Characterizations of $B^s_{p,q}(G),L^s_{p,q}(G),W^s_p(G)$ and certain other function spaces. Applications

The anisotropic Sobolev, Nikol'skii, Besov, and Lizorkin–Triebel spaces are considered on irregular domains (in particular, on open sets) as well as closely related spaces defined by the best local polynomial approximation in different metrics. The questions of the equivalence of norms, dependence of the space structure on the defining parameters, boundedness of pointwise multipliers, description of maximal subalgebras, and also the questions concerning the reflexivity, translation continuity, and comleteness of these spaces are studied.