# Bivariate functions of bounded variation: Fractal dimension and fractional integral

## TL;DR

This paper investigates the fractal dimensions of graphs of bivariate functions with bounded variation and explores how fractional integrals affect these dimensions, extending classical concepts to multivariate and fractional contexts.

## Contribution

It introduces a study of Hausdorff and box dimensions for bivariate functions of bounded variation and analyzes the impact of Riemann-Liouville fractional integrals on these dimensions.

## Key findings

- Hausdorff and box dimensions of bivariate functions of bounded variation are characterized.
- Fractional integrals of such functions preserve bounded variation in the Arzela sense.
- Dimensions of the graphs of fractional integrals are explicitly determined.

## 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 Arzela 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.