Concentration analysis in Banach spaces
2016 (English)In: Communications in Contemporary Mathematics, ISSN 0219-1997, Vol. 18, no 3, 1550038Article in journal (Refereed) PublishedText
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 Delta-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 l(p)-spaces, but not in L-p(R-N), p not equal 2 Delta-convergence appears naturally in the context of fixed point theory for non-expansive maps. The paper also studies the connection of Delta-convergence with the Brezis-Lieb lemma and gives a version of the latter without an assumption of convergence a.e.
Place, publisher, year, edition, pages
2016. Vol. 18, no 3, 1550038
Weak topology, Delta-convergence, Banach spaces, concentration compactness, cocompact imbeddings, profile decompositions, Brezis-Lieb lemma
IdentifiersURN: urn:nbn:se:uu:diva-294640DOI: 10.1142/S0219199715500388ISI: 000373281200004OAI: oai:DiVA.org:uu-294640DiVA: diva2:932543