## Dorronsoro’s theorem in Heisenberg groups

### Fässler, Katrin ; Orponen, Tuomas

### In: Bulletin of the London Mathematical Society, 2020, vol. 52, no. 3, p. 472–488

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... More

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- 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 versus horizontal Poincaré inequalities for real‐valued functions on the Heisenberg group, originally due to Austin–Naor–Tessera and Lafforgue–Naor.