Concentration analysis in Banach spaces

2015/02/02 by Sergio Solimini, Solimini, Sergio, Cyril Tintarev +1
Mathematics · #46B10 #46B20 #46B50 #46B99 #FOS: Mathematics #Functional Analysis (math.FA) #math.FA #msc:46B10 #msc:46B20 #msc:46B50 #msc:46B99

paper · pdf · doi:10.48550/arxiv.1502.00414

arxiv created 2015/02/02 · arxiv updated 2015/02/03

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 Δ-convergence by T. C. Lim 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 ℓ^p-spaces, but not in L^p(\mathbb R^N), p≠2. Δ-convergence appears naturally in the context of fixed point theory for non-expansive maps. The paper also studies connection of Δ-convergence with Brezis-Lieb Lemma and gives a version of the latter without an assumption of convergence a.e.

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