A study of dimension estimates in the context of spherical Talbot effect and Besov mappings

This thesis focuses on dimension estimates. We present our results in two projects in this well-studied topic - one in the context of dispersive partial differential equations (PDEs) on the sphere and the other under the umbrella of dimension distortion estimates in geometric measure theory. In our first project concerning dispersive PDEs, we extend the results of \cite{ErdShak} on $\T$ to the linear Schr\"{o}dinger equation on \sph² with varying initial data. In particular, we use the Besov scale to obtain fractal dimension bounds for the graph of the solution to this equation with not only arbitrary initial data but also initial data whose spherical orthogonal expansions assume specific forms. As a result of this study, we establish the tools to further understand the graph of the solution to dispersive PDEs on \sph², and give evidence for the Talbot effect on this manifold. In the second project, our study investigates the effect of Besov mappings on subsets and subspaces of \Rⁿ. Given f: \Omega \to \R^k where $n,k \in \N$, \Omega \subseteq \Rⁿ, with additional assumptions on the space $f$ belongs to, many mathematicians have studied Hausdorff dimension distortion of subsets and subspaces under such a mapping. Especially in the case of subspace dimension distortion for Sobolev mappings, the history is rich with results from Gehring and V\"{a}is\"{a}l\"{a}, Kaufman, as well as Balogh, Tyson and Wildrick. Using ideas from results proven for Sobolev mappings, Hencl and Honz\'{i}k extended the study into mappings of the Triebel-Lizorkin type \hF^s_p,q, a generalization of the Sobolev scale. Since the Besov scale \hB^s_p,q is another generalization of the Sobolev scale, it is natural to expect similar results for mappings in such space. Our results addresses this line of questioning for both subsets and subspaces, as well as the question of sharpness of the bounds in some circumstances.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2022

Author dc:creator

• Huynh, Chi Ngoc Yen

Contributors dc:contributor

• Erdogan, M. Burak
• Tyson, Jeremy
• Tzirakis, Nikolaos
• Albin, Pierre

• dimension estimates
• Talbot effect
• Besov mappings

Statement dc:rights

• Copyright 2022 Chi Huynh

Language dc:language

en, eng

Handle dc:identifier

https://hdl.handle.net/2142/116051

OAI record oai:identifier

https://www.ideals.illinois.edu/oai-pmh?verb=GetRecord&metadataPrefix=etdms&identifier=oai%3Awww.ideals.illinois.edu%3A2142%2F116051