Lipschitz and Holder mappings into jet space Carnot groups

Jung, Derek

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https://hdl.handle.net/2142/102425 Copy

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

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http://hdl.handle.net/2142/102425

Copyright and License Information

Copyright 2018 Derek Jung

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