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 G\'erard for the embedding of the homogeneous
Sobolev space X=H˙s into the Y=Lp in d dimensions with
0<s=d/2−d/p, and then generalized by Jaffard 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, H\"older
and BMO spaces.Comment: 24 page