Representations and properties of a new family of ω -Caputo fractional derivatives

In the most general case of ω -weights, some normed functional spaces X ω(a, b)(1 ≤ p ≤ ∞) , AC γ,ω[a, b] and a generalization of the fractional integro-differentiation operator are introduced and analyzed. The boundedness of the ω -weighted fractional operator over X ω(a, b) is proved. Some theorems and lemmas on the properties of the invertions of the mentioned operator and several representations of functions from AC γ,ω[a, b] are established. A general ω -weighted Caputo fractional derivative of order α is studied over AC γ,ω[a, b] . Some representations and other properties of this fractional derivative are proved. Some conclusions are presented.


Introduction
Recently, many researchers are dealing with some well-known equations by means of some generalized fractional integro-differential operators, see, e.g., [5,8,10,15]. These kind of extensions imply several expected and unexpected properties of the solutions of the considered equation, we refer to [3,6,18]. Besides, in many papers some special classes of functions have been introduced to apply the apparatus of the fractional integrodifferential equations in different fields of knowledge: engineering, physics, chemistry, mathematics, etc., see, e.g., [4,12,17]. This is the base for studying in this paper more general fractional derivatives. These derivatives will be considered for functions belonging to several spaces AC n γ,ω [a, b] , which are some subsets of the set AC[a, b] of absolutely continuous functions on [a, b] , see, e.g., [13,18]. Notice that that the space AC [a, b] coincides with the space of primitives of Lebesgue summable functions, see, e.g., [18, p. 3], then absolutely continuous functions have a summable derivative f ′ (z) almost everywhere. The converse is no true. Indeed, this is one of the most important facts of the theory of fractional calculus to define fractional integro-differentiation operators with good enough representations and properties. In general terms, this paper gives some ω -weighted extensions of the results in [18, Chapter 1. Sec. 2] and [12,13]. Furthermore, we give some other new results that appear naturally while it was proving the ω -extension.
The paper is organized as follows: In Section 2 we give some definitions on the fractional integrodifferentiation operators. Section 3 is devoted to the representation of any function f ∈ AC n γ,ω [a, b] [a, b]. Some theorems on inverse properties of the ω -Caputo fractional derivatives and fractional integrals are established. In Section 5 we give a boundedness result on ω -weighted fractional integral in a weighted space X p ω (a, b) , while in Section 6 we finish the paper with some conclusions. The goal of this paper is to present and study some properties of a Caputo fractional derivative with respect to another function. With this idea, we generalize some previous works dealing with the Caputo fractional derivatives. We study the main properties of this operator.

ω -weighted fractional integrals and derivatives
Below we recall some classical definitions on fractional integration and fractional derivative of a function.
Moreover, other kind of fractional integrals and derivatives of a function f with respect to another function h have been defined and studied in different articles and books, see, e.g., [3,15,18]. In this article, we will consider the same form of fractional integro-differentiation operators, but with respect to a weight ω that turns into those in [3,7,15,17,18] for some particular weights. The consideration of these weights ω will bring out more expected and unknown results. These kind of generalized fractional integrals and derivatives are partially studied in more general settings in [1,3,12,16,17] and the references therein.
From now on, we assume that Ω is the class of those absolutely continuous functions ω(x) on (a, b) , such that Besides, N is the set of natural numbers, C is the set of complex numbers and ⌈x⌉ is the ceiling function.
Below we introduce the ω -weighted fractional integro-differentiation operators and give some examples of them.

Definition 2.2
If ω ∈ Ω , α ∈ C (Re α > 0) and f ∈ L(a, b), we define the left-sided (right-sided) fractional integrals of order α and the left-sided (right-sided) fractional derivatives of order α of a function f with respect to a weight ω as: 3)

4)
and setting γ = 1 Notice that in the above definition we are not taking α ∈ C with Re α ∈ N , since n = ⌈Re α⌉ = Re α . Hence, below we always distinguish both cases just setting α ∈ N or n = ⌈Re α⌉ .
Semigroup and commutative properties hold for Re α, Re β > 0: For proving the above properties is just necessary to take into account the change of integration region and Beta's function representation.
Now some examples. First, we recall the Pochhammer k -symbol [9].

Definition 2.3
Let x ∈ C , k ∈ R and n ∈ N + , the Pochhammer k -symbol is given by

Preliminary results
We begin this section introducing the spaces of functions that will be considered through this paper.

Definition 3.1 We define
where ω ∈ Ω and n ∈ N.
It is easy to see that under some suitable weights ω , the above space coincide with those define and treat in [12,13,15,18].
Below we give a characterization of any function over the space AC n γ,ω [a, b].

Theorem 3.2 If ω ∈ Ω, then any function f ∈ AC n γ,ω [a, b] if and only if f can be represented as
.
where A 0 = (γ n−1 f )(a) . Multiplying both sides of (3.1) by ω ′ (x) and integrating over (a, x) , we obtain where A 1 = (γ n−2 f )(a) . Again, multiplying both sides of the above equation by ω ′ (x) and integrating over (a, x), we get If the same process is repeated n − 3 times, we arrive at  L 1 (a, b).
Proof By the representation of Theorem 3.2 we can write I α a,ω f as follows By the semigroup properties of I α a,ω and the calculation of the second fractional integral above by means of the substitution u = ω(t)−ω(a) ω(x)−ω(a) and taking into account that 1 − u = ω(x)−ω(t) ω(x)−ω(a) and the properties of the Beta's function we obtain Under the last representation it remains to prove that γ n−1 I α a,ω f (x) ∈ AC [a, b]. We now calculate the first term of (3.2). Indeed, by the Leibniz's rule for differentiation under the integral sign we have Notice that [a, b]). This implies that the first term of (3.2) belongs to AC [a, b] . On the other hand, for the second term of (3.2) we just need to estimate γ n−1 ω(x) − ω(a) α+n−k−1 for k = 0, . . . , n − 1 since the other terms are constants. Indeed,

Main results
We begin this section showing the representations of the fractional derivative D α a,ω f (x) and D α ω,b f (x) for any function in the space AC n γ,ω [a, b] .