The Intermediate Case of Regularity in the Problem of Differentiation of Multiple Integrals

The paper deals with generalization of Lebesgue and Jessen–Marcinkiewicz–Zygmund theorems of the differentiation of multiple integrals for the intermediate case of regularity of the system of sets. The application of the result to the Fourier–Haar series and to orthorecursive expansions with respect to system of indicators of multi-dimensional intervals is considered.