Mixed fractional integration operators

We study mixed Riemann ¿ Liuoville integrals of functions of two variables in Hölder spaces of different orders in each variables. We consider Hölder spaces defined both by first order differences in each variable and also by the mixed second order difference, the main interest being in the evaluation of the latter for the mixed fractional integral in both the cases where the density of the integral belongs to the Hölder class defined by usual or mixed differences. The obtained results extend the well known theorem of Hardy ¿ Littlewood for one ¿ dimensional fractional integrals to the case of mixed Hölderness. We cover also the weighted case with power weights.