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

Huynh, Chi Ngoc Yen

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

Description

Title

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

Author(s)

Huynh, Chi Ngoc Yen

Issue Date

2022-07-06

Director of Research (if dissertation) or Advisor (if thesis)

• Erdogan, M. Burak
• Tyson, Jeremy

Doctoral Committee Chair(s)

Tzirakis, Nikolaos

Committee Member(s)

Albin, Pierre

Department of Study

Mathematics

Discipline

Mathematics

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(s)

• dimension estimates
• Talbot effect
• Besov mappings

Abstract

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 $\sph^2$ 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 $\sph^2$, 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 $\R^n$. Given $f: \Omega \to \R^k$ where $n,k \in \N$, $\Omega \subseteq \R^n$, 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.

Graduation Semester

2022-08

Type of Resource

Thesis

Copyright and License Information

Copyright 2022 Chi Huynh

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