CONCENTRATION COMPACTNESS

Book from 2007: Hilbert space theory and applications to Sobolev spaces, including Sobolev spaces with magnetic potential, periodic manifolds, and Lie groups

Corrections and additions (2007 book)

Book from 2020: Banach space theory and applications to Moser-Trudinger and Strichartz embeddings, Besov and Triebel-Lizorkin spaces, affine Sobolev inequalities, etc.

Corrections and additions (2020 book)

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Concentration compactness (which more rightfully should be called theory of cocompact imbeddings and profile decompositions) deals with convergence defined relative to "domesticating" actions of non-compact groups and other transformations in Banach spaces. See Terence Tao's blog with his view of concentration compactness.

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 L¹is 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.

In the more recent use any refinement of Sobolev imbedding in terms of Besov spaces may be called compensated compactness.