Mappings with bounded distortion and with finite distortion on Carnot groups
The goal of the article is to develop a theory of mappings with bounded distortion (quasiregular mappings) on a Carnot group equipped with a Carnot-Carathéodory metric, starting from the weakest analytical assumption. A Carnot group is said to be a connected simply connected nilpotent Lie group \(\mathbb G\) whose Lie algebra is decomposed into the direct sum \(V_1\oplus\dots\oplus V_m\) of vector spaces such that \(\dim V_1\geq 2\), \([V_1,V_k]=V_{k+1}\) for \(1\leq k\leq m-1\) and \([V_1,V_m]=\{0\}\); the Carnot-Carathéodory distance \(d(x,y)\) between \(x,y\in\mathbb G\) is defined as the infimum of the lengths of the curves tangent to \(V_1\) a.e. [see \textit{P.~Pansu}, Ann. Math., II. Ser. 129, No. 1, 1-60 (1989; Zbl 0678.53042)]. Definition 1. A continuous mapping \(\varphi: \Omega \to \mathbb G\) is a mapping with bounded distortion on a domain \(\Omega\) of a Carnot group if \(\varphi\) belongs to \(HW_{\nu,\text{loc}}^1(\Omega)\), \(\nu\) is the Hausdorff dimension of \(\mathbb G\), and, for a.e. \(x\in\Omega\), the formal horizontal differential \(|D_H \varphi(x)|\) determined by the matrix \(X_k\varphi\) (\(X_k\), \(1\leq\dim V_1\), is the left-invariant basis of \(V_1\)) satisfies the inequality \[ |D_H \varphi(x)|^{\nu}\leq KJ(x,\varphi). \] Foundation of the theory is based on a broader class of mappings with finite distortion. Definition 2. A mapping \(\varphi: \Omega \to \mathbb G\) is a mapping with finite distortion on a domain \(\Omega\) of a Carnot group if \(\varphi\) belongs to \(HW_{\nu,\text{loc}}^1(\Omega)\), its Jacobian \(J(x,\varphi)\) is nonnegative, and \(D_H \varphi(x)\neq 0\) implies \(J(x,\varphi)\geq 0\) for a.e. \(x\in\Omega\). It is proven that mappings with finite distortion are continuous, quasimonotone, \(P\)-differentiable, and satisfy the \(N\)-Luzin property. This makes it possible to establish a connection between solutions to quasilinear subelliptic equations and mappings with bounded distortion in the case when the \(\nu\)-sub-Laplacian \(-\text{div}_*(|\nabla_{\mathcal L}u(x)|^{\nu-2}\nabla_{\mathcal L}u(x))=0\) on a Carnot group has a singular solution of class \(C^2\). For mappings with bounded distortion on Carnot groups, it is established that they are open and discrete, enjoy the \(\mathcal N^{-1}\) property, and have nondegenerate Jacobian. Normality of the class of mappings under study is proven. The author also proves a semicontinuity theorem for the distortion coefficient under locally uniform convergence of a sequence of mappings with bounded distortion, and a Liouville theorem for arbitrary Heisenberg groups under minimal smoothness assumptions. Note that the approach to constructing a theory and the majority of the proofs in the article are novel for a Euclidean space as well. The method is based on the change-of-variable formula for the Lebesgue integral for \(HW^1_{\nu,\text{loc}}\) classes and on other results obtained by the author in [\textit{S.~K.~Vodop'yanov}, \textit{A.~D.~Ukhlov}, Sib. Math. J. 37, No. 1, 62-78 (1996; Zbl 0870.43005)], [\textit{S.~K.~Vodop'yanov}, Sib. Math. J. 37, No. 6, 1113-1136 (1996; Zbl 0876.30020)], and [\textit{S.~K.~Vodop'yanov}, Sib. Math. J. 41, No. 1, 19-39 (2000)]. See related topics in [\textit{J.~Heinonen}, \textit{I.~Holopainen}, J. Geom. Anal. 7, No. 1, 109-148 (1997; Zbl 0905.30018)]where, in the very beginning, the authors stipulate a too strong regularity condition in the definition of a mapping with bounded distortion.
scientific article; zbMATH DE number 1439065