HÖLDER CONTINUITY AND BOX DIMENSION FOR THE MIXED RIEMANN–LIOUVILLE FRACTIONAL INTEGRAL

Abstract

In this paper, we investigate the Hölder continuity and the estimate of the Box dimension of [Formula: see text], which is called the mixed Riemann–Liouville fractional integral of the continuous function [Formula: see text]. We focus on the case that [Formula: see text] is [Formula: see text]th-order Hölder continuous, where [Formula: see text]. By using the approximated integral, we obtain that, [Formula: see text] is [Formula: see text]th-order Hölder continuous, and the Box dimension of the graph of [Formula: see text] is less than or equal to [Formula: see text], provided that [Formula: see text]. Here [Formula: see text]. By using Stein’s Lemma, we prove that [Formula: see text] is [Formula: see text]th-order Hölder continuous, provided that [Formula: see text], and the Box dimension of the graph of [Formula: see text] is less than or equal to [Formula: see text]. Moreover, we also illustrate that the latter conclusion is sharp.