HoLDER CONTINUITY AND BOX DIMENSION FOR THE MIXED RIEMANN-LIOUVILLE FRACTIONAL INTEGRAL

In this paper, we investigate the Holder continuity and the estimate of the Box dimension of R(alpha 1,alpha 2)f (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 mu th-order Ho spexpressioncing diexpressioneresis lder continuous, where mu is an element of (0, 1). By using the approximated integral, we obtain that, R alpha 1,alpha 2f (x, y) is alpha 1-mu th-order Ho spexpressioncing diexpressioneresis lder continuous, and the Box dimension of the graph of R alpha(1),alpha(2)f (x, y) is less than or equal to 3- (alpha) /(1-mu), provided that alpha + mu < 1. Here alpha = min{alpha 1, alpha 2}. By using Stein's Lemma, we prove that R alpha(1),alpha(2)f (x, y) is (alpha + mu)th-order Ho spexpressioncing diexpressioneresis lder continuous, provided that alpha + mu < 1, and the Box dimension of the graph of R alpha(1),alpha(2)f (x, y) is less than or equal to 3 - alpha - mu. Moreover, we also illustrate that the latter conclusion is sharp.