The numbers and of a function are defined by
In order to answer the question whether it is possible to derive results on mapping properties of operators of fractional integration in generalized Hölder spaces in terms of the direct numerical interval for the exponents of the weight, with boundaries depending on the indices and , the authors establish the following main theorem, which provides the equivalence of the integral Zygmund conditions to certain direct numerical inequalities for the aforesaid indices.
Theorem. A function belongs to , , iff and it belongs to , , iff , so that in the case , there holds the formula and for and for any there exist constants and such that ,
Based on this theorem, the authors further establish a series of new theorems on the action of fractional integrals in the generalized Hölder spaces for various types of weights including one- dimensional and multi-dimensional cases.