A general wavelet-based profile decomposition in the critical embedding of function spaces

We characterize the lack of compactness in the critical embedding of functions spaces X ⊂ Y having similar scaling properties in the following terms: a sequence (un)n≥0 bounded in X has a subsequence that can be expressed as a finite sum of translations and dilations of functions (ϕl)l>0 such that the remainder converges to zero in Y as the number of functions in the sum and n tend to +∞. Such a decomposition was established by Gerard in [13] for the embedding of the homogeneous Sobolev space X = Ḣs into the Y = Lp in d dimensions with 0 < s = d/2 - d/p, and then generalized by Jaffard in [15] to the case where X is a Riesz potential space, using wavelet expansions. In this paper, we revisit the wavelet-based profile decomposition, in order to treat a larger range of examples of critical embedding in a hopefully simplified way. In particular, we identify two generic properties on the spaces X and Y that are of key use in building the profile decomposition. These properties may then easily be checked for typical choices of X and Y satisfying critical embedding properties. These includes Sobolev, Besov, Triebel-Lizorkin, Lorentz, Holder and BMO spaces.