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Dorronsoro’s theorem in Heisenberg groups. (English) Zbl 1457.46046

Summary: 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 [J. R. Dorronsoro, Proc. Am. Math. Soc. 95, 21–31 (1985; Zbl 0577.46035)]. 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 T. Austin et al. [Groups Geom. Dyn. 7, No. 3, 497–522 (2013; Zbl 1284.46019)] and V. Lafforgue and A. Naor [Isr. J. Math. 203, 309–339 (2014; Zbl 1312.46032)].

MSC:

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
42B35 Function spaces arising in harmonic analysis
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References:

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