The weighted isoperimetric-type and Sobolev-type inequalities for hypersurfaces in Carnot groups

This paper mainly studies hypersurfaces immersed in a sub-Riemannian step kk Carnot group GG with a density efef, where ff is a smooth function on GG. We give the first variation formula about the weighted horizontal area of hypersurfaces. Furthermore on the base of some work we derive the weighted isoperimetric-type and Sobolev-type inequalities for hypersurfaces.