Geodesic
Imbedding theorems for Banach spaces of infinitely differentiable functions
Sbornik. Mathematics, Tome 56 (1987) no. 1, pp. 63-78

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Algebraic conditions are obtained for imbedding of the spaces $$ W^{\infty}\{a_n,p,r\}_{(G)}=\biggl\{u(x)\in C^\infty(G):\sum_{n=0}^\infty a_n\|D^nu\| _r^p\infty\biggr\}, $$ where $G$ can be a closed interval, the line, a ray, or the circle. The imbedding conditions depend on the form of the domain. Bibliography: 15 titles. 
@article{SM_1987_56_1_a4,
     author = {G. S. Balashova},
     title = {Imbedding theorems for {Banach} spaces of infinitely differentiable functions},
     journal = {Sbornik. Mathematics},
     pages = {63--78},
     publisher = {mathdoc},
     volume = {56},
     number = {1},
     year = {1987},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1987_56_1_a4/}
}
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G. S. Balashova. Imbedding theorems for Banach spaces of infinitely differentiable functions. Sbornik. Mathematics, Tome 56 (1987) no. 1, pp. 63-78. http://geodesic.mathdoc.fr/item/SM_1987_56_1_a4/