Интерполяционные те оремы для классов фун кций с доминирующей смеша нной производной

If a function belongs to two functional spaces with a dominating mixed derivative, then it also belongs to the intermediate spaces (in the sense of the order of differentiation and the integrability exponent). An interpolation theorem is proved for the operators on such spaces. A linear operator is considered which is bounded on each of the two periodic functional spaces with a dominating mixed derivative. Boundedness of the operator on the intermediate functional spces is proved and the corresponding estimates of the norms of the operator are deduced.