Intrinsic Lipschitz Graphs Within Carnot Groups

Differentiability and ApproximateDifferentiability for Intrinsic LipschitzFunctions in Carnot Groups and a RademacherTheorem

“…If M, H are complementary subgroups of G and one of them is a normal subgroup then G is said to be the semi-direct product of M and H. If both M and H are normal subgroups then G is said to be the direct product of M and H. The elements m ∈ M and h ∈ H such that g = mh are unique because of M ∩ H = {e} and are denoted as components of g along M and H or as projections of g on M and H. Proposition 2.2.2 (see [7,17]…”

“…The notion of intrinsic graphs within Carnot groups (connected, simply connected, strati ed nilpotent Lie groups) and, speci cally of intrinsic Lipschitz graphs has been introduced with di erent degrees of generality in [21], [7], [23]. In [17] the authors provide a comprehensive presentation of this theory.…”

“…Hence, understanding the notions of C and of Lipschitz submanifolds within Carnot groups is preliminary in order to develop a satisfactory theory of intrinsic recti able sets. We refer the reader to [17] for a thorough discussion of the problem that lies behind our notion of Lipschitz graph.…”

“…Let us sketch here the main points of the discussion in [17]. First of all, we stress that considering Euclidean regular submanifolds may be sometimes too general and sometimes too restrictive.…”

“…Intrinsic graphs appeared naturally in [21], [7], [23] in relation with non critical level sets of di erentiable functions from G to R k . Indeed, the implicit function theorem within groups ( [18], [22], [19], [12], [13]) can be rephrased stating that level sets are always, locally, intrinsic graphs.…”