Bivariate functions of bounded variation: Fractal dimension and fractional integral

Abstract

In contrast to the univariate case, several definitions are available for the notion of bounded variation for a bivariate function. This article is an attempt to study the Hausdorff dimension and box dimension of the graph of a continuous function defined on a rectangular region in R2, which is of bounded variation according to some of these approaches. We show also that the Riemann–Liouville fractional integral of a function of bounded variation in the sense of Arzelá is of bounded variation in the same sense. Further, we deduce the Hausdorff dimension and box dimension of the graph of the fractional integral of a bivariate continuous function of bounded variation in the sense of Arzelá.