Hardy, Hardy-Sobolev and Caffarelli-Kohn-Nirenberg inequalities on general Lie groups

Michael Ruzhansky, Nurgissa Yessirkegenov

Published 2018-10-20Version 1

In this paper we obtain two-weight Hardy inequalities on general metric measure spaces possessing polar decompositions. Moreover, we also find necessary and sufficient conditions for the weights for such inequalities to be true. As a consequence, we establish the Hardy, Hardy-Sobolev, Caffarelli-Kohn-Nirenberg, Gagliardo-Nirenberg inequalities and their critical versions on general connected Lie groups, which include both unimodular and non-unimodular cases in compact and noncompact settings. As a byproduct, it also gives, as a special case, an alternative proof for Sobolev embedding theorems on general (non-unimodular) Lie groups. We also obtain the corresponding uncertainty type principles.