AbstractThe 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}}-CKN inequalities are sharp for all parameter values except one exceptional case.
To show this, generalized Kelvin transforms {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_{\sigma}} is also derived.
Lastly, these techniques are shown to be valid for establishing CKN-type inequalities with monomial and horizontal norm weights.