Differentiability and Approximate Differentiability for Intrinsic Lipschitz Functions in Carnot Groups and a Rademacher Theorem

Bruno Franchi; Marco Marchi; Raul Paolo Serapioni

doi:10.2478/agms-2014-0010

Differentiability and Approximate Differentiability for Intrinsic Lipschitz Functions in Carnot Groups and a Rademacher Theorem

Analysis and Geometry in Metric Spaces ◽

10.2478/agms-2014-0010 ◽

2014 ◽

Vol 2 (1) ◽

Author(s):

Bruno Franchi ◽

Marco Marchi ◽

Raul Paolo Serapioni

Keyword(s):

Lipschitz Functions ◽

Carnot Groups ◽

Rademacher Theorem ◽

Approximate Differentiability

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On rectifiable measures in Carnot groups: representation

Calculus of Variations and Partial Differential Equations ◽

10.1007/s00526-021-02112-4 ◽

2021 ◽

Vol 61 (1) ◽

Author(s):

Gioacchino Antonelli ◽

Andrea Merlo

Keyword(s):

Large Class ◽

Hausdorff Measure ◽

Carnot Group ◽

Lipschitz Functions ◽

Carnot Groups ◽

Geodesic Sphere ◽

Area Formula ◽

Definition Of ◽

Lipschitz Graphs ◽

Almost All

AbstractThis 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.

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