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

Abstract

A Carnot group G is a connected, simply connected, nilpotent Lie group with stratified Lie algebra.We study intrinsic Lipschitz graphs and intrinsic differentiable graphs within Carnot groups. Both seem to bethe natural analogues inside Carnot groups of the corresponding Euclidean notions. Here ‘natural’ is meantto stress that the intrinsic notions depend only on the structure of the algebra of G. We prove that one codimensionalintrinsic Lipschitz graphs are sets with locally finite G-perimeter. From this a Rademacher’s typetheorem for one codimensional graphs in a general class of groups is proved.

Keywords

• carnot groups
• rectifiable sets
• intrinsic lipschitz functions
• rademacher’s theorem