CONCENTRATION COMPACTNESS
Functional-Analytic Grounds and Applications
Kyril Tintarev & Karl-Heinz Fieseler
Imperial College Press, 2007
Concentration compactness (which more rightfully should be called theory of cocompact imbeddings) deals with convergence defined relative to "domesticating" actions of non-compact groups. See Terence Tao's blog with his view of concentration compactness.
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Disambiguation: Concentration Compactness is not Compensated Compactness!
Both terms emerged almost at the same time and both are connected to
convergence issues, but otherwise they have very little in common.
Compensated compactness is basically a collection of results verifying
that a bounded subset of L1 is in fact a subset of a the Hardy space H1. See the definitive article on the matter:
Coifman, R.;
Lions, P.-L.;
Meyer, Y.;
Semmes, S. Compensated compactness and Hardy spaces.
J. Math. Pures Appl. (9) 72 (1993), no. 3, 247--286.
Im the more recent use any refinement of Sobolev imbedding in terms of Besov spaces may be called compensated compactness.