Abstract
AbstractThe main result of the present article is a Rademacher-type theorem for intrinsic Lipschitz graphs of codimension$k\leq n$in sub-Riemannian Heisenberg groups${\mathbb H}^{n}$. For the purpose of proving such a result, we settle several related questions pertaining both to the theory of intrinsic Lipschitz graphs and to the one of currents. First, we prove an extension result for intrinsic Lipschitz graphs as well as a uniform approximation theorem by means of smooth graphs: both of these results stem from a new definition (equivalent to the one introduced by B. Franchi, R. Serapioni and F. Serra Cassano) of intrinsic Lipschitz graphs and are valid for a more general class of intrinsic Lipschitz graphs in Carnot groups. Second, our proof of Rademacher’s theorem heavily uses the language of currents in Heisenberg groups: one key result is, for us, a version of the celebrated constancy theorem. Inasmuch as Heisenberg currents are defined in terms of Rumin’s complex of differential forms, we also provide a convenient basis of Rumin’s spaces. Eventually, we provide some applications of Rademacher’s theorem including a Lusin-type result for intrinsic Lipschitz graphs, the equivalence between${\mathbb H}$-rectifiability and ‘Lipschitz’${\mathbb H}$-rectifiability and an area formula for intrinsic Lipschitz graphs in Heisenberg groups.
Highlights
We prove an extension result for intrinsic Lipschitz graphs as well as a uniform approximation theorem by means of smooth graphs: both of these results stem from a new definition
Some of the tools we develop for proving our main result are worth mentioning; we prove an extension result for intrinsic Lipschitz graphs as well as the fact that they can be uniformly approximated by smooth graphs
Both results stem from what can be considered as another contribution of the present article; that is, a new definition of intrinsic Lipschitz graphs that is equivalent to the original one, introduced by B
Summary
The celebrated Rademacher’s theorem [82] states that a Lipschitz continuous function f : Rh → Rk is differentiable almost everywhere in Rh; in particular, the graph of f in Rh+k has an h-dimensional tangent plane at almost all of its points. Some applications – namely, a Lusin-type result and an area formula for intrinsic Lipschitz graphs – are provided here as well; we believe that further consequences are yet to come concerning, for instance, rectifiability and minimal submanifolds in Heisenberg groups. Some of the tools we develop for proving our main result are worth mentioning; we prove an extension result for intrinsic Lipschitz graphs as well as the fact that they can be uniformly approximated by smooth graphs Both results stem from what can be considered as another contribution of the present article; that is, a new definition of intrinsic Lipschitz graphs that is equivalent to the original one, introduced by B.
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