On the Brezis-Lieb Lemma without pointwise convergence

TL;DR

This paper extends the Brezis-Lieb lemma to cases without pointwise convergence, establishing conditions under which the gap between integrals can be evaluated, especially for p ≥ 3, and explores applications to gradients.

Contribution

It provides a new version of the Brezis-Lieb lemma applicable without pointwise convergence for p ≥ 3, including vector-valued functions, and shows the limitations for p < 3.

Findings

The same lower bound for the gap holds for p ≥ 3 under weak convergence conditions.

The statement fails for p < 3.

Application to a Brezis-Lieb lemma for gradients.

Abstract

Brezis-Lieb lemma is a refinement of Fatou lemma providing an evaluation of the gap between the integral for a sequence and the integral for its pointwise limit. This note studies the question if such gap can be evaluated when there is no a.e. convergence. In particular, it gives the same lower bound for the gap in L^p as the gap in the Brezis-Lieb lemma (including the case vector-valued functions) provided that p is greater or equal than 3 and the sequence converges both weakly and weakly in the sense of a duality map. It also shows that the statement is false if p<3. An application is given in form of a Brezis-Lieb lemma for gradients.