Concentration analysis in Banach spaces

Abstract

The concept of a profile decomposition formalizes concentration compactness arguments on the functional-analytic level, providing a powerful refinement of the Banach–Alaoglu weak-star compactness theorem. We prove existence of profile decompositions for general bounded sequences in uniformly convex Banach spaces equipped with a group of bijective isometries, thus generalizing analogous results previously obtained for Sobolev spaces and for Hilbert spaces. Profile decompositions in uniformly convex Banach spaces are based on the notion of [Formula: see text]-convergence by Lim [Remarks on some fixed point theorems, Proc. Amer. Math. Soc. 60 (1976) 179–182] instead of weak convergence, and the two modes coincide if and only if the norm satisfies the well-known Opial condition, in particular, in Hilbert spaces and [Formula: see text]-spaces, but not in [Formula: see text], [Formula: see text]. [Formula: see text]-convergence appears naturally in the context of fixed point theory for non-expansive maps. The paper also studies the connection of [Formula: see text]-convergence with the Brezis–Lieb lemma and gives a version of the latter without an assumption of convergence a.e.