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Lipschitz and Holder mappings into jet space Carnot groups
Jung, Derek
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https://hdl.handle.net/2142/102425
Description
- Title
- Lipschitz and Holder mappings into jet space Carnot groups
- Author(s)
- Jung, Derek
- Issue Date
- 2018-11-08
- Director of Research (if dissertation) or Advisor (if thesis)
- Tyson, Jeremy
- Doctoral Committee Chair(s)
- Wu, Jang-Mei
- Committee Member(s)
- Fernandes, Rui
- Kaufman, Robert
- Department of Study
- Mathematics
- Discipline
- Mathematics
- Degree Granting Institution
- University of Illinois at Urbana-Champaign
- Degree Name
- Ph.D.
- Degree Level
- Dissertation
- Keyword(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 $J^k(\R)$."
- Graduation Semester
- 2018-12
- Type of Resource
- text
- Permalink
- http://hdl.handle.net/2142/102425
- Copyright and License Information
- Copyright 2018 Derek Jung
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Graduate Dissertations and Theses at Illinois PRIMARY
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