Isoperimetric and Sobolev Inequalities for Carnot-Carathkodory Spaces and the Existence of Minimal Surfaces zyx NICOLA GAROFALO AND DUY-MINH NHIEU zyxw Purdue University 1. Introduction and Statement of the Results After Hormander's fundamental paper on hypoellipticity zyx [54], the study of par- tial differential equations arising from families of noncommuting vector fields has developed significantly. In this paper we study some basic functional and geo- metric 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 zyxwvuts C R" be a Caccioppoli set (a measurable set having a locally finite perimeter); then one has the isoperimetric inequality where P(E) denotes De Giorgi's perimeter and C, > 0 is such that equality is attained in (1.1) when E is a Euclidean ball. It is well-known that (1.1) is equivalent to the Gagliardo-Nirenberg inequality Corresponding to (1.1) and (1.2), one has the following difficult results. Again, let E C zyxwvu Iw" be a Caccioppoli set. Then, for any Euclidean ball B = B(x0, R) one has the relative isoperimetric inequality where now P(E;B) denotes the perimeter of E relative to B. It turns out that (1.3) is equivalent to the following scale-invariant Sobolev embedding: zyx 11-1 R - 1 (- (u - us[ 5 dx) " zyxwvu 5 C, - I Dul dx, IBI IBI Communications on Pure and Applied Mathematics, Vol. XLIX, 1081-1 144 (1996) zyx 0 1996 John Wiley & Sons, Inc. CCC zyxw 001 0-3640/96/101081-64