Concentration analysis in Banach spaces

TL;DR

This paper introduces a generalized profile decomposition framework in uniformly convex Banach spaces using $ riangle$-convergence, extending previous results and connecting to fixed point theory and the Brezis-Lieb Lemma.

Contribution

It proves existence of profile decompositions in uniformly convex Banach spaces with isometries, generalizing prior results from Sobolev and Hilbert spaces, and explores $ riangle$-convergence properties.

Findings

Profile decompositions exist in uniformly convex Banach spaces with isometries.

$ riangle$-convergence coincides with weak convergence under the Opial condition.

A version of the Brezis-Lieb Lemma without almost everywhere convergence is established.

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 $\Delta$-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 $\ell^{p}$-spaces, but not in $L^{p}(\mathbb R^{N})$, $p\neq2$. $\Delta$-convergence appears naturally in the context of fixed point theory for non-expansive maps. The paper also studies connection of $\Delta$-convergence with Brezis-Lieb Lemma and gives a version of the latter without an assumption of convergence a.e.