The paper gives the following improvement of the Trudinger-Moser inequality:2000 Mathematics Subject Classification. 35J61, 35J75, 35A23. Key words and phrases. Trudinger-Moser inequality, borderline Sobolev imbeddings, singular elliptic operators, remainder terms, spectral gap, virtual bound state, Hardy-Sobolev-Mazya inequality.

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 ∆-convergence by T. C. Lim [20] 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 = 2. ∆-convergence appears naturally in the context of fixed point theory for non-expansive maps. The paper also studies connection of ∆-convergence with Brezis-Lieb Lemma and gives a version of the latter without an assumption of convergence a.e.

Abstract. Let M be a non-compact homogeneous Riemannian manifold, and let Ω be a compact subgroup of isometries of M . We show, under general conditions, that the Ω-invariant subspace AΩ of a normed vector space A → L q (M ) is compactly embedded into L q (M ) if and only if the group Ω has no orbits with a uniformly bounded diameter in a neighborhood of infinity. Mathematics Subject Classification (2010). Primary 46B50; Secondary 46E35, 46N20.

We show that the Moser functional J (u) = R (e 4⇡u 2 1) dx on the set B = {u 2 H 1 0 () : kruk 2  1}, where ⇢ R 2 is a bounded domain, fails to be weakly continuous only in the following exceptional case. Defineup to translations and up to a remainder vanishing in the Sobolev norm. In other words, the weak continuity fails only on translations of concentrating Moser functions. The proof is based on a profile decomposition similar to that of Solimini [16], but with different concentration operators, pertinent to the two-dimensional case.

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