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<p>We study various notions of the Schwarzian derivative for contact mappings in the Heisenberg group <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\({\mathbb {H}}_{1}\)</EquationSource> </InlineEquation> and introduce two definitions: (1) <i>the CR Schwarzian derivative</i> based on the conformal connection approach studied by Osgood and Stowe and, recently, by Son; (2) <i>the classical type Schwarzian</i> refering to the well-known complex analytic definition. In particular, we take into consideration the effect of conformal rigidity and the limitations it imposes. Moreover, we study the kernels of both Schwarzians and the cocycle conditions. Inspired by ideas of Chuaqui et al. (J. Anal. Math. 91:329–351, 2003), Hernández and Martín (J. Geom. Anal. 25(1):64–91, 2015) and Hernández and Venegas (Complex Anal. Oper. Theory 13(4):1783–1793, 2019), we introduce the <i>Preschwarzian</i> for mappings in <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\({\mathbb {H}}_{1}\)</EquationSource> </InlineEquation>. Furthermore, we study results in the theory of subelliptic PDEs for the horizontal Jacobian and related differential expressions for harmonic mappings and the <i>gradient harmonic mappings</i>, the latter notion introduced here in the setting of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\({\mathbb {H}}_{1}\)</EquationSource> </InlineEquation>.</p>

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Schwarzians on the Heisenberg Group

  • 期刊论文

  • 发表日期：

    2026年3月9日

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  • Tomasz Adamowicz,
  • Ben Warhurst

摘要

We study various notions of the Schwarzian derivative for contact mappings in the Heisenberg group \({\mathbb {H}}_{1}\) and introduce two definitions: (1) the CR Schwarzian derivative based on the conformal connection approach studied by Osgood and Stowe and, recently, by Son; (2) the classical type Schwarzian refering to the well-known complex analytic definition. In particular, we take into consideration the effect of conformal rigidity and the limitations it imposes. Moreover, we study the kernels of both Schwarzians and the cocycle conditions. Inspired by ideas of Chuaqui et al. (J. Anal. Math. 91:329–351, 2003), Hernández and Martín (J. Geom. Anal. 25(1):64–91, 2015) and Hernández and Venegas (Complex Anal. Oper. Theory 13(4):1783–1793, 2019), we introduce the Preschwarzian for mappings in \({\mathbb {H}}_{1}\) . Furthermore, we study results in the theory of subelliptic PDEs for the horizontal Jacobian and related differential expressions for harmonic mappings and the gradient harmonic mappings, the latter notion introduced here in the setting of \({\mathbb {H}}_{1}\) .