Differential properties of the functions $f \in V_{p, \theta}^{<m+\alpha ; N>}(G ; s)$

The first integral representation of the functions of many variables defined in the regions (star regions relative to the points of a given ball) $G \subset E_{n}$ belongs to academician S.L. Sobolev. S.L.Sobolev developed a method of integral representations of functions from well-known functional spaces constructed by him $W_{p}^{r}(G)$ and proved the basic theorems of embedding these spaces with further applications to the theory of partial differential equations.
Further development of the method of integral representations of the theory of spaces of differentiable functions of many variables is associated with the name of V.P. Ilyin. He proved a fundamentally new integral representation of functions of many variables at any point $x \in E_{n}$.
The study investigates the "weight" spaces of functions $f=f(x)$, points of $x=\left(x_{1}, \ldots, x_{s}\right) \in E_{n}(1 \leq s \leq n)$ many bundles of variables $x_{k}=\left(x_{k, 1}, \ldots, x_{k, n_{k}}\right) \in E_{n_{k}} \quad(k=1,2, \ldots, s) \quad$ defined in the domain $G \subset E_{n}=E_{n_{1}} \times \cdots \times E_{n_{s}}\left(n=n_{1}+\cdots+n_{s}\right)$ satisfying the condition "variable $\Psi(x, h)$-semihorn". These constructed "weight" spaces of the type of generalized $B$-spaces depend on the parameter $s\left(1 \leq s \leq n=n_{1}+\cdots+n_{s}\right)$, which in the case $s=1$ generalize the known "weight" spaces $B_{p}^{\eta, . ., r_{n}}\left(G, \rho^{\alpha}\right)-\mathrm{O} . \mathrm{V} .$ Besov, and in the case $s=1$, generalize the known spaces of $S_{p, 0}^{r} B\left(G, \rho^{\alpha}\right)$ functions with a dominant shifted derivative, in the case of power "weights" given in the works of A.J.Dzhabrailov. A.D.Dzhabrailov proved new integral representations of functions of many variables, with the help of which he managed to build a general theory of spaces of functions with a dominant mixed derivative $S_{p}^{r} W(G)$ and $S_{p, \theta}^{r} B(G)$, with further development of the method of integral representations in the theory of the embedding theorem of these spaces.
A new functional space is also constructed by the method of integral representations [1] based on a new integral representation of smooth functions at points $x \in E_{n}$.