$C^{1,\alpha}$-rectifiability in low codimension in Heisenberg groups

TL;DR

This paper introduces a new notion of higher order rectifiability in Heisenberg groups, characterizing when sets can be covered by regular surfaces with a certain smoothness, based on tangent paraboloids.

Contribution

It defines a higher order rectifiability concept in Heisenberg groups and establishes a sufficient condition involving tangent paraboloids for low-codimensional sets.

Findings

Introduces a notion of $C^{1,eta}$-rectifiability in Heisenberg groups.

Proves that the existence of approximate tangent paraboloids implies rectifiability.

Provides a characterization of rectifiable sets via tangent paraboloids.

Abstract

A natural notion of higher order rectifiability is introduced for subsets of Heisenberg groups $\mathbb{H}^n$ in terms of covering a set almost everywhere by a countable union of $(\mathbf{C}_H^{1,\alpha},\mathbb{H})$-regular surfaces, for some $0 < \alpha \leq 1$. We prove that a sufficient condition for $C^{1,\alpha}$-rectifiability of low-codimensional subsets in Heisenberg groups is the almost everywhere existence of suitable approximate tangent paraboloids.