Improved Adams-type inequalities and their extremals in dimension 2 m

In this paper, we prove the existence of an extremal function for the Adams-Moser-Trudinger inequality on smooth, bounded, open subsets in dimension 2m, m ≥ 1. Moreover, we extend this result to improved versions of Adams' inequality of Adimurthi-Druet type. Our strategy is based on blow-up analysis for sequences of subcritical extremals and introduces several new techniques and constructions. The most important one is a new procedure for obtaining capacity-type estimates on annular regions.