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

Long Tian ()
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Long Tian: School of Mathematics and Statistics, Nanjing University of Science and Technology, Nanjing 210094, P. R. China

FRACTALS (fractals), 2023, vol. 31, issue 01, 1-13

Abstract: In this paper, we investigate the Hölder continuity and the estimate of the Box dimension of Rα1,α2f(x,y), which is called the mixed Riemann–Liouville fractional integral of the continuous function f(x,y). We focus on the case that f(x,y) is μth-order Hölder continuous, where μ ∈ (0, 1). By using the approximated integral, we obtain that, Rα1,α2f(x,y) is α 1−μth-order Hölder continuous, and the Box dimension of the graph of Rα1,α2f(x,y) is less than or equal to 3 − α 1−μ, provided that α + μ < 1. Here α =min{α1,α2}. By using Stein’s Lemma, we prove that Rα1,α2f(x,y) is (α + μ)th-order Hölder continuous, provided that α + μ < 1, and the Box dimension of the graph of Rα1,α2f(x,y) is less than or equal to 3 − α − μ. Moreover, we also illustrate that the latter conclusion is sharp.

Keywords: Mixed Riemann–Liouville Fractional Integral; Hölder Continuity; Box Dimension (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1142/S0218348X23500044

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