THE MIXED FRACTIONAL DIFFERENTIAL OPERATORS IN HÖLDER SPACES

We study mixed fractional differentiation operators in the form Marchaud of functions 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 fractional differentiation operators in the form Marchaud 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 differentials to the case of mixed Hölderness.