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].