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 Title: Lipschitz and Holder mappings into jet space Carnot groups Author(s): Jung, Derek Director of Research: Tyson, Jeremy Doctoral Committee Chair(s): Wu, Jang-Mei Doctoral Committee Member(s): Fernandes, Rui; Kaufman, Robert Department / Program: Mathematics Discipline: Mathematics Degree Granting Institution: University of Illinois at Urbana-Champaign Degree: Ph.D. Genre: Dissertation Subject(s): Sub-Riemannnian Geometry Jet spaces Carnot groups Geometric Analysis Abstract: For $k,n\ge 1$, the jet space $J^k(\R^n)$ is the set of $k^{th}$-order Taylor polynomials of functions in $C^k(\R^n)$. Warhurst constructs a Carnot group structure on $J^k(\R^n)$ such that the jets of functions in $C^{k+1}(\R^n)$ are horizontal. Like in all Carnot groups, one can define a Carnot-Carath\'eodory metric on $J^k(\R^n)$ by minimizing lengths of horizontal paths. Unfortunately, exact forms or even the regularities of geodesics connecting generic pairs of points are not known for $J^k(\R^n)$. After describing the Carnot group structure of $J^k(\R^n)$, we will prove that there exists a biLipschitz embedding of $\mathbb{S}^n$ into $J^k(\R^n)$ that does not admit a Lipschitz extension to $\mathbb{B}^{n+1}$. This strengthens a result of Rigot and Wenger \cite{RW:LNE} and generalizes a result for $\mathbb{H}^n$ of Dejarnette, Haj{\l}asz, Lukyanenko, and Tyson. We will then consider a problem related to Gromov's conjecture on the H\"older equivalence of Carnot groups. We will prove that for all $m\ge 2$ and $\epsilon>0$, there does not exist an injective, locally $(\frac{1}{2}+\epsilon)$-H\"older mapping $f:\R^m\to J^k(\R)$ that is locally Lipschitz as a mapping into $\R^{k+2}$. This builds on a result of Balogh, Haj{\l}asz, and Wildrick for $\mathbb{H}^n$. We will conclude by proposing analogues of horizontal and vertical projections for $J^k(\R)$. We prove Marstrand-type results for these mappings. This continues efforts of Balogh, Durand-Cartagena, F\"assler, Mattila, and Tyson over the past decade to prove Marstrand-type theorems in a sub-Riemannian setting. We will study the metric structure of $J^k(\R^n)$, focusing primarily on the model filiform jet spaces $J^k(\R)$. Issue Date: 2018-11-08 Type: Thesis URI: http://hdl.handle.net/2142/102425 Rights Information: Copyright 2018 Derek Jung Date Available in IDEALS: 2019-02-06 Date Deposited: 2018-12
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