Maximal noncompactness of limiting Sobolev embeddings

Abstract

We develop a new method suitable for establishing lower bounds on the ball measure of noncompactness of operators acting between considerably general quasinormed function spaces. This new method removes some of the restrictions oft-presented in the previous work. Most notably, the target function space need not be disjointly superadditive nor equipped with a norm. Instead, a property that is far more often at our disposal is exploited, namely the absolute continuity of the target quasinorm. We use this new method to prove that limiting Sobolev embeddings into spaces of Brezis--Wainger type are so-called maximally noncompact, i.e., their ball measure of noncompactness is the worst possible.