Some imbedding theorems for the function spaces – $\mathscr L_{r,p,\theta}^{\lambda,\varphi,b_s}(G)$

One considers imbedding type theorems for the spaces – $\mathscr L_{r,p,\theta}^{\lambda,\varphi,b_s}(G)$ of real functions, defined on a domain $G$ of the $n$-dimensional Euclidean space $E^n$. As opposed to the known spaces of this type, the power function $t^a$, characterizing the degree of the smoothness of the functions, is replaced here by a function $\varphi(t)$, arbitrary in a certain sense.