On mappings with bounded distortion on Heisenberg group
Let \(\mathbb H^1\) be the Heisenberg group [see \textit{A. Koráni} and \textit{H. Reimann}, Invent. Math. 80, 309-338 (1985; Zbl 0567.30017)]. The author continues to study mappings with bounded distortion on \(\mathbb H^1\) which were introduced by him in [Sib. Math. J. 40, No. 4, 682-694 (1999)]. Definition. A mapping \(f\: U\to \mathbb H^1\) from an open set \(U\subset\mathbb H^1\) into \(\mathbb H^1\) is a mapping with bounded distortion if a) \(f\) is continuous; b) \(f\in \text{HW}_{\text{loc}}^{1,4}(U)\); c) there exists a constant \(K<\infty\) such that \(\|Hf_*(q) \|^4\leq KJ(q,f)\) almost everywhere on \(U\). Here \(\|Hf_*(q) \|=\max_{\xi\in HT_q, |\xi|=1} |Hf_*(q)\xi |\) is the operator norm of the linear mapping \(Hf_*(q)\). An equivalent class of quasiconformal mappings was studied by \textit{S.~K.~Vodop'yanov} [Sib. Math. J. 37, No. 6, 1113-1136 (1996; Zbl 0876.30020) and Sib. Math. J. 40, No. 4, 644-677 (2000)]. The main result of the article is as follows: Theorem. If \(f\: U\to\mathbb H^1\) is a continuous contact mapping of class \(\text{HW}_{\text{loc}}^{1,s}(U)\), with \(s>4\), then \(f\) is \(\mathcal P\)-differentiable almost everywhere on \(U\) and its \(\mathcal P\)-differential coincides with the formal \(\mathcal P\)-differential. For the definition of \(\mathcal P\)-differentiability see \textit{P. Pansu}, Ann. Math., II. Ser. 129, No. 1, 1-60 (1991; Zbl 0678.53042). The author also establishes some properties of the mappings with bounded distortion which ensue from some results of the papers [\textit{N. S. Dairbekov}, Sib. Math. J. 40, No. 4, 682-694 (1999), \textit{S.~K~Vodop'yanov}, Sib. Math. J. 37, No. 6, 1113-1136 (1996; Zbl 0876.30020), and \textit{S.~K.~Vodop'yanov} and \textit{A.~D.~ Ukhlov}, Sib. Math. J. 37, No. 1, 62-78 (1996; Zbl 0870.43005)].
scientific article; zbMATH DE number 1534464