A characterization of Nikolskii–Besov classes via integration by parts

In this note we give a characterization of Nikolskii–Besov classes of functions of fractional smoothness (see [1–3]) by means of a nonlinear integration by parts formula in the form of a certain nonlinear inequality. This characterization is motivated by the recent papers [4–6] on distributions of polynomials in Gaussian random variables, where it has been shown that the distribution densities of nonconstant polynomials in Gaussian random variables belong to Nikolskii–Besov classes. Our main result is a generalization of the classical description of the class BV of functions of bounded variation in terms of integration by parts.