An isoperimetric inequality and the geometric Sobolev embedding for vector fields

The classical embedding theorem of Sobolev W (R) ↪→ L(R) was originally proved in the case 1 < p < n, with q = np n−p . It was only in the late fifties that by means of an elegant integral inequality Gagliardo and Nirenberg were independently able to obtain the limiting case p = 1 and prove that: (∗) W (R) ↪→ L n n−1 (R) (see, e.g., [S]). Meanwhile, in his fundamental work [DG] De Giorgi introduced the notion of total variation of an L distribution and laid down his theory of generalized perimeters. Soon after, Fleming and Rishel [FR] gave a beautiful geometric new proof of (∗) based on Federer’s co-area formula and the celebrated isoperimetric inequality: (∗∗) P (E) ≥ cn|E| n−1 n , where P (E) denotes the perimeter according to De Giorgi and cn = nΓ( 12 )Γ( n 2 +1) − 1 n . It turns out that, in fact, (∗) is equivalent to (∗∗), see e.g. [T]. The purpose of this note is to announce an optimal embedding theorem similar to (∗) for the Sobolev spaces associated to some general families of vector fields satisfying certain geometric conditions. A priori, we do not need to require C∞ smoothness of the vector fields. To keep some unity of presentation, however, we have confined our discussion to the case of C∞ vector fields satisfying Hormander’s condition on the Lie algebra [H]. For instance, operators of Baouendi–Grushin type