Weighted inequalities for the fractional Laplacian and the existence of extremals

Abstract

In this paper, we obtain improved versions of Stein–Weiss and Caffarelli–Kohn–Nirenberg inequalities, involving Besov norms of negative smoothness. As an application of the former, we derive the existence of extremals of the Stein–Weiss inequality in certain cases, some of which are not contained in the celebrated theorem of Lieb [Sharp constants in the Hardy–Littlewood–Sobolev and related inequalities, Ann. of Math. (2) 118(2) (1983) 101–116].