On the Brezis-Lieb Lemma without pointwise convergence

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

On a version of Trudinger-Moser inequality with M\"obius shift invariance

The paper raises a question about the optimal critical nonlinearity for the Sobolev space in two dimensions, connected to loss of compactness, and discusses the pertinent concentration compactness framework. We study properties of the improved version of the Trudinger-Moser inequality on the open unit disk $B\subset\R^2$, recently proved by G. Mancini and K. Sandeep. Unlike the original Trudinger-Moser inequality, this inequality is invariant with respect to M\"obius automorphisms of the unit disk, and as such is a closer analogy of the critical nonlinearity $\int |u|^{2^*}$ in the higher dimension than the original Trudinger-Moser nonlinearity.Comment: This version gives the credit to an independently proved result, missed in the early version, and corrects an error in one of the proof