New interpolation spaces and strict Hölder regularity for fractional abstract Cauchy problem

We know that interpolation spaces in terms of analytic semigroup have a significant role into the study of strict Hölder regularity of solutions of classical abstract Cauchy problem (ACP). In this paper, we first construct interpolation spaces in terms of solution operators in fractional calculus and characterize these spaces. Then we establish strict Hölder regularity of mild solutions of fractional order ACP.


Introduction
Let (X, · ) be a Banach space. First, we take into consideration of the following abstract Cauchy problem: where F : [0, T] → X, v 0 ∈ X, and A : D(A) ⊂ X → X is densely defined operator satisfying following conditions: (i) θ := {μ ∈ C : μ = 0, | arg(μ)| < θ } ⊂ ρ(A), (ii) (μI -A) -1 L(X) ≤ N |μ| , ∀μ ∈ θ , for some θ ∈ ( π 2 , π), N > 0. The sets ρ(A), L(X) are denoted as resolvent set of A and the set of all bounded linear operators on X, respectively. Such operator A, is known as sectorial operator, and it generates the analytic semigroup {T(t)} t≥0 on X (cf. [1]). The study of strict solution of (1.1) is very well known. For instance, we refer to [1,Chap. 4], [2] and the references therein. In these cited papers, the authors proved strict Hölder regularity of the mild solution of (1.1) under suitable space and time regularity of the initial datum. More precisely, if F ∈ C θ ([0, T]; X), where 0 < θ < 1, v 0 ∈ D(A) with Av 0 + F(0) ∈ D A (θ , ∞), an interpolation space, then (1.1) has strict solution. The results have been obtained by using the advantage of representation of interpolation space D A (θ , ∞) in terms of analytic semigroup. Here we are interested in the investigation of similar results for the following fractional order ACP on X: where c D α t represents Caputo fractional differential operator of order α ∈ (0, 1). Over the last few decades, the study in the area of fractional calculus has influenced the researchers owing to its generous applicability in the branch of science and engineering. Specifically, we refer the work [3] for some substantial applications of fractional differential equations.
There have been intensive investigation on existence, uniqueness and regularity of solution for variety of generalized model of (1.2). For the enthusiastic reader, we refer to [4][5][6] for linear autonomous case, [7,8] for semilinear autonomous case, [9] and some work cited therein for the non-autonomous case with delay. But the study of strict regularity of the problem (1.2) is very rare for the case α ∈ (0, 1). Bazhlekova [10], in her precious thesis, studied strict L p -regularity by introducing the concept of solution operators and using the resolvent representation of classical interpolation space D A (θ , p) for 1 < p < ∞. The case p = ∞ (i.e., strict Hölder regularity case) has been studied first by Ph. Clément et al. in their pioneer work [11]. Using some suitable transformation, the author split the problem (1.2) into two abstract problems, one is having homogeneous force function (i.e., F = 0) but with inhomogeneous initial data, and another is having inhomogeneous force function F(t) -F(0) but with homogeneous initial data. The author used method of sum to investigate the regularity of the latter one, in which the problem is converted to an operator equation of the form Bu + Cu = v on the spaceX = {v ∈ C([0, T]; X) : v(0) = 0} and applied a suitable theorem [11,Theorem 8]. Whereas, the regularity of the first one is solved by exploiting resolvent representation of the interpolation space D A (θ , ∞). Finally, combining the both results, the author obtained the strict Hölder regularity of the problem (1.2). Also, the work of Li liu et al. [12] is devoted to examining maximal regularity property of the weighted Hölder space C θ 0 ([0, T]; X) for the problem (1.2) by utilizing the concept of α-times resolvent families introduced in [13 In contrast to this above-mentioned work, for the first time we are going to prove that the classical real interpolation space D A (θ , p) can also be represented in terms of S α (t), T α (t), named as "solution operators" of fractional ACP. Guswanto, in [14], explicitly introduced these pair of operators on X, defined as follows: where ϒ ⊂ ρ(A), an anticlockwise oriented path. He also demonstrated the topological properties of these operators which are somewhat similar to those in classical case; see [15]. These similarities create a great advantage to study the existence, uniqueness and regularity properties of solution of (1.2) in the same fashion as of classical case. However, there are some disadvantages as these families do not satisfy semigroup properties and the operator T α (t) has singularity near t = 0. On the other hand, the study of strict solutions come into effect when one wants to get the differentiability of the solutions up to t = 0. In fact, it is known that the α-Caputo derivative of classical solutions of (1.2) cannot be extended up to t = 0, even under the conditions v 0 ∈ D(A), F ∈ C θ ([0, T]; X) (see, e.g., [16]). Thanks to interpolation spaces which play crucial role to fill this gap between classical solutions and strict solutions. It is evident that interpolation spaces can be expressed in terms of semigroup. However, no such expression in terms of S α (t), T α (t) is available in the literature so far. Motivated by this, we analyze the following points: (I) constructing interpolation spaces in terms of solution operators S α (t), T α (t) of fractional ACP and characterize these spaces, (II) establishing the strict Hölder regularity (or maximal Hölder regularity) results for the problem (1.2) utilizing this new representation of interpolation space. It is worth to mention that this new finding creates not only a great opportunity to study the strict regularity results of the problem (1.2) in the similar fashion as of classical case, but also provide a new sight in the area of fractional calculus.
This paper is organised as follows. In Sect. 2, we present preliminary results on some Hardy-type inequalities, a short introduction of real interpolation spaces and properties of the solution operators. In Sect. 3, we construct interpolation spaces in terms of solution operators and characterize these spaces, which is one of the goals of this paper. In the last section, we establish the sufficient conditions to investigate the strict Hölder regularity of mild solutions of (1.2), and provide an example to illustrate the results.

Some notations, preliminary results and interpolation spaces
Let I ⊂ R be an interval. Traditionally, for a compact interval I, C(I; X) represents the space of X-valued continuous functions on I endowed with the usual supremum norm. The Hölder space C θ (I; X) is defined by is known as Riemann-Liouville integral of order α ∈ (0, 1).
Then the Caputo derivative of order α ∈ (0, 1) is defined as
Before constructing interpolation spaces in terms of S α (t), T α (t), we draft a base by recalling the classical real interpolation spaces. For Banach spaces X and Y , L(X; Y ) stands for the set of all bounded linear operators from X to Y . For X = Y , L(X; X) is indicated as L(X).

Real interpolation spaces
In this subsection, we recall interpolation space for the case Y → X, though for general theory of interpolation spaces, we direct [20] for the interested reader. Let (X, . A classical method known as K -method to produce a class of real interpolation spaces is recalled now.
For x ∈ X, t > 0, set Let 0 < θ < 1. For 1 ≤ p ≤ ∞, define the following spaces: with the norm x (X,Y ) θ ,p = φ L p * (0,∞) . Then (X, Y ) θ,p are known as real interpolation spaces. One important thing is to note that K(t, x; X, Y ) ≤ x X as Y → X. Hence, to prove x ∈ (X, Y ) θ,p , it is adequate to prove that t → φ(t) ∈ L p * (0, a), and then x (X,Y ) θ ,p becomes equivalent to x X + φ L p * (0,a) , for any fixed a > 0.
A well-known result is that interpolation space (X, D(A)) θ,p can also be represented in terms of semigroup as follows: In such a case, it is written as E ∈ J θ (X, Y ).
In such case, it is written as For any θ ∈ (0, 1), the domain of fractional power of the sectorial operator, D(A θ ) is not an interpolation space but belongs to the class J θ (X, D(A)) ∩ K θ (X, D(A)). However, in a Hilbert space X, if A is densely defined positive self-adjoint operator, then D(A θ ) is an interpolation space (cf. [1]). Now, we recall the properties of the solution operators S α (t), T α (t) in fractional calculus in order to find the motivation behind constructing new interpolation spaces. ([14]) Let S α (t) be defined as (1.3). Then the following results hold.

Lemma 2.11
(i) S α (t) ∈ L(X) and (2.5) Remark 2.12 Since A is densely defined, closed operator, therefore (2.5) holds for all x ∈ X. Thus, the operator equation Lemma 2.13 ([14]) Let T α (t) be defined as (1.4). Then the following results hold. ∞); L(X)), and ∃K n = K n (α) > 0 such that Note that S α (t) is bounded near zero. Hence, by defining S α (0) = I, the identity operator on X and taking note of the property given in Lemma 2.11(iv), we can say that the family of bounded linear operators {S α (t)} t≥0 is continuous for t ≥ 0 in the strong operator topology. Lemma 2.14 Let A : D(A) ⊂ X → X be a densely defined sectorial operator and S α (t), T α (t) be the fractional solutions operator defined as in (1.3), (1.4), respectively. Then the following identity holds: Proof Consider the following representations of S α (t), T α (t) (see, [5, p. 213]): where M α is a probability density function having the property ∞ 0 s r M α (s) ds = (1+r) (1+αr) , r > -1.
Since t → T α (t) ∈ C ∞ ((0, ∞); L(X)), using the Lebesgue dominated convergence theorem, we have the following integral representation of T α (t): Finally, using the identity T (n) (t) = A n T(t), t > 0, n ∈ N, analyticity of t → S α (t) and the Lebesgue dominated convergence theorem, we get

Construction and characterization of the interpolation spaces in terms of S α (t), T α (t)
In this section, we introduce two classes of interpolation spaces in terms of solution operators. We show that these interpolation spaces are identical with the classical real interpolation space. Let 0 < θ < 1, 1 ≤ p ≤ ∞. We define the following classes: equipped with the norm T α y θ,p = y X + ψ α L p * (0,1) .
From now onward, we will use C as a positive constant appeared in any estimation instead of mentioning it precisely and the norm · in place of the norm · X on X.
and the respective norms defined on these spaces are equivalent.
Next we show the another formation of S α D A (θ , p) in the next result.

Strict regularity of mild solutions
This section is devoted to establishing strict Hölder regularity (i.e., the p = ∞ case) results using the interpolation space constructed in the last section. Before that we shall present the necessary and sufficient condition for a mild solution to be a strict solution of the problem (1.2) in the case 0 < α < 1. For the case 1 < α < 2, we refer to [21].
Consider the operator A :  We denote C θ b (R n ) as the space of all bounded and Hölder continuous functions endowed with the usual Hölder norm. We also define C 2+θ b (R n ) = {f : D β f exists, D β f ∈ C b (R n )∀multi-index β with |β| ≤ 2, and D β f ∈ C θ b (R n ) for |β| = 2}. By Theorem 3.  Therefore, u 0 ∈ C 2+2θ b (R n ) implies that u 0 ∈ D(A 0 ) and A 0 u 0 ∈ T α D A (θ , ∞).

Conclusion
We constructed two classes of interpolation spaces in terms of solution operators of fractional abstract Cauchy problem, and showed that these classes coincide with the real interpolation space. Moreover, it is found that these newly formulated interpolation spaces are useful to prove the strict Hölder regularity of a mild solution of a fractional ACP in the semigroup fashion as of the classical case.