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.

中文翻译：

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海森堡群中的多伦索罗定理

1980年代的Dorronsoro定理量化了这样一个事实，即欧几里得空间上的实值Sobolev函数几乎可以在所有地方，并且在足够小的范围内都可以通过仿射函数来近似。我们证明了海森堡组中多伦索罗定理的一个变体：水平Sobolev空间中的函数可以通过与最后一个变量无关的仿射函数来近似。作为一种应用，我们推论出海森堡群上实值函数的某些纵向和横向庞加莱不等式的新证明，这最初是由于奥斯汀-纳尔-特塞拉和拉夫福格-纳尔的缘故。