On rectifiable measures in Carnot groups: representation

Abstract

This paper deals with the theory of rectifiability in arbitrary Carnot groups, and in particular with the study of the notion of $$\mathscr {P}$$ P -rectifiable measure. First, we show that in arbitrary Carnot groups the natural infinitesimal definition of rectifiabile measure, i.e., the definition given in terms of the existence of flat tangent measures, is equivalent to the global definition given in terms of coverings with intrinsically differentiable graphs, i.e., graphs with flat Hausdorff tangents. In general we do not have the latter equivalence if we ask the covering to be made of intrinsically Lipschitz graphs. Second, we show a geometric area formula for the centered Hausdorff measure restricted to intrinsically differentiable graphs in arbitrary Carnot groups. The latter formula extends and strengthens other area formulae obtained in the literature in the context of Carnot groups. As an application, our analysis allows us to prove the intrinsic $$C^1$$ C 1 -rectifiability of almost all the preimages of a large class of Lipschitz functions between Carnot groups. In particular, from the latter result, we obtain that any geodesic sphere in a Carnot group equipped with an arbitrary left-invariant homogeneous distance is intrinsic $$C^1$$ C 1 -rectifiable.

Nowhere differentiable intrinsic Lipschitz graphs

“…Observe also that no tangent measure can be flat, that is, supported on a homogeneous subgroup. In particular, is purely ‐unrectifiable, that is, for every submanifold of class (see, for example, [3, § 2.5 and 6.1]).…”

On rectifiable measures in Carnot groups: Marstrand-Mattila rectifiability criterion

“…The characterization of the k-rectifiability of a measure through the existence of the k-density in Euclidean spaces was one of the great achievement of Geometric Measure Theory [27]. Another Preiss's type result has been proved by A. Lorent [20] in 3 ∞ . Recently, the second named author has accomplished to prove the analogue of Theorem 1.3 for the 3-density, which requires a deeper understanding of 3-uniform measures in the first Heisenberg group H 1 , see [24,25].…”

“…The previous infinitesimal characterization of rectifiable measures in the Euclidean spaces is at the core of the definition of P-rectifiable measures, which have been introduced by the second named author in [25, Definition 3.1 & Definition 3.2], in the setting of Carnot groups, and which have been studied by the two authors in [5,4]. We stress that the present paper is the second of two companion papers derived from [5].…”

“…We stress that the present paper is the second of two companion papers derived from [5]. For a more thorough introduction we refer the reader to the introductions of [5,4].…”

“…This is the second of two companion papers derived from [5]. The present work consists of an elaboration of Sections 2 and 5 of the Preprint [5], while the first of the two (that will appear as [3]) is an elaboration of Sections 2, 3, 4, and 6 of [5]. In order to avoid redundancy, we wrote the complete proofs of some of the ancillary results of Section 2 of [5] only in the first of the two.…”