Let \(f: [a,b]\times[c,d]\to\mathbb{R}\), where \(0\leq a < b\) and \(0\leq c < d\) be a function. If the integral \[ \mathcal{I}^\gamma f(x,y) := \frac{1}{\Gamma(\gamma_1)\Gamma(\gamma_2)}\,\int_a^x \,\int_c^y (x-u)^{\gamma_1-1}(y-v)^{\gamma_2-1} f(u,v) du dv, \] where \(\gamma :=(\gamma_1, \gamma_2)\) with \(\gamma_1,\gamma_2 >0\), exists, then it is termed the mixed Riemann-Liouville fractional integral of \(f\).
The authors provide estimates for the box and Hausdorff dimension of the graphs of mixed Riemann-Liousville fractional integrals for various choices of functions \(f\). Moreover, given that the function \(f\) has box dimension 2, the box dimension of \(\mathcal{I}^\gamma f\) is estimated.
Unbounded variational functions are constructed on \([0,1]\times [0,1]\) and it is proved that the graphs of their mixed Riemann-Liouville fractional integrals have Hausdorff and box dimension equal to 2 for \(0< \gamma_1, \gamma_2 < 1\).
Some graphs of mixed Riemann-Liouville fractional integrals are also provided.