Lipschitz and Holder mappings into jet space Carnot groups

Sub-Riemannnian Geometry;Jet spaces;Carnot groups;Geometric Analysis

Jung, Derek

关键词: Sub-Riemannnian Geometry;    Jet spaces;    Carnot groups;    Geometric Analysis;   

Others  :  https://www.ideals.illinois.edu/bitstream/handle/2142/102425/JUNG-DISSERTATION-2018.pdf?sequence=1&isAllowed=y

美国|英语

来源: The Illinois Digital Environment for Access to Learning and Scholarship

【 摘 要 】

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 $J^k(\R)$.