Sharp Caffarelli–Kohn–Nirenberg-Type Inequalities on Carnot Groups

Abstract

Abstract The main purpose of this paper is to establish several general Caffarelli–Kohn–Nirenberg (CKN) inequalities on Carnot groups G (also known as stratified groups). These CKN inequalities are sharp for certain parameter values. In case G is an Iwasawa group, it is shown here that the L 2 {L^{2}} -CKN inequalities are sharp for all parameter values except one exceptional case. To show this, generalized Kelvin transforms K σ {K_{\sigma}} are introduced and shown to be isometries for certain weighted Sobolev spaces. An interesting transformation formula for the sub-Laplacian with respect to K σ {K_{\sigma}} is also derived. Lastly, these techniques are shown to be valid for establishing CKN-type inequalities with monomial and horizontal norm weights.