Embedding theorems for Banach spaces of infinitely differentiable functions of several variables (Q1813392)

Sobolev spaces of infinite order, ''Dubinskij spaces'' [see \textit{J. A. Dubinskij}, Sobolev spaces of infinite order and differential equations, Leipzig (1986: Zbl 0616.46028)]are considered: \[ W^ \infty\{a_ \alpha,p,r\}_{(G)}=\{u(x)\in C^ \infty(G);\;\rho(u)=\sum_{|\alpha|=0}^ \infty a_ \alpha\| D^ \alpha u\|_ r^ p<\infty\}; \] \(\alpha=(\alpha_ 1,\dots,\alpha_ n)\), \(a_ \alpha\geq0\), \(p,r\geq1\), \(\|\cdot\|_ r=\|\cdot\|_{L_ r(G)}\). In one of her previous papers, the authoress has established simple sufficient conditions which guarantee the embedding, or compact embedding, respectively, \[ W^ \infty\{a_ \alpha,p,r\}_{(G)}\subset W^ \infty\{b_ \alpha,p,r\}_{(G)} \] [Mat. Sb., N. Ser. 128(170), No. 1, 61-81 (1985; Zbl 0615.46029)]. The aim of the present text is to construct such sequences \(\{a_ \alpha'\}\), for which the equivalence \[ W^ \infty\{a_ \alpha,p,r,\}_{(G)}=W^ \infty\{a_ \alpha',p,r\}_{(G)} \] holds; by this way one can extend the class of cases in which the embedding theorems mentioned above are applicable. This problem is solved in the case \(G\) is the \(n\)-dimensional torus; this means, the functions \(u(x)\) are \(2\pi\)-periodical with respect to all variables, and \(G=(-\pi,\pi)^ n\). To this end, two different ways are used: convex regularisation of sequences, or representation of functions via Fourier series. Some examples of applications are given, too. As an auxiliary result, the author establishes some estimates of \(L_ 2\) norm, or \(L_ p\) norm, respectively, of mixed derivatives \(D^ \alpha u\) with the help of ''non-mixed'' ones.