The Lusin theorem and horizontal graphs in the Heisenberg group

TL;DR

This paper extends Lusin's theorem to higher-order derivatives in the Heisenberg group, showing measurable functions can be approximated by derivatives of smooth functions with controlled continuity, and constructs horizontal tangent surfaces in this setting.

Contribution

It provides a stronger version of Lusin's theorem for higher derivatives and applies it to construct surfaces with horizontal tangent spaces in the Heisenberg group.

Findings

Higher-order derivatives can be approximated by smooth functions with weaker continuity.

Constructed surfaces in the Heisenberg group with horizontal tangent spaces almost everywhere.

Extended classical approximation results to a sub-Riemannian geometric context.

Abstract

In this paper we prove that every collection of measurable functions $f_\alpha$, $|\alpha|=m$ coincides a.e. with $m$th order derivatives of a function $g\in C^{m-1}$ whose derivatives of order $m-1$ may have any modulus of continuity weaker than that of a Lipschitz function. This is a stronger version of earlier results of Lusin, Moonens-Pfeffer and Francos. As an application we construct surfaces in the Heisenberg group with tangent spaces being horizontal a.e.