). Then, 𝐹_𝑤(𝑡)=𝐿𝑣₀+𝑓(𝑡),𝑥₀=𝐿𝑣₀+𝑓(0) and 𝑤=𝑤₀+𝑤₁=𝑄₇𝑥₀+𝑄₅𝑓−𝑄₂𝑓. Consequently, Lemma 42 and Corollary 44 are unneeded, and the proof of Theorem 48 simplifies as follows. First, due to 𝛾∈(5−3𝛼−2𝛽,1) we have 𝜒_{𝛼,𝛽,𝛾}∈((3−2𝛼−𝛽)/𝛼,1), and the interval 𝐼_{𝛼,𝛽,𝜒𝛼,𝛽,𝛾} is well defined. Hence, let 𝛿∈𝐼_{𝛼,𝛽,𝜒𝛼,𝛽,𝛾} being fixed. Since (cf. (200)) 𝑓∈𝐶^𝜇(𝐼_𝑇;𝑋), 𝜇∈[𝛿+𝜇_𝛼,𝛽,1)⊊((3−2𝛼−𝛽)/𝛼,1), reasoning as in the last part of the proof of Lemma 45 we get 𝑄₂𝑓∈𝐶^𝛿₀(𝐼_𝑇;𝑋). Moreover (see the proof of Lemma 46), since 𝑥₀∈𝑌^𝑟_𝛾, 𝛾∈(5−3𝛼−2𝛽,1)⊆(2−𝛼−𝛽,1) and 𝑓∈𝐶^𝜇₀(𝐼_𝑇;𝑋), 𝜇∈[𝛿+𝜇_𝛼,𝛽,1)⊆[𝛿+𝜚_𝛼,𝛽,1), 𝜚_𝛼,𝛽=(2−𝛼−𝛽)/𝛼, Corollary 38 and Lemma 32 applied with (𝑥,𝛿₇)=(𝑥₀,𝛿) and (𝑔₅,𝛿₅)=(𝑓,𝛿+𝜚_𝛼,𝛽) yield 𝑄₇𝑥₀,𝑄₅𝑓∈𝐶^𝛿₀(𝐼_𝑇;𝑋). Summing up, we find that 𝑤∈𝐶^𝛿₀(𝐼_𝑇;𝑋). The assertion then follows from 𝑣=𝐿⁻¹𝑤+𝑣₀ and (cf. (206)) 𝐷_𝑡𝑀𝑣=𝑤+𝐿𝑣₀+𝑓.

Remark 55. We refer to [19, Theorem 5.3] for a result of both time and space regularity for problem (163). There, provided that 𝜓 and 𝛿 are opportunely chosen and the data satisfy assumptions similar to those in Theorem 54, it is shown that 𝐷_𝑡𝑀𝑣∈𝐶^𝛿(𝐼_𝑇;(𝑋,𝒟(𝐴))_𝜓,𝑟), and that the higher is the order 𝜓 of the interpolation space where we look for space regularity, the lower is the Hölder exponent 𝛿 of regularity in time. Notice that 𝐿𝑣=𝐷_𝑡𝑀𝑣−𝑓 has no space regularity, unless 𝑓 has too.

Theorem 56. Assume (161) and 𝑣₀∈𝒟(𝐿), and let 5𝛼+2𝛽>6 in (H2). Assume that ℎ_𝑖2∈𝐶^𝜎𝑖2(𝐼_𝑇;𝐂), 𝑦_𝑖2∈𝑌^𝑟_𝛾𝑖2, and 𝐿𝑣₀∈𝑌^𝑟_𝜑, where 𝜎_𝑖2∈((3−2𝛼−𝛽)/𝛼,1), 𝛾_𝑖2,𝜑∈(5−3𝛼−2𝛽,1), 𝑖₂=1,…,𝑛₂, and 𝑟∈[1,∞]. Let 𝛾=min_{𝑖2=1,…,𝑛2}{𝛾_𝑖2,𝜑} and 𝜏=min_{𝑖2=1,…,𝑛2}{𝜎_𝑖2,𝜒_{𝛼,𝛽,𝛾}}, where 𝜒_{𝛼,𝛽,𝛾}=(𝛼+𝛽+𝛾−2)/𝛼. Then, for every fixed 𝛿∈𝐼_{𝛼,𝛽,𝜏}, problem (164) admits a unique strict global solution 𝑣∈𝐶^𝛿(𝐼_𝑇;𝒟(𝐿)) satisfying 𝑣(0)=𝑣₀ and such that 𝐿𝑣,𝐷_𝑡𝑀𝑣∈𝐶^𝛿(𝐼_𝑇;𝑋).

Proof. Let 𝜆₀=𝑘_𝑖1=𝑓=0, 𝑖₁=1,…,𝑛₁, in problem (160) and formulae (169), (174), and (179)–(182). Then, 𝐹_𝑤(𝑡)=∑^𝑛2_𝑖2=1ℎ_𝑖2(𝑡)𝑦_𝑖2+𝐿𝑣₀,𝑥₀=∑^𝑛2_𝑖2=1ℎ_𝑖2(0)𝑦_𝑖2+𝐿𝑣₀ and 𝑤=𝑤₀+𝑤₁=𝑄₇𝑥₀+∑^𝑛2_𝑖2=1𝑄₅ℎ_𝑖2−∑^𝑛2_𝑖2=1𝑄₄(ℎ_𝑖2,𝑦_𝑖2). Therefore, as in Theorem 54, we do not need Lemma 42 and Corollary 44, and the proof of Theorem 48 simplifies as follows. Again, 𝛾=min_{𝑖2=1,…,𝑛2}{𝛾_𝑖2,𝜑}∈(5−3𝛼−2𝛽,1) implies that 𝜒_{𝛼,𝛽,𝛾}∈((3−2𝛼−𝛽)/𝛼,1), so that 𝜏=min_{𝑖2=1,…,𝑛2}{𝜎_𝑖2,𝜒_{𝛼,𝛽,𝛾}}∈((3−2𝛼−𝛽)/𝛼,1), and the interval 𝐼_{𝛼,𝛽,𝜏} is well defined. Let 𝛿∈𝐼_{𝛼,𝛽,𝜏} be fixed. First (see the proof of Lemma 45), since 𝛾_𝑖2∈(5−3𝛼−2𝛽,1)⊆(3−2𝛼−𝛽,1), Lemma 30 applied with (𝑔₄,𝑦,𝛿₄,𝛾)=(ℎ_𝑖2,𝑦_𝑖2,𝛿,𝛾_𝑖2) yields 𝑄₄(ℎ_𝑖2,𝑦_𝑖2)∈𝐶^𝛿₀(𝐼_𝑇;𝑋), 𝑖₂=1,…,𝑛₂. On the other side (see the proof of Lemma 46), since 𝑥₀∈𝑌^𝑟_𝛾 and ℎ_𝑖2∈𝐶^𝛿₀(𝐼_𝑇;𝑌^𝑟_𝛾𝑖2)↪𝐶^𝛿₀(𝐼_𝑇;𝑌^𝑟_𝛾), 𝛾∈(5−3𝛼−2𝛽,1)⊆(2−𝛼−𝛽,1), from Lemma 37 and Corollary 38 applied with (𝑔₅,𝛿₅)=(ℎ_𝑖2,𝛿) and (𝑥,𝛿₇)=(𝑥₀,𝛿) we deduce that 𝑄₅ℎ_𝑖2,𝑄₇𝑥₀∈𝐶^𝛿₀(𝐼_𝑇;𝑋),𝑖₂=1,…,𝑛₂. Summing up, we find that 𝑤∈𝐶^𝛿₀(𝐼_𝑇;𝑋), and the assertion again follows from 𝑣=𝐿⁻¹𝑤+𝑣₀ and (cf. (206)) 𝐷_𝑡𝑀𝑣=𝑤+𝐿𝑣₀+∑^𝑛2_𝑖2=1ℎ_𝑖2𝑦_𝑖2.

6. An Application to a Concrete Case

Theorem 48 is here applied to determine the right functional framework where to search for the solution of an inverse problem arising in the theory of heat conduction for materials with memory. To this purpose, let Ω⊊𝐑^𝑁,𝑁∈𝐍, be a bounded domain with boundary 𝜕Ω of class 𝐶^1,1 (cf. [36, p. 94]). If Ω represents a rigid thermal body with memory, then the linearized theory of heat flow yields the following equations linking the internal energy 𝑒, the heat flux 𝐪=(𝑞₁,…,𝑞_𝑁), and the temperature Θ (cf. [32, 37–40]): 𝑒(𝑡,𝑥)=𝑒₀+𝑎(0,𝑥)Θ(𝑡,𝑥)+^𝑡₀𝐷_𝑡𝑎()Θ()𝑞𝑡−𝑠,𝑥𝑠,𝑥d𝑠,_𝑗(𝑡,𝑥)=−^𝑟1_𝑖=1𝑏_𝑖(0)𝐶_𝑖,𝑗𝑥;𝐷_𝑥Θ()−𝑡,𝑥_{𝑟1𝑖=1𝑡0}𝐷_𝑡𝑏_𝑖(𝑡−𝑠)𝐶_𝑖,𝑗𝑥;𝐷_𝑥Θ()𝐷𝑠,𝑥d𝑠,𝑗=1,…,𝑁,_𝑡𝑒(𝑡,𝑥)=−div_𝑥𝐪(𝑡,𝑥)+𝑔(𝑡,𝑥)=−^𝑁_𝑗=1𝐷_𝑥𝑗𝑞_𝑗(𝑡,𝑥)+𝑔(𝑡,𝑥). (208) Here 𝑡∈𝐼_𝑇, 𝐼_𝑇=[0,𝑇], 𝑇>0, 𝑥=(𝑥₁,…,𝑥_𝑁)∈Ω, 𝑟₁∈𝐍, 𝑒₀∈𝐑, and 𝐷_𝑡=𝜕/𝜕𝑡, whereas the 𝐶_𝑖,𝑗(𝑥;𝐷_𝑥)’s represent the first-order linear differential operators 𝐶_𝑖,𝑗x;𝐷_𝑥=^𝑁_𝑘=1𝑐_{𝑖,𝑗,𝑘}(𝑥)𝐷_𝑥𝑘,𝑥∈Ω,𝑖=1,…,𝑟₁,𝑗=1,…,𝑁, (209) where 𝑐_{𝑖,𝑗,𝑘}∈𝐶¹(Ω;𝐑) and 𝐷_𝑥𝑘=𝜕/𝜕𝑥_𝑘, 𝑖=1,…,𝑟₁, 𝑗,𝑘=1,…,𝑁. According to the terminology of [39, 40], the functions 𝑎, 𝑏_𝑖, 𝑖=1,…,𝑟₁, and 𝑔 are called, respectively, the energy-temperature relaxation function, the heat conduction relaxation functions, and the heat supply function and we assume that they satisfy the following conditions: 𝐷^𝑘_𝑡𝑎(⋅,𝑥)∈𝐶𝐼_𝑇𝑎()𝐷;𝐑,𝑘=0,1,2,0,𝑥≥0,𝑥∈Ω,_𝑘𝑡𝑏_𝑖∈𝐶𝐼_𝑇;𝐑,𝑘=0,1,𝑖=1,…,𝑟₁,𝑔∈𝐶¹𝐼_𝑇×Ω;𝐑. (210)(211) Notice that, different from [32, 37–40], here the energy-temperature relaxation function 𝑎 is assumed to depend also on the spatial variable 𝑥∈Ω. In physical terms, this is equivalent to say that Ω represents a rigid inhomogeneous material with memory. Furthermore, in contrast with the quoted papers where only the cases 𝑟₁=1 and 𝐶_1,𝑗(𝑥;𝐷_𝑥)=𝐷_𝑥𝑗 are treated, here we have assumed that the history record of Ω is kept by an arbitrary number 𝑟₁∈𝐍 of heat conduction relaxation functions and that the 𝐶_𝑖,𝑗’s are the more general first-order differential operators defined in (209).

By setting 𝑎_𝑗,𝑘=^𝑟1_𝑖=1𝑏_𝑖(0)𝑐_{𝑖,𝑗,𝑘}∈𝐶¹Ω;𝐑,𝑗,𝑘=1,…,𝑁, (212) from (208) and (209), it thus follows that the temperature Θ must satisfy the following equation: 𝑎(0,𝑥)𝐷_𝑡Θ(𝑡,𝑥)+𝐷_𝑡𝑎()Θ()+0,𝑥𝑡,𝑥_𝑡0𝐷²_𝑡𝑎()Θ()()=𝑡−𝑠,𝑥𝑠,𝑥d𝑠−𝑔𝑡,𝑥_{𝑁𝑗,𝑘=1}𝐷_𝑥𝑗𝑎_𝑗,𝑘(𝑥)𝐷_𝑥𝑘Θ(+𝑡,𝑥)_{𝑟1𝑖=1𝑡0}𝐷_𝑡𝑏_𝑖(𝑡−𝑠)^𝑁_𝑗=1𝐷_𝑥𝑗𝐶_𝑖,𝑗𝑥;𝐷_𝑥Θ(𝑠,𝑥)d𝑠. (213) Let us now assume that 𝑎 is of the following special form: 𝑎(𝑡,𝑥)=²_𝑛=1𝑚_𝑛(𝑥)𝑢_𝑛(𝑡),(𝑡,𝑥)∈𝐼_𝑇×Ω, (214) where the functions 𝑚_𝑛 and 𝑢_𝑛, 𝑛=1,2, satisfy the following conditions (cf. (210)): 𝑚_𝑛∈𝐿_∞(Ω)𝑚,𝑛=1,2,₁≥0,𝑚₂𝑢>0,_𝑛∈𝐶²𝐼_𝑇𝑢;𝐑,𝑛=1,2,₂(0)=0,𝑢₁(0)>0,𝐷_𝑡𝑢₂(0)>0. (215)(216) Here, 𝐿_𝑞(Ω)=𝐿_𝑞(Ω;𝐑),𝑞∈[1,∞], is the usual 𝐿_𝑞 space with norm ‖⋅‖_𝑞;Ω (cf. [36, Chapter 7]). Using 𝑚₂,𝑢₁(0),𝐷_𝑡𝑢₂(0)>0, for 𝑡∈𝐼_𝑇 and 𝑥∈Ω we now set 𝑎₀(𝑥)=−𝑢₁(0)⁻¹𝑚₂(𝑥)𝐷_𝑡𝑢₂(0)𝑎<0,_𝑗,𝑘(𝑥)=𝑢₁(0)⁻¹𝑎_𝑗,𝑘(𝑥)𝐿,𝑗,𝑘=1,…,𝑁,𝑥;𝐷_𝑥=^𝑁_𝑗,𝑘=1𝐷_𝑥𝑗𝑎_𝑗,𝑘(𝑥)𝐷_𝑥𝑘+𝑎₀(𝑥),𝐿_𝑖𝑥;𝐷_𝑥=𝑢₁(0)^𝑁−1_𝑗=1𝐷_𝑥𝑗𝐶_𝑖,j𝑥;𝐷_𝑥,𝑖=1,…,𝑟₁,𝐿_𝑟1+𝑛𝑥;𝐷_𝑥=𝐿_𝑟1+𝑛(𝑥)=𝑢₁(0)⁻¹𝑚_𝑛(𝑥),𝑘𝑛=1,2,_𝑖(𝑡)=𝐷_𝑡𝑏_𝑖(𝑡),𝑖=1,…,𝑟₁,𝑘_𝑟1+𝑛(𝑡)=−𝐷²_𝑡𝑢_𝑛(𝑡)𝑔()=𝑢,𝑛=1,2,𝑡,𝑥₁(0)⁻¹𝑔(),𝜆𝑡,𝑥₀=−𝑢₁(0)⁻¹𝐷_𝑡𝑢₁(0)∈𝐑. (217)(218)(219)(220)(221)(222)(223) Then, since (214)–(216) yield 𝑎(0,𝑥)=𝑚₁(𝑥)𝑢₁(0) and 𝐷^𝑘_𝑡𝑎(𝑡,𝑥)=∑²_𝑛=1𝑚_𝑛(𝑥)𝐷^𝑘_𝑡𝑢_𝑛(𝑡), 𝑘=1,2, if we multiply both sides of (213) by [𝑢₁(0)]⁻¹ and use (218)–(223), we are led to the following basic differential equation for the temperature Θ, where 𝑛₁=𝑟₁+2: 𝐷_𝑡𝑚₁(𝑥)Θ(𝑡,𝑥)=𝜆₀𝑚₁(𝑥)Θ(𝑡,𝑥)+𝐿𝑥;𝐷_𝑥Θ()+𝑔()+𝑡,𝑥𝑡,𝑥_{𝑛1𝑖=1𝑡0}𝑘_𝑖(𝑡−𝑠)𝐿_𝑖𝑥;𝐷_𝑥Θ(𝑠,𝑥)d𝑠,𝑡∈𝐼_𝑇,𝑥∈Ω. (224) We endow this differential equation with the initial condition Θ(0,𝑥)=Θ₀(𝑥),𝑥∈Ω, and the Dirichlet boundary condition Θ(𝑡,𝑥)=0, 𝑡∈𝐼_𝑇, 𝑥∈𝜕Ω.

We now suppress the dependence on 𝑥∈Ω, and we transform (224) in a degenerate integrodifferential Cauchy problem in a Banach space 𝑋. To this purpose, for every fixed 𝑞∈(1,∞) and observing that 𝑚_𝑛∈𝐿_∞(Ω) implies that ‖𝑚_𝑛𝑢‖_𝑞;Ω≤‖𝑚_𝑛‖_∞;Ω‖𝑢‖_𝑞;Ω for every 𝑢∈𝐿_𝑞(Ω), 𝑛=1,2, we set 𝑋=𝒟(𝑀)=𝒟𝐿_𝑟1+𝑛=𝐿_𝑞(Ω)𝒟(𝐿),𝑛=1,2,=𝑊²_𝑞(Ω)∩𝑊^∘1_𝑞(Ω),𝒟𝐿_𝑖=𝑊²_𝑞(Ω),𝑖=1,…,𝑟₁,𝑀,𝐿_𝑟1+𝑛∈ℒ(𝑋),𝑀𝑢=𝑚₁𝐿𝑢,_𝑟1+𝑛𝑢=𝐿_𝑟1+𝑛(𝑥)(𝐿)𝑢,𝑢∈𝑋,𝑛=1,2,𝐿∶𝒟⊆𝑋⟶𝑋,𝐿𝑢=𝐿𝑥;𝐷_𝑥(𝐿),𝐿𝑢,𝑢∈𝒟_𝑖∶𝒟𝐿_𝑖𝐿⊆𝑋⟶𝑋,_𝑖𝑢=𝐿_𝑖𝑥;𝐷_𝑥𝑢,𝑢∈𝒟𝐿_𝑖,𝑖=1,…,𝑟₁. (225)(226)(227)(228)(229) Here (cf. [36, Chapter 7]), 𝑊^𝑘_𝑞(Ω)=𝑊^𝑘_𝑞(Ω;𝐑), 𝑘∈𝐍∪{0}, 𝑞∈(1,∞), denotes the usual Sobolev space endowed with the norm ‖⋅‖_𝑘,𝑞;Ω ((𝑊⁰_𝑞(Ω),‖⋅‖_0,𝑞;Ω)=(𝐿_𝑞(Ω),‖⋅‖_𝑞;Ω)), whereas 𝑊^∘𝑘_𝑞(Ω) denotes the completion of 𝐶^∞₀(Ω;𝐑) in 𝑊^𝑘_𝑞(Ω), 𝐶^∞₀(Ω;𝐑) being the set of all real-valued infinitely differentiable functions having compact support in Ω. We further assume that there exists positive constant Λ_𝑖, 𝑖=0,…,𝑟₁, such that for every (𝑥,𝜉)∈Ω×𝐑^𝑁 the following inequalities hold: ^𝑁_𝑗,𝑘=1𝑐_{𝑖,𝑗,𝑘}(𝑥)𝜉_𝑗𝜉_𝑘≥Λ_𝑖𝜉²,𝑖=1,…,𝑟₁,_{𝑟1𝑗,𝑘=1}𝑏_𝑖(0)Λ_𝑖≥Λ₀, (230) where |𝜉|²=∑^𝑁_𝑙=1𝜉²_𝑙. Therefore, from (212), (218), and (230) we get ^𝑁_𝑗,𝑘=1𝑎_𝑗,𝑘(𝑥)𝜉_𝑗𝜉_𝑘=𝑢₁(0)^𝑟−11_𝑖=1𝑏_𝑖(0)^𝑁_𝑗,𝑘=1𝑐_{𝑖,𝑗,𝑘}𝜉_𝑗𝜉_𝑘≥𝑢₁(0)⁻¹Λ₀𝜉². (231) From (225)–(231) it follows that 𝑀,𝐿, and 𝐿_𝑖, 𝑖=1,…,𝑛₁, are closed linear operators from 𝑋 to itself, and the relation 𝒟(𝐿)⊊⋂^𝑛1_𝑖=1[𝒟(𝑀)∩𝒟(𝐿_𝑖)]=𝑊²_𝑞(Ω) holds. In addition, due to (212), (217), (218), and (231), from [36, Theorem 9.15 and Lemma 9.17], it follows that for every fixed 𝑞∈(1,∞) the operator 𝐿 admits an inverse operator 𝐿⁻¹∈ℒ(𝑋;𝑊²_𝑞(Ω)). Hence, a fortiori, 𝐿⁻¹∈ℒ(𝑋) and so condition (161) is satisfied (observe also that 𝐿⁻¹∈ℒ(𝑋;𝑊²_𝑞(Ω)) implies that the norms ‖⋅‖_2,𝑞;Ω and ‖⋅‖_𝒟(𝐿)=‖⋅‖_𝑞;Ω+‖𝐿⋅‖_𝑞;Ω are equivalent on 𝒟(𝐿). In fact, if 𝑣∈𝒟(𝐿), then ‖𝑣‖_2,𝑞;Ω=‖𝐿⁻¹𝐿𝑣‖_2,𝑞;Ω≤‖𝐿⁻¹‖_{ℒ(𝑋;𝑊}²_𝑞(Ω))‖𝑣‖_𝒟(𝐿)≤ 𝐶‖𝐿⁻¹‖_{ℒ(𝑋;𝑊}²_𝑞(Ω))‖𝑣‖_2,𝑞;Ω, 𝐶 being a positive constant depending on max_{𝑗,𝑘=1,…,𝑁}‖𝑎_𝑗,𝑘‖_{𝐶1(Ω;𝐑)}). The closed graph theorem then yield 𝑀𝐿⁻¹,𝐿_𝑖𝐿⁻¹∈ℒ(𝑋),𝑖=1,…,𝑛₁. Moreover (cf. [19, formula (77)], and [41, formula (2.16)]), the following estimate holds (of course, here 𝑋=𝐿_𝑞(Ω;𝐑) is replaced with the more general 𝑋=𝐿_𝑞(Ω;𝐂)): 𝑀(𝜆𝑀−𝐿)⁻¹_ℒ(𝑋)≤𝐶(|𝜆|+1)^−𝛽,∀𝜆∈Σ₁1,𝛽=𝑞, (232) where Σ₁={𝑧∈𝐂∶ℜe𝑧≥−𝑐(|ℑm𝑧|+1),ℑm𝑧∈𝐑}, 𝑐 being a suitable positive constant depending on 𝑞 and ‖𝑚₁‖_∞;Ω. Hence, condition (H2) is satisfied with 𝑋=𝐿^𝑞(Ω) and (𝛼,𝛽)=(1,1/𝑞). Notice that, since 𝑚₁ may have zeros in Ω, 𝑀⁻¹ is in general a m. l. operator, so that 𝐴=𝐿𝑀⁻¹ is determined by (cf. (162)): 𝒟(𝐴)=𝑀(𝒟(𝐿))=𝑚₁(𝐿,(𝐿)𝑣∶𝑣∈𝒟)𝐴𝑢=𝐿𝑣∶𝑣∈𝒟suchthat𝑢=𝑚₁𝑣,𝑢∈𝒟(𝐴). (233) Using the convolution operator 𝒦 in (104) in which for the bilinear operator 𝒫 we take the scalar multiplication in 𝑋, from (224)–(229) we finally obtain that the temperature Θ(𝑡)=Θ(𝑡,⋅) solves the following degenerate integrodifferential Cauchy problem in 𝑋: 𝐷_𝑡(𝑀Θ(𝑡))=𝜆₀Θ(𝑡)+𝑀+𝐿_𝑛1𝑖=1𝒦𝑘_𝑖,𝐿_𝑖Θ(𝑡)+𝑔(𝑡),𝑡∈𝐼_𝑇,Θ(0)=Θ₀. (234) Now, assume for a moment that we are interested in solving the inverse problem of recovering both the temperature Θ and the memory kernels 𝑘₁,…,𝑘_𝑟1 in (234). Clearly, due to (222), if we recover 𝑘₁,…,𝑘_𝑟1, then the heat conduction relaxation functions 𝑏₁,…,𝑏_𝑟1 will be known too, unless of the 𝑟₁ arbitrary constants 𝑏_𝑖(0), 𝑖=1,…,𝑟₁. Indeed, 𝑏₁(𝑡)=𝑏_𝑖(0)+∫^𝑡₀𝑘_𝑖(𝑠)d𝑠, 𝑡∈𝐼_𝑇. To solve such an inverse problem, we need 𝑟₁ additional informations other than the initial condition Θ(0)=Θ₀, which, in general, suffices only to guarantee the well-posedness of the direct problem of recovering Θ in (234). Suppose then that the following additional pieces of information are given: Ψ_𝑗[𝑀Θ(𝑡)]=𝑔_𝑗(𝑡),𝑡∈𝐼_𝑇,𝑗=1,…,𝑟₁, (235) where Ψ_𝑗∈𝑋^∗=ℒ(𝑋;𝐑) and 𝑔_𝑗∈𝐶^2+𝜈𝑗(𝐼_𝑇;𝐑), 𝜈_𝑗∈(0,1),𝑗=1,…,𝑟₁. We will search for a solution vector (Θ,𝑘₁,…,𝑘_𝑟1) of the inverse problem (234) and (235) such that Θ∈𝐶^1+𝛿(𝐼_𝑇;𝒟(𝐿)) and 𝑘_𝑗∈𝐶^𝜂𝑗(𝐼_𝑇;𝐑), 𝑗=1,…,𝑟₁, with the Hölder exponents 𝛿 and 𝜂_𝑗,𝑗=1,…,𝑟₁, to be made precise in the sequel. We stress that here we will not solve completely the mentioned inverse problem. For, its detailed treatment would lead us out of the aims of this paper. Our intention here is only to highlight how the main results of Section 5 allow to determine the correct functional framework in which the solution of the inverse problem has to be searched. However, a complete treatment of the inverse problem will be the object of a future paper.

Assuming that Θ∈𝐶^1+𝛿(𝐼_𝑇;𝒟(𝐿)) solves (234), we introduce the new unknown 𝑣(𝑡,𝑥)=𝐷_𝑡Θ(𝑡,𝑥)⟺Θ(𝑡,𝑥)=Θ₀(𝑥)+^𝑡₀𝑣(𝑠,𝑥)d𝑠. (236) Then, differentiating (234) with respect to time and using 𝐷_𝑡𝒦𝑘_𝑖,𝐿_𝑖Θ(𝑡)=𝐷_𝑡^𝑡₀𝑘_𝑖(𝑡−𝑠)𝐿_𝑖Θ(𝑠)d𝑠=𝐷_𝑡^𝑡₀𝑘_𝑖(𝑠)𝐿_𝑖Θ(𝑡−𝑠)d𝑠=𝑘_𝑖(𝑡)𝐿_𝑖Θ(0)+^𝑡₀𝑘_𝑖(𝑠)𝐿_𝑖𝐷_𝑡Θ(𝑡−𝑠)d𝑠=𝑘_𝑖(𝑡)𝐿_𝑖Θ₀+^𝑡₀𝑘_𝑖(𝑡−𝑠)𝐿_𝑖𝑣(𝑠)d𝑠, (237) we find that 𝑣∈𝐶^𝛿(𝐼_𝑇;𝒟(𝐿)) solves the following degenerate integrodifferential problem: 𝐷_𝑡((𝑡=𝜆𝑀𝑣))₀𝑀+𝐿𝑣(𝑡)+^𝑛1_𝑖=1𝒦𝑘_𝑖,𝐿_𝑖𝑣(𝑡)+𝑘_𝑖(𝑡)𝑦_𝑖+𝑓(𝑡),𝑡∈𝐼_𝑇,𝑀𝑣(0)=𝑀𝑣₀, (238) where 𝑦_𝑖=𝐿_𝑖Θ₀,𝑖=1,…,𝑛₁,𝑓=𝐷_𝑡𝑔 and 𝑀𝑣₀=[𝜆₀𝑀+𝐿]Θ₀+𝑔(0,⋅) (indeed, since 𝑀 is the multiplication operator by the function 𝑚₁ independent of 𝑡, from the differential equation in (234) with 𝑡=0 we get 𝑀𝑣(0)=𝑀𝐷_𝑡Θ(0)=[𝜆₀𝑀+𝐿]Θ(0)+𝑔(0)). Of course, (238) is the special case (𝑖₁,𝑖₂,𝑛₂)=(𝑖,𝑖,𝑛₁), ℎ_𝑖=𝑘_𝑖, 𝑖=1…,𝑛₁, of problem (160).

Conversely, assume that 𝑣∈𝐶^𝛿(𝐼_𝑇;𝒟(𝐿)) solves (238). Then, the function Θ defined by (236) belongs to 𝐶^1+𝛿(𝐼_𝑇;𝒟(𝐿)) and solves (234). Indeed, using the fact that 𝑚₁ does not depend on time and that 𝑀, 𝐿, and 𝐿_𝑖, 𝑖=1,…,𝑛₁, are closed, we obtain 𝐷_𝑡(𝑀Θ(𝑡))−𝜆₀Θ(𝑡)−𝑀+𝐿_𝑛1𝑖=1𝒦𝑘_𝑖,𝐿_𝑖Θ(𝑡)−𝑔(𝑡)=𝐷_𝑡𝑀Θ₀+^𝑡₀𝑣(𝑠)−𝜆d𝑠₀𝑀+𝐿Θ₀+^𝑡₀𝑣(𝑠)−d𝑠_{𝑛1𝑖=1𝑡0}𝑘_𝑖(𝑡−𝑠)𝐿_𝑖Θ₀+^𝑠₀𝑣(𝜉)−𝑔(0)−d𝜉d𝑠_𝑡0𝐷_𝑠𝑔(𝑠)(𝑡)−𝜆d𝑠=𝑀𝑣₀𝑀+𝐿Θ₀−^𝑡₀𝜆₀𝑣(𝑠)−𝑀+𝐿d𝑠_{𝑛1𝑖=1𝑡0}𝑘_𝑖(𝑡−𝑠)𝐿_𝑖Θ₀−d𝑠_{𝑛1𝑖=1𝑡0}𝑘_𝑖(𝑡−𝑠)^𝑠₀𝐿_𝑖𝑣(𝜉)−𝑔(0)d𝜉d𝑠−^𝑡₀𝑓(𝑠)d𝑠. (239) Now, observe that 𝑀𝑣(𝑡)=𝑀𝑣₀+^𝑡₀𝐷_𝑠((𝑠=𝜆𝑀𝑣))d𝑠₀𝑀+𝐿Θ₀+𝑔(0)+^𝑡₀𝐷_𝑠(𝑀𝑣(𝑠))d𝑠,_𝑡0𝑘_𝑖(𝑡−𝑠)𝐿_𝑖Θ₀d𝑠=^𝑡₀𝑘_𝑖(𝑠)𝐿_𝑖Θ₀d𝑠=^𝑡₀𝑘_𝑖(𝑠)𝑦_𝑖d𝑠,𝑖=1,…,𝑛₁, (240)(241) whereas an application of Fubini’s theorem combined with the changes of variables 𝜉=𝑠−𝑟, 𝑟−𝑠=𝜏 and 𝑡−𝑠=𝜁 easily yields for every 𝑖=1,…,𝑛₁ the following: ^𝑡₀𝑘_𝑖(𝑡−𝑠)^𝑠₀𝐿_𝑖𝑣(𝜉)=d𝜉d𝑠_𝑡0𝑘_𝑖(𝑡−𝑠)^𝑠₀𝐿_𝑖𝑣()=𝑠−𝑟d𝑟d𝑠_{𝑡0𝑡𝑠}𝑘_𝑖(𝑡−𝑟)𝐿_𝑖𝑣()=𝑟−𝑠d𝑟d𝑠_{𝑡00𝑡−𝑠}𝑘_𝑖(𝑡−𝑠−𝜏)𝐿_𝑖𝑣(𝜏)=d𝜏d𝑠_𝑡0𝒦𝑘_𝑖,𝐿_𝑖𝑣()=𝑡−𝑠d𝑠_𝑡0𝒦𝑘_𝑖,𝐿_𝑖𝑣(𝜁)d𝜁=^𝑡₀𝒦𝑘_𝑖,𝐿_𝑖𝑣(𝑠)d𝑠. (242) Therefore, replacing (240)–(242) in (239), it follows for every 𝑡∈𝐼_𝑇 that 𝐷_𝑡(𝑀Θ(𝑡))−𝜆₀𝑀+𝐿Θ(𝑡)−^𝑛1_𝑖=1𝒦𝑘_𝑖,𝐿_𝑖Θ(𝑡)−𝑔(𝑡)=^𝑡₀𝐷_𝑠(𝑀𝑣(𝑠))−𝜆₀𝑣(𝑠)−𝑀+𝐿_𝑛1𝑖=1𝒦𝑘_𝑖,𝐿_𝑖𝑣(𝑠)+𝑘_𝑖(𝑠)𝑦_𝑖−𝑓(𝑠)d𝑠, (243) and the latter integral is equal to zero by virtue of (238). Since from (236) it follows that Θ(0)=Θ₀, we have thus shown that (234) and (238) are equivalent. Such an equivalence is the first step in solving the mentioned inverse problem of recovering the vector (Θ,𝑘₁,…,𝑘_𝑟1) with the help of the additional information (235).

Let us now apply the linear functional Ψ_𝑗, 𝑗=1…,𝑟₁, to (238). Using Ψ_𝑗𝐷^𝑘_𝑡(𝑀𝑣(𝑡))=Ψ_𝑗𝑀𝐷_𝑡^𝑘+1Θ(𝑡)=𝐷_𝑡^𝑘+1Ψ_𝑗[𝑀Θ(𝑡)]=𝐷_𝑡^𝑘+1𝑔_𝑗(𝑡),𝑘=0,1, (244) we thus find the following system of 𝑟₁ equations for the 𝑟₁ unknown 𝑘₁,…,𝑘_𝑟1: ^𝑟1_𝑖=1Ψ_𝑗𝑦_𝑖𝑘_𝑖(𝑡)=𝑁_𝑗(𝑡)−Ψ_𝑗[𝐿𝑣]−^𝑛1_𝑖=1Ψ_𝑗𝒦𝑘_𝑖,𝐿_𝑖𝑣(𝑡),𝑗=1,…,𝑟₁, (245) where we have set (recall that 𝑘_𝑟1+𝑛=−𝐷_𝑡𝑢_𝑛, 𝑛=1,2, are known) 𝑁_𝑗(𝑡)=𝐷_𝑡−𝜆₀𝐷_𝑡𝑔_𝑗(𝑡)−Ψ_𝑗𝑓(𝑡−_2𝑛=1)Ψ_𝑗𝑦_𝑟1+𝑛𝑘_𝑟1+𝑛(𝑡),𝑗=1,…,𝑟₁. (246) Therefore, if the matrix 𝒰∶=𝒰^{𝑦1,…,𝑦𝑟1}_{Ψ1,…,Ψ𝑟1}=(Ψ_𝑖[𝑦_𝑗])_{𝑖,𝑗=1,…,𝑟1} has determinant det𝒰≠0, then from Cramer’s formula it follows that the solution (𝑘₁,…,𝑘_𝑟1) of (245) is given by 𝑘_𝑗(𝑡)=[det𝒰]^𝑟−11_𝑘=1𝑁_𝑘(𝑡)−Ψ_𝑘[]−𝐿𝑣_𝑛1𝑖=1Ψ_𝑘𝒦𝑘_𝑖,𝐿_𝑖𝑣(𝑡)𝒰_𝑘,𝑗=∶𝑅_𝑗𝑣,𝑘₁,…,𝑘_𝑟1(𝑡),𝑗=1,…,𝑟₁, (247) with 𝒰_𝑘,𝑗, 𝑘,𝑗=1,…,𝑟₁, being the cofactor of the element Ψ_𝑘[𝑦_𝑗] of 𝒰 (with the convention that 𝒰_1,1=1 in the case of 𝑟₁=1). We have thus found a system of 𝑟₁ fixed-point equations for the 𝑟₁ unknown 𝑘₁,…,𝑘_𝑟1.

Now, let 𝑌^𝑟_𝜓∈{(𝑋,𝒟(𝐴))_𝜓,𝑟,𝑋_𝐴^𝜓,𝑟}, 𝜓∈(0,1), 𝑟∈[1,∞], where 𝐴 is as in (233). Assume that 𝑣₀ in the initial condition 𝑀𝑣(0)=𝑀𝑣₀ belongs to 𝒟(𝐿) and that 𝑘_𝑖∈𝐶^𝜂𝑖𝐼_𝑇,𝐑,𝑓∈𝐶^𝜇𝐼_𝑇;𝑋,𝜂_𝑖1,𝜇∈𝑞^′,1,𝑖=1,…,𝑛₁,𝑦_𝑖∈𝑌^𝑝_𝛾𝑖,𝑣₁+𝑓(0)∈𝑌^𝑝_𝜑,𝛾_𝑖1,𝜑∈𝑞^′,[],1𝑝∈1,∞,𝑖=1,…,𝑛₁, (248) where 𝑣₁=(𝜆₀𝑀+𝐿)𝑣₀ and 𝑞^′ is the conjugate exponent of 𝑞∈(1,∞). Then (cf. (179) with (𝑖₁,𝑖₂,𝑛₂)=(𝑖,𝑖,𝑛₁), (𝛼,𝛽,𝑍)=(1,1/𝑞,𝐑), and ℎ_𝑖=𝑘_𝑖, 𝑖=1,…,𝑛₁), problem (238) is equivalent to the fixed-point equation 𝑤(𝑡)=𝑅𝑤,𝑘₁,…,𝑘_𝑟1(𝑡)+¹_𝑙=0𝑤_𝑙𝑘₁,…,𝑘_𝑟1(𝑡)=∶𝑇𝑤,𝑘₁,…,𝑘_𝑟1(𝑡), (249) where 𝑤=𝐿(𝑣−𝑣₀) and 𝑤₀𝑘₁,…,𝑘_𝑟1=𝑄₇𝑥₀+^𝑛1_𝑖=1𝑄₆𝑘_𝑖,𝐿_𝑖𝑣₀+𝑄₅𝑘_𝑖+𝑄₅𝑤𝑓,₁𝑘₁,…,𝑘_𝑟1=−^𝑛1_𝑖=1𝑄₃𝑘_𝑖,𝐿_𝑖𝑣₀+𝑄₄𝑘_𝑖,𝑦_𝑖−𝑄₂𝑅𝑓,𝑤,𝑘₁,…,𝑘_𝑟1=𝜆₀𝑄₅𝐴⁻¹𝑤−𝑄₂𝐴⁻¹𝑤+^𝑛1_𝑖=1𝑄₆𝑘_𝑖,𝑆_𝑖𝑤−𝑄₃𝑘_𝑖,𝑆_𝑖𝑤. (250) Here, the 𝑄_𝑗’s, 𝑗=2,…,6, are defined by (106)–(110), 𝑆_𝑖=𝐿_𝑖𝐿⁻¹, and the functions 𝑓, 𝑘_𝑖 and 𝑄₇𝑥₀ are defined by 𝑓(𝑡)=𝑓(𝑡)−𝑓(0), 𝑘_𝑖(𝑡)=[𝑘_𝑖(𝑡)−𝑘_𝑖(0)]𝑦_𝑖, and [𝑄₇𝑥₀](𝑡)=(e^𝑡𝐴−𝐼)𝑥₀, respectively, where (cf. (174)) 𝑥₀=∑^𝑛1_𝑖=1𝑘_𝑖(0)𝑦_𝑖+𝑣₁+𝑓(0).

Then, since 𝑣=𝐿⁻¹𝑤+𝑣₀, if we set 𝑅_𝑗(𝑤,𝑘₁,…,𝑘_𝑟1)=𝑅_𝑗(𝐿⁻¹𝑤+𝑣₀,𝑘₁,…,𝑘_𝑟1), 𝑗=1,…,𝑟₁, and Ξ𝑤,𝑘₁,…,𝑘_𝑟1=𝑇𝑤,𝑘₁,…,𝑘_𝑟1,𝑅₁𝑤,𝑘₁,…,𝑘_𝑟1,…,𝑅_𝑟𝑖𝑤,𝑘₁,…,𝑘_𝑟1, (251) from (247) and (249) we deduce that to solve the inverse problems (234) and (235) for the unknown vector (Θ,𝑘₁,…,𝑘_𝑟1), it suffices to show that the fixed-point equation 𝑤,𝑘₁,…,𝑘_𝑟1=Ξ𝑤,𝑘₁,…,𝑘_𝑟1 (252) has a unique solution. In general, this will be done by proving that Ξ is a contraction map in the Banach space 𝑍_{𝛿,𝜂1,…,𝜂𝑟1}=𝐶^𝛿𝐼_𝑇;𝑋×𝐶^𝜂1𝐼_𝑇;𝐑×⋯×𝐶^𝜂𝑟1𝐼_𝑇,𝑓;𝐑₀,𝑓₁,…,𝑓_{𝑟1𝑍1𝑟1𝛿,𝜂,…,𝜂}=𝑓_{0𝛿,0,𝑇;𝑋}+𝑓_{1𝜂1,0,𝑇;𝐑}+⋯+𝑓_{𝑟1𝜂𝑟1,0,𝑇;𝐑}, (253) at least for opportunely chosen Hölder exponents 𝛿∈(0,1) and 𝜂_𝑖∈(1/𝑞^′,1), 𝑖=1,…,𝑟₁, and, eventually, sufficiently small values of 𝑇>0. It is just in the choice of 𝛿 and the 𝜂_𝑖’s that the main result of Section 5 plays a key role. The Hölder exponents have to be chosen so that the direct problem (234) in which the 𝑘_𝑖’s are assumed to be known is well posed. Due to the shown equivalence between problems (234) and (238), the well-posedness of the direct problem (234) is then a consequence of Theorem 48 and formula (236). More precisely, recalling Remark 50 for the case 𝛼=1, an application of that theorem yields the following maximal time regularity result for the solution Θ of (234).

Theorem 57. Let 𝑋, 𝒟(𝑀), 𝒟(𝐿), and 𝒟(𝐿_𝑖), 𝑖=1,…,𝑛₁, 𝑛₁=𝑟₁+2, 𝑟₁∈𝐍, be defined by (225) and (226) with 𝑞∈(1,2). Let 𝑀,𝐿, and 𝐿_𝑖, 𝑖=1,…,𝑛₁, be defined by (227)–(229) through (209), (212), and (215)–(221), and let (230) and (231) be satisfied. Further, let (𝐴,𝒟(𝐴)) be defined by (233), and let 𝑌^𝑟_𝜓∈{(𝑋,𝒟(𝐴))_𝜓,𝑟,𝑋_𝐴^𝜓,𝑟}, 𝜓∈(0,1), 𝑟∈[1,∞]. Let 𝜂_𝑖∈(1/𝑞^′,1) and 𝛾_𝑖,𝜑∈(2/𝑞^′,1), 𝑖=1,…,𝑛₁, and assume that 𝑘_𝑖∈𝐶^𝜂𝑖𝐼_𝑇;𝐑,𝑖=1,…,𝑛₁,Θ₀(𝐿),𝜆∈𝒟₀𝑀+𝐿Θ₀+𝑔(0,⋅)=𝑀𝑣₀𝑓𝑜𝑟𝑠𝑜𝑚𝑒𝑣₀(𝐿),𝐿∈𝒟_𝑖Θ₀∈𝑌^𝑟_𝛾𝑖,𝑣₁+𝐷_𝑡𝑔(0,⋅)∈𝑌^𝑟_𝜑,𝑖=1,…,𝑛₁,𝑟∈[1,∞], (254) where 𝑘_𝑖, 𝑖=1,…,𝑛₁,𝑔 and 𝜆₀ are defined by (222) and (223) through (211) and (216), whereas 𝑣₁=(𝜆₀𝑀+𝐿)𝑣₀. Let 𝛾=min_{𝑖=1,…,𝑛1}{𝛾_𝑖,𝜑} and 𝜏=min_{𝑖=1,…,𝑛1}{𝜂_𝑖,𝛾−1/𝑞^′}, and let 𝐼_{1,1/𝑞,𝜏}⊆(1/𝑞^′,1/2) be the interval defined by (cf. (207) with 𝛽=1/𝑞) 𝐼_{1,1/𝑞,𝜏}=1𝑞^′1,𝜏,𝑖𝑓𝜏∈𝑞^′,12,𝐼_{1,1/𝑞,𝜏}=1𝑞^′,121,𝑖𝑓𝜏∈2,1. (255) Then, for every fixed 𝛿∈𝐼_{1,1/𝑞,𝜏} problem (234), or, equivalently, problem (224), admits a unique strict solution Θ∈𝐶^1+𝛿(𝐼_𝑇;𝒟(𝐿)) satisfying 𝐷_𝑡Θ(0)=𝑣₀ and such that 𝐷_𝑡𝑀Θ,𝐿Θ∈𝐶^1+𝛿(𝐼_𝑇;𝑋), provided that 𝐷_𝑡𝑔∈𝐶^𝜇(𝐼_𝑇;𝑋), 𝜇∈[𝛿+1/𝑞^′,1).

Proof. Apply Theorem 48 with (𝑖₁,𝑖₂,𝑛₂)=(𝑖,𝑖,𝑛₁), (𝛼,𝛽,𝑍)=(1,1/𝑞,𝐑), and ℎ_𝑖=𝑘_𝑖, 𝑖=1,…,𝑛₁, to the equivalent problem (238). Since 𝑀 is the multiplication operator by the function 𝑚₁ independent of 𝑡, the assertion then follows from 𝐷_𝑡Θ=𝑣∈𝐶^𝛿(𝐼_𝑇;𝒟(𝐿)), 𝐷_𝑡Θ(0)=𝑣(0), 𝐷_𝑡𝐿Θ=𝐿𝑣∈𝐶^𝛿(𝐼_𝑇;𝑋) and 𝐷²_𝑡𝑀Θ=𝐷_𝑡𝑀𝑣∈𝐶^𝛿(𝐼_𝑇;𝑋).

Larger values of 𝑞 in Theorem 57 can be obtained assuming more smoothness and some order of vanishing for the function 𝑚₁. In fact, let 𝑚₁∈𝐶¹(Ω) be such that the following estimate holds for some positive constant 𝐾: ∇𝑚₁(𝑥)∶=^𝑁_𝑗=1𝐷_𝑥𝑗𝑚₁(𝑥)^21/2≤𝐾𝑚₁(𝑥)^𝜗,𝑥∈Ω,𝜗∈(0,1). (256) Then (232) holds with 𝛽=1/𝑞 being replaced by (cf. [41, formulae (3.23) and (4.41)]): 1𝛽=(),22−𝜗,if𝑞∈2−𝜗,2𝛽=𝑞()[).2−𝜗,if𝑞∈2,∞ (257) (precisely, in [41, formula (3.23)] it is shown that (|𝜆|+1)‖𝑀𝑢‖^{𝑞(2−𝜗)/2}_𝑞;Ω≤𝐶_𝑞[‖𝑓‖_𝑞;Ω‖𝑀𝑢‖^{−1+𝑞(2−𝜗)/2}_𝑞;Ω+‖𝑓‖^{𝑞(2−𝜗)/2}_𝑞;Ω], where 𝑢=(𝜆𝑀−𝐿)⁻¹𝑓 and 𝑞∈[2,∞). Using (cf. [41, formula (2.15)]) ‖𝑀𝑢‖_𝑞;Ω≤‖𝑚₁‖_∞;Ω‖𝑢‖_𝑞;Ω≤𝐶‖𝑚₁‖_∞;Ω‖𝑓‖_𝑞;Ω, we thus find that (|𝜆|+1)‖𝑀𝑢‖^{𝑞(2−𝜗)/2}_𝑞;Ω≤𝐶_𝑞[(𝐶‖𝑚₁‖_∞;Ω)^{−1+𝑞(2−𝜗)/2}+1]‖𝑓‖^{𝑞(2−𝜗)/2}_𝑞;Ω; that is, ‖𝑀(𝜆𝑀−𝐿)⁻¹‖_ℒ(𝑋)≤{𝐶_𝑞[(𝐶‖𝑚₁‖_∞;Ω)^{−1+𝑞(2−𝜗)/2}+1]}^{2/[𝑞(2−𝜗)]}(|𝜆|+1)^{−2/[𝑞(2−𝜗)]}). Under (256) we thus find the following better result, where 𝑞 may be greater than two.

Theorem 58. Let (256) holds, and let 𝑋, (𝑀,𝒟(𝑀)), (𝐿,𝒟(𝐿)), (𝐿_𝑖,𝒟(𝐿_𝑖)), 𝑖=1,…,𝑛₁, be as in Theorem 57, but with 𝑞∈(2−𝜗,2)∪[2,4/(2−𝜗)). Let (254) be fulfilled, but with 𝜂_𝑖∈(1−𝛽,1) and 𝛾_𝑖,𝜑∈(2−2𝛽,1), 𝑖=1,…,𝑛₁, where 𝛽 is as in (257). Let 𝛾=min_{𝑖=1,…,𝑛1}{𝛾_𝑖,𝜑} and 𝜏=min_{𝑖=1,…,𝑛1}{𝜂_𝑖,𝛽+𝛾−1}, and let 𝐼_1,𝛽,𝜏 be as in (207). Then, for every fixed 𝛿∈𝐼_1,𝛽,𝜏 problem (234), or, equivalently, problem (224), admits a unique strict solution Θ∈𝐶^1+𝛿(𝐼_𝑇;𝒟(𝐿)) satisfying 𝐷_𝑡Θ(0)=𝑣₀ and such that 𝐷_𝑡𝑀Θ,𝐿Θ∈𝐶^1+𝛿(𝐼_𝑇;𝑋), provided that 𝐷_𝑡𝑔∈𝐶^𝜇(𝐼_𝑇;𝑋), 𝜇∈[𝛿+1−𝛽,1).

Proof. It suffices to observe that for every 𝜗∈(0,1) and 𝑞∈(2−𝜗,2)∪[2,4/(2−𝜗)), the number 𝛽 in (257) satisfies 𝛽>1/2. Hence, proceeding as in the proofs of Theorem 57, except for replacing there 𝛽=1/𝑞 with 𝛽 as in (257), we get the assertion.

Appendix

Here we clarify why the definition of 𝑄₂ in [20] has to be modified in accordance to that in this paper. To avoid confusion with the present notation, we will denote the operator 𝑄₂ in [20] with 𝑆₂. Precisely, in [20, formula (4.12)], 𝑆₂ was defined as follows:𝑆₂𝑔₂(𝑡)∶=^𝑡₀(−𝐴)^1∘e^{(𝑡−𝑠)𝐴}𝑔₂(𝑠)d𝑠,𝑡∈[0,𝑇], (A.1) and considered as acting on functions 𝑔₂∈𝐶^𝛿2₀([0,𝑇];𝑋), 𝛿₂∈((3−2𝛼−𝛽)/𝛼,1), 3𝛼+𝛽>3. Even though 𝑔₂(0)=0, formula (A.1) may have no sense, since 𝑆₂𝑔₂(𝑡)_𝑋≤𝑐̃_𝛼,𝛽,1𝑔_{2𝛿2,0,𝑡;𝑋}^𝑡₀(𝑡−𝑠)^{(𝛽−2)/𝛼}𝑠^𝛿2d𝑠, (A.2) and the integral on the right is not convergent, the exponent (𝛽−2)/𝛼 being less or equal than −1. It is for this reason that 𝑔₂(𝑠) in (A.1) has to be replaced with the increment 𝑔₂(𝑠)−𝑔₂(𝑡) as in formula (106) (see inequality (118)) and to introduce the operator 𝑄₅ as in (109). Of course, as a consequence, the definitions of 𝑄₃ and 𝑄₄ in [20, Lemmas 4.6 and 4.8] as 𝑆₂𝒦(𝑔₃₁,𝑔₃₂) and 𝑆₂(𝑔₄𝑦), respectively, have to be changed too in accordance with the present formulae (107) and (108) containing the increments 𝒦(𝑔₃₁,𝑔₃₂)(𝑠)−𝒦(𝑔₃₁,𝑔₃₂)(𝑡) and [𝑔₄(𝑠)−𝑔₄(𝑡)]𝑦. To this purpose, we want to make clear that, contrarily to [20, Lemma 4.4], the statement and the proof of [20, Lemma 4.8] is correct, since there the function inside the integral on the right-hand side of (A.1) takes its values in an opportune intermediate space 𝑋_𝐴^𝜃,𝑟. However, the correctness of that lemma does not suffice to proceed as in [20, Section 5] to solve problem (160) with 𝑛₁=𝑛₂=1.

For the reader’s convenience we thus now indicate how to change the definitions of the functions 𝑤_𝑗, 𝑗=0,1, and the operator 𝑅𝑤 in [20, formulae (5.8)–(5.10)], and we state the amended version of [20, Theorems 5.6 and 5.7]. First, according to [20] where only this case was treated, let 𝑛₁=𝑛₂=1 in problem (160), and write 𝑘, ℎ, 𝑦 in place of 𝑘₁, ℎ₁ and 𝑦₁, respectively. Then, under the same assumptions on the vector (𝛼,𝛽,𝑘,ℎ,𝑓) as those in the present Section 5, it can be shown that problem (160) with 𝑛₁=𝑛₂=1 is equivalent to the fixed-point equation (179), where (cf. (180)–(182)) 𝑤₀=𝑄₇𝑥₀+𝑄₆𝑘,𝐿₁𝑣₀+𝑄₅ℎ+𝑄₅𝑤𝑓,₁=−𝑄₃𝑘,𝐿₁𝑣₀−𝑄₄ℎ,𝑦−𝑄₂𝑓,𝑅𝑤∶=𝜆₀𝑄₅𝐴⁻¹𝑤−𝑄₂𝐴⁻¹𝑤+𝑄₆(𝑘,𝑆𝑤)−𝑄₃(𝑘,𝑆𝑤). (A.3) Here, 𝑥₀=𝑣₁+ℎ(0)𝑦+𝑓(0), 𝑣₁=(𝜆₀𝑀+𝐿)𝑣₀, is the value at 𝑡=0 of the function 𝐹_𝑤 defined by (169) with 𝑛_𝑙=𝑛₂=1, 𝑄₇𝑥₀,𝑓 and ℎ are defined, respectively, by (e^𝑡𝐴−𝐼)𝑥₀,𝑓(𝑡)−𝑓(0) and [ℎ(𝑡)−ℎ(0)]𝑦, 𝑆 is the operator 𝐿₁𝐿⁻¹∈ℒ(𝑋), and the 𝑄_𝑗’s, 𝑗=2,…,6, are as in (106)–(110). Formulae (A.3) replace the definitions of 𝑤₀, 𝑤₁ and 𝑅𝑤 in [20, formulae (5.8)–(5.10)]. Therefore, from Lemmas 42, 45, and 46 and Corollary 44 with 𝑛₁=𝑛₂=1 we obtain the following version of Theorem 48.

Theorem A.1. Assume (161) and 𝑣₀∈𝒟(𝐿), and let 5𝛼+2𝛽>6 in (H2). Assume that 𝑘∈𝐶^𝜂(𝐼_𝑇;𝑍), ℎ∈𝐶^𝜎(𝐼_𝑇;𝐂), 𝑦∈𝑌^𝑟_𝜃, and (𝜆₀𝑀+𝐿)𝑣₀+𝑓(0)∈𝑌^𝑟_𝜑, where 𝜂,𝜎∈((3−2𝛼−𝛽)/𝛼,1), 𝜃,𝜑∈(5−3𝛼−2𝛽,1), and 𝑟∈[1,∞]. Let 𝛾=min{𝜃,𝜑} and 𝜏=min{𝜂,𝜎,(𝛼+𝛽+𝛾−2)/𝛼}. Then, for every fixed 𝛿∈𝐼_{𝛼,𝛽,𝜏} the problem 𝐷_𝑡(𝑀𝑣(𝑡))=𝜆₀𝑀+𝐿𝑣(𝑡)+𝒦𝑘,𝐿₁𝑣(𝑡)+ℎ(𝑡)𝑦+𝑓(𝑡),𝑡∈𝐼_𝑇,𝑀𝑣(0)=𝑀𝑣₀ (A.4) admits a unique strict solution 𝑣∈𝐶^𝛿(𝐼_𝑇;𝒟(𝐿)) satisfying 𝑣(0)=𝑣₀ and such that 𝐿𝑣,𝐷_𝑡𝑀𝑣∈𝐶^𝛿(𝐼_𝑇;𝑋), provided that 𝑓∈𝐶^𝜇(𝐼_𝑇;𝑋), 𝜇∈[𝛿+(3−2𝛼−𝛽)/𝛼,1).

Theorem A.1 substitutes [20, Theorem 5.6 and 5.7]. Notice that, differently than [20], here only one statement occurs. In fact, the more suitable procedure followed in this paper makes the separation in [20] of two distinct intervals in which 𝛾 may vary totally unneeded. Finally, letting 𝑛₁=𝑛₂=1 in Theorems 52, 5.14, 54, and 56, we obtain the correct versions of [20, Theorems 5.11, 53, and 5.16] for the subcases of (A.4) corresponding to the choices 𝜆₀=ℎ=0, 𝜆₀=𝑓=0, 𝜆₀=𝑘=ℎ=0, and 𝜆₀=𝑘=𝑓=0, respectively. For saving space, we leave this easy task to the reader.