Geodesic
A Note on Fourier Transforms and Imbedding Theorems
Canadian mathematical bulletin, Tome 10 (1967) no. 5, pp. 695-698

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It is well known that Sobolev′s Lemma on the continuity of functions possessing L2 distributional derivatives of sufficiently high order is a simple consequence of elementary properties of the Fourier transform in L2 (e.g. [1, p. 174]). (In fact this statement remains true if 2 is replaced by p, 1 ≤ p ≤ 2). In this note we show that imbedding theorems of the type Wm, p ⊂Lq can also be obtained using Fourier transforms and an elementary lemma which reduces the cases p > 2 to the case p = 2. The simplicity of this approach is obtained at the expense of a slight loss of generality in the imbedding theorem. 
Adams, Robert A. A Note on Fourier Transforms and Imbedding Theorems. Canadian mathematical bulletin, Tome 10 (1967) no. 5, pp. 695-698. doi: 10.4153/CMB-1967-071-7
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     author = {Adams, Robert A.},
     title = {A {Note} on {Fourier} {Transforms} and {Imbedding} {Theorems}},
     journal = {Canadian mathematical bulletin},
     pages = {695--698},
     year = {1967},
     volume = {10},
     number = {5},
     doi = {10.4153/CMB-1967-071-7},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1967-071-7/}
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