Analysis of the Lack of Compactness in the Critical Sobolev Embeddings

Let (un) be a bounded sequence inHs, p(Rd) (0<s<d/p). We show that (un) has a subsequence (u′n) such that eachu′ncan be expressed as a finite sum (plus a remainder) of translations/dilations of functionsφmand such that the remainder has arbitrary small norm inLq(1/q=(1/p)−(s/d)). This generalizes a result obtained by Patrick Gérard for the casep=2.