# Dorronsoro's theorem in Heisenberg groups

## TL;DR

This paper extends Dorronsoro's theorem to Heisenberg groups, showing that horizontal Sobolev functions can be approximated by affine functions independent of the last variable, with applications to Poincaré inequalities.

## Contribution

It proves a variant of Dorronsoro's theorem in Heisenberg groups, introducing a new approximation method for horizontal Sobolev functions and providing alternative proofs for Poincaré inequalities.

## Key findings

- Horizontal Sobolev functions can be approximated by affine functions in Heisenberg groups.
- New proofs for vertical vs. horizontal Poincaré inequalities are established.
- The approximation is independent of the last variable in the Heisenberg group context.

## Abstract

A theorem of Dorronsoro from the 1980s quantifies the fact that real-valued Sobolev functions on Euclidean spaces can be approximated by affine functions almost everywhere, and at all sufficiently small scales. We prove a variant of Dorronsoro's theorem in Heisenberg groups: functions in horizontal Sobolev spaces can be approximated by affine functions which are independent of the last variable. As an application, we deduce new proofs for certain vertical vs. horizontal Poincar\'e inequalities for real-valued functions on the Heisenberg group, originally due to Austin-Naor-Tessera and Lafforgue-Naor.