CHAPTER 1 Introduction In the last decade there has been an explosion of interest in the theory of Carnot-Caratheodory spaces (CC spaces, henceforth), and in the ramifications of this subject into analysis and geometry. We recall that a CC space is a Riemannian manifold (M, g), which has been endowed with a distance d different from the Riemannian metric attached to the tensor g. Such distance d is the control metric associated with a sub-bundle H of the tangent bundle TM. Loosely speaking, if X = {Xt,..., Xm} denotes a system of non-commuting vector fields which (locally) generates H, then one defines d(x, y) by a minimization procedure which selects among all curves in M which join x to y, only those whose tangent vector belongs to span{Xi, ...,X m }. The ensuing metric space (M,d) is called a CC space, or also a sub-Riemannian space. Excellent references on the subject are the books [VSC92], [Be96], [Gro98], [Mon02]. We also refer the reader to the forthcoming book [G02], which more directly treats the connections of CC geometry with partial differential equations. In this work we study the following general question: to characterize the traces of Sobolev functions in a CC space with respect to a measure supported on a lower dimensional manifold. Our primary motivation is the study of boundary value problems, arising for instance in CR geometry, for various linear and nonlinear equations of sub-elliptic type. The common trend of these equations is lack of ellipticity. But their leading part can be expressed by the sum of squares of smooth vector fields satisfying a certain algebraic assumption known as the finite rank condition on the Lie algebra, see (1.4). Thanks to a fundamental result of Hormander [H67], such condition implies the hypoelliticity of the relevant operator, see also [OR73]. Another important motivation is the connection of the questions studied here with the newly forming theory of perimeters and minimal surfaces in CC spaces. We recall that the existence of minimal surfaces was established in [GN96], where the problem of their regularity was also posed. This aspect, however, is only barely touched upon in the present work, and will be systematically investigated in forthcoming studies. In this connection we mention the interesting recent papers [FSSOl], [Pa(II)04], [FSS03(I)], [FSS03(II)], and also the preprint [DGN04(II)]. Among many others devoted to the subject, the classical books [Ne67], [LaU68], [LiMa72], [Tre75], [Gr85], [Tro87], underline the fundamental role played by trace theorems in the theory of boundary value problems for partial differential equations. The reader should also consult the pioneering papers by Gagliardo [Ga57], [Ga58], [Ga59], and Stein [St61]. The more recent papers [JK95], [FMM98] and [MiMa04] contain sharp results for the solvability of boundary value problems for the non-homogeneous Laplace equation in the framework of Besov spaces.
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