On Gaussian Nikolskii–Besov classes

In this note we study Nikolskii–Besov classes of functions of fractional smoothness on finitedimensional and infinite-dimensional spaces with Gaussian measures. We prove the equivalence of two characterizations of these classes: one is based on a certain nonlinear integration by parts formula and the other one is given in terms of the Ornstein–Uhlenbeck semigroup. In addition, we obtain a new Poincaré-type inequality. The case of Lebesgue measure has been considered in [1] (see also [2, 3]).