Nontriviality of Sobolev spaces of infinite order for a full Euclidean space and a torus

In this paper we obtain necessary and sufficient conditions for nontriviality of the spaces
$$ W^\infty\{a_\alpha,p_\alpha\}\equiv\biggl\{u(x):\rho(u)\equiv\sum_{|\alpha|=0}^\infty a_\alpha\|D^\alpha u\|^{p_\alpha}_{L_{r_\alpha}}<\infty\biggr\} $$
in the cases where the functions $u(x)$ are defined on $\mathbf R^n$ or are periodic with period $2\pi$ with respect to $n$ variables.
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