Implicit Function Theorem in Carnot–carathéodory Spaces

In this paper, we study the notion of regular surface in the Carnot–Carathéodory spaces and we prove the implicit function theorem in this setting. We fix at every point of R a subset of the tangent space, called horizontal tangent plane, and we assume that it has a basis X1, . . . , Xm of C∞ vector fields satisfying the Hörmander condition of hypoellipticity, (see [16]). This choice induces a natural metric structure on R of subriemannian type (see [20]), in which the notion of tangent space is replaced by the horizontal tangent subspace. Most results of real analysis and geometric measure theory have been extended in this setting: the properties of regular functions, C X Hölder spaces, Sobolev spaces defined in terms of given vectors, as well as the definition of BVX functions and the notion of perimeter in the sense of De Giorgi. However, in this setting, the notion of regular surface is not completely clear. The first definition, given by Federer (see [10]), was that a regular surface in R is the image of an open set of Rn−1 through a lipschitz continuous function. However, the Heisenberg group turned out to be completely non-rectifiable in this sense, (see [1]). A more natural definition of regular surface was given in [12] in the Heisenberg group, and successively in the Carnot group, (see [13–15]): a regular