A Note on Function Space and Boundedness of the General Fractional Integral in Continuous Time Random Walk

The general fractional calculus becomes popular in continuous time random walk recently. However, the boundedness condition of the general fractional integral is one of the fundamental problems. It wasn’t given yet. In this short communication, the classical norm space is used, and a general boundedness theorem is presented. Finally, various long–tailed waiting time probability density functions are suggested in continuous time random walk since the general fractional integral is well defined.


Introduction
A fractional integral was proposed with general kernels in [1][2][3] I ,g a + f (t)∶= ∫ t a 1 Γ( ) (g(t) − g(s)) −1 g � (s)f (s)ds, 1 3 where is a positive real number, g(t) is strictly increasing function on [a, b] and g(a) ≥ 0 . Due to the new features in comparison with the standard fractional derivatives, much attention has been paid to theoretical research and applications, for example, fractional calculus of variations [4], the Laplace transform [5,6], exact solution [7,8] and numerical methods [9]. Motivated by the continuous time random walk understood by means of the standard fractional calculus [10,11], suppose a long-tailed waiting time probability density function [12] which is more general than the power law function Then, one comes across a general time-fractional Fokker-Planck equation with the general fractional integral where T is the temperature, P(x, t) is the probability density function, K is the diffusion coefficient, F(x) represents external force field and k b represents the Boltzmann constant. So the physical meaning of the general fractional derivative or the kernel function g(t) was provided. More details were shown in [12]. However, one of the fundamental problems is still not addressed yet. That is, what is the function space of the general fractional integral and what are the constraint conditions with respect to g(t)?
A norm of the general fractional integral was recently given (see [13]). However, the condition c = 1 p and g(t) = t is too strong there. For a general function g(t) ≠ t , it cannot be reduced to the Lebesgue space L p [a, b] (1 ≤ p) . In order to address this problem and improve the existing results, we reconsider the classical norm used by Kilbas et al for the Riemann-Liouville integral [3]. We present the boundedness of the general fractional integral operator in X p c (a, b) and we give the constraint conditions of g(t).

General Fractional Integral
Let us revisit the general fractional integral in X p c (a, b) with the norm. The following proves that the norm defined in X p c (a, b) satisfies three axioms.

Lemma 2.2 For arbitrary f(t) and h(t)
The trigonometric inequality is true. In fact, from Minkowski inequality ‖f + h‖ L p ≤ ‖f ‖ L p + ‖h‖ L p , we have Moreover, since f(t) and h(t) ∈ X p c (a, b) we know that ‖f ‖ X p c < ∞ and ‖h‖ X p c < ∞ . Therefore, from the above formula, we derive that t integrable. Thus, the inequality (6) holds and For p = ∞ , the norm (5) also can be easily verified. ◻ The general fractional integral can be derived by the n-fold integral method from [3]. Suppose n ∈ ℕ = {1, 2, ⋯} and g(t) ∈ C 1 [a, b] is a strictly increasing function and g(a) ≥ 0 for t ∈ [a, b] , the n-fold integral reads As a result, for any positive real number , we give the definition general fractional integral in X p c (a, b) space as follows.
For some specific functions, we obtain some well known definitions of fractional integrals, for example, the Riemann-Liouville integral [3] when g(t) = t , the Hadamard integral when g(t) = ln t [3] and the generalized integral [14] when g(t) = t +1 , the fractional integral with exponential memory [15,16] when The general fractional integral operator I ,g a + is bounded in the space X p c (a, b) (1 ≤ p ≤ ∞) . We assume a > 0 and g(a) > 0 for simplicity in the sequel. (4) and (8), we have Since f (t) ∈ X p c (a, b) and g(t) ∈ C 1 [a, b] is a strictly increasing function, then

Theorem 2.4 (Boundedness theorem)
. Hence, we can apply the general Minkowski inequality [2] and give It follows from that g −1 (ug(s)) ≤ F(u)s , d(g −1 (ug(s))) ds ≤ F(u) , u ≥ 1 and c ≥ 1 p that and Due to F(u) ∈ C [1, g(b) g(a) ] , we note that K = ∫ As a result, this completes the proof. ◻ . It can be derived from Theorem 2.4.

Corollary 2.5 Let
, then the general fractional integral is bounded, that is where the constant K is defined by (9).

Conclusion
We can verify that Theorem 2.4 can hold for several known fractional integrals: • when g(s) = s (The Riemann-Liouville integral [3]), we can find a function F(u) = u such that g −1 (ug(s)) ≤ F(u)s and d(g −1 (ug(s))) ds ≤ F(u) , hence K is defined by (9) where F(u) = u. • when g(s) = ln s (s > 1) (The Hadamard integral [3]), we also can choose ln a b u−1 such that those two conditions hold, hence K is given where • when g(s) = s +1 +1 ( > −1) (The generalized fractional integral [14]), we can use F(u) = u 1 +1 and two constraint conditions with respect to F(u) and g(s) hold.
• when g(s) = e s ( ≥ 1 a ) (The fractional integral with exponential memory [16]), we can take F(u) = u to satisfy the conditions of Theorem 2.4.
As a result, we can conclude that the function space X p c and the general fractional integral are well-defined when we use the classical norm (Definition 2.1). The above g(t) with constraints conditions can be employed as long-tailed waiting time probability density function. On the other hand, they are well-defined in the general fractional calculus.

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