Embedding theorems for subspaces in spaces of fast decaying functions

By means of the family ${\mathfrak M} = \{{M_{\nu}}\}_{\nu=1}^{\infty}$ of separately radial convex functions $M_{\nu}: {\mathbb{R}}^n \to {\mathbb{R}}$ we define the space $GS({\mathfrak M})$ of type $W_M$, which is a natural generalization of the space $W_M$ introduced in works by B.L. Gurevich, I.M. Gelfand, and G.E. Shilov. By a certain rule, each function $M_{\nu}$ is associated with a non–negative separately radial convex function $h_{\nu}$ in ${\mathbb{R}}^n$. The properties of the functions $h_{\nu}$ allows one to form, by the family ${\mathcal H} = \{{h_{\nu}}\}_{\nu=1}^{\infty}$, the space ${\mathbb S}_{\mathcal H}$, which is the inner inductive limit of countably–normed spaces ${\mathbb S}(h_{\nu})$ of the functions $f \in C^{\infty}({\mathbb{R}}^n)$ with the finite norms
$$\| f \|_{m, \nu} = \sup_{x \in {\mathbb{R}}^n, \beta \in {\mathbb{Z}}_+^n, \atop \alpha \in {\mathbb{Z}}_+^n: \| \alpha \| \le m} \frac {\| x^{\beta}(D^{\alpha}f)(x) \|}{\beta! e^{-h_{\nu}(\beta)}}, \qquad m \in {\mathbb{Z}}_+ .$$

We consider the problem on finding conditions on ${\mathfrak M}$, which ensure continuous embedding of the spaces $GS({\mathfrak M})$ and ${\mathbb S}_{\mathcal H}$ one to the other.