The Estimates of HÖLDER Index and the Box Dimension for the Hadamard Fractional Integral
Abstract
This paper focuses on the Hölder continuity and the Box Dimension to the pth Hadamard Fractional Integral (HFI) on a given interval [α,β]. We use Hpϕ(x) to denote it. In this paper, two different methods are used to study this problem. By using the approximation method, we obtain that for ϕ(x) ∈ℋq([α,β]) with p ∈ (0, 1) and q ∈ (0, 1 − p), if α > 0, then Hpϕ(x) is γth Hölder continuous in (α,β] with γ = p (1−q)(1+p)−p2, and is p 1+pth Hölder continuous on [α,β]. Moreover, the Box Dimension of the graph of Hpϕ(x) on the interval [α,β] is less than or equal to 2 − (1+2p)γ (1+p)γ+(1+2p). If α = 0, then Hpϕ(x) is locally γth Hölder continuous in (0,β] with the same γ, and the Box Dimension of Hpϕ(x) on [0,β] is less than or equal to 2 − γ 1+γ. By using another method, we imply that, for ϕ(x) ∈ℋq([α,β]) with q ∈ (0, 1 − p) and 0 < p < 1, if α > 0, then Hpϕ(x) is (p + q)th Hölder continuous, and thus the Box Dimension of the graph of Hpϕ(x) is no more than 2 − (p + q); if α = 0, then Hpϕ(x) is locally (p + q)th Hölder continuous in (0,β], and is qth Hölder continuous at 0. Then the Box Dimension of the graph to Hpϕ(x) on [0,β] is less than or equal to 2 −(1+q)(p+q) 1+2q+p. We also give two examples to show that the above Hölder indexes given by the second method are optimal.
 Publication:

Fractals
 Pub Date:
 2021
 DOI:
 10.1142/S0218348X21500729
 Bibcode:
 2021Fract..2950072T
 Keywords:

 The Hadamard Fractional Integral;
 The Hölder Index;
 The Box Dimension