KATUGAMPOLA FRACTIONAL INTEGRAL AND FRACTAL DIMENSION OF BIVARIATE FUNCTIONS

The paper proves that for 0 < a < b, 0 < c < d, −1 < p, q ≤ 0, and 0 < α, β < 1, the upper box dimension of the graph of the mixed left Katugampola fractional integral of a continuous f is at most 3 − min{α, β} (Theorem 3.10) . Its proof decomposes the increment into four terms (J1–J4), estimates each via changes of variables to obtain a Hölder-type bound |ΔF| ≤ C(h^α + k^β), and then invokes a general Hölder-to-dimension result (Corollary 2.7) to conclude dim_B ≤ 3 − min{α, β} . The candidate solution derives separate Hölder estimates in x and y by explicitly integrating the kernel via u = x^{p+1} − s^{p+1} (and analogously in y), combines them to a joint Hölder estimate with exponent γ = min{α, β}, and then applies a standard covering argument for Hölder graphs to deduce the same dimension bound. This matches the paper’s strategy in substance (establish Hölder continuity, then apply a Hölder-graph dimension bound), with minor stylistic differences (direct covering argument vs. citing Corollary 2.7).