A Note on Fourier Transforms and Imbedding Theorems

Extract

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 W^{m, p} ⊂L^q 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.