The limit theorem for mappings with bounded distortion on the Heisenberg group and the local homeomorphism theorem (Q2773610)

The notion of the Heisenberg group \(\mathbb H^1\) was introduced by \textit{A.~Korányi} and \textit{H.~M.~Reimann} in [Invent. Math. 80, 309-338 (1985; Zbl 0567.30017)]. Previously, the author introduced some class of mappings with bounded distortion on the Heisenberg group, see [Sib. Math. J. 40, No. 4, 682-694 (1999)] and [Sib. Math. J. 41, No. 1, 40-48 (2000; Zbl 0962.30008)]. In the article under review, he proves a theorem about the limit of a sequence of mappings with bounded distortion on the Heisenberg group and a local homeomorphism theorem for mappings with distortion coefficient close to one.NEWLINENEWLINE More precisely, a mapping \(f\:U\to\mathbb H\) from an open set \(U\subset\mathbb H\) into \(\mathbb H\) is said to be a mapping with bounded distortion or a quasiregular mapping if: (a) \(f\) is continuous, (b) \(f\in HW^{1,4}_{\text{loc}}\); (c) \(f\) is a contact mapping, (d) there is a constant \(K<\infty\) such that the inequality \(\| Hf_*(q)\| ^4\leq KJ(q,f)\) holds almost everywhere in \(U\). Here \(\| Hf_*(q)\| =\max_{\eta\in HT_q,| \eta| =1}| Hf_*(q)\eta| \), \(Hf_*(q)\) is the formal horizontal differential at \(q\in \mathbb H\), \(J(q,f)=\det{Hf_*(q)}\), and \(HW^{1,4}\) is the horizontal Sobolev space.NEWLINENEWLINE The main results are as follows:NEWLINENEWLINE Theorem 1. Let \(U\) be a domain in \(\mathbb H\) and let \(f_j\:U\to\mathbb H\), \(j=1,2\dots\), be a sequence of mappings with distortion \(K\) converging locally uniformly to a mapping \(f\:U\to\mathbb H\) on \(U\). Then \(f\) is a mapping with bounded distortion and \(K(f)\leq K\).NEWLINENEWLINE Theorem 2. There exists \(K_0>0\) such that each nonconstant mapping with bounded distortion \(K_0\), defined on a domain \(U\) of the Heisenberg group, is injective on every ball \(B=B_R(a)\) such that the ball \(B=B_{9R}(a)\) lies in \(U\).NEWLINENEWLINE The proof of Theorem 1 uses essentially condition (c) of the definition of the mapping with bounded distortion and some results related to the change-of-variable formula on nilpotent groups obtained by \textit{S.~K.~Vodop'yanov} and \textit{A.~D.~Ukhlov} in [Sib. Math. J. 37, No. 1, 62-78 (1996; Zbl 0870.43005)]. Note that the result of Theorem 1 was proven by \textit{S.~K.~Vodop'yanov} in [Sib. Math. J. 40, No. 4, 644-677 (1999)] for the mappings with bounded distortion on general Carnot groups which need not satisfy condition (c). The proof of Theorem~2 is based on Iwaniec's proof for the theorem on the radius of injectivity for the mappings with bounded distortion in Euclidean space, see [\textit{T.~Iwaniec}, Proc. Am. Math. Soc. 100, 61-69 (1987; Zbl 0622.30015)], and an analog of the Liouville theorem for Carnot groups for the mappings with bounded distortion, see Theorem 12 in [\textit{S.~K.~Vodop'yanov}, Sib. Math. J. 40, No. 4, 644-677 (1999)] and [\textit{L.~Capogna}, Commun. Pure Appl. Math. 50, No. 9, 867-889 (1997; Zbl 0886.22006)].