Concentration–Compactness principle for the sharp Adams inequalities in bounded domains and whole space [formula omitted

Abstract

We prove an improvement of the sharp Adams inequality in W0m,nm(Ω) where Ω is a bounded domain in Rn inspired by Lions Concentration–Compactness principle for the sharp Moser–Trudinger inequality. Our method gives an alternative approach to a Concentration–Compactness principle in W0m,nm(Ω) recently established by do Ó and Macedo. Furthermore, we obtain a sharp threshold for m odd improving the one of do Ó and Macedo. Our approach is also successfully applied to whole space Rn to establish the improvements of the sharp Adams inequalities in Wm,nm(Rn). This type of improvement is still unknown, in general, except the case m=1 due to do Ó, de Souza, de Medeiros and Severo. Our method is a further development of the one of Černy, Cianchi and Hencl combining with some estimates for the decreasing rearrangement of a function in terms of the one of its higher order derivatives.