Isoperimetric and Sobolev inequalities for Carnot-Carath�odory spaces and the existence of minimal surfaces

After Hormander's fundamental paper on hypoellipticity [54], the study of partial differential equations arising from families of noncommuting vector fields has developed significantly. In this paper we study some basic functional and geometric properties of general families of vector fields that include the Hormander type as a special case. Similar to their classical counterparts, such properties play an important role in the analysis of the relevant differential operators (both linear and nonlinear). To motivate our results, we recall some classical inequalities. Let E C R" be a Caccioppoli set (a measurable set having a locally finite perimeter); then one has the isoperimetric inequality