Restricted projection families and weighted Fourier restriction

Harris, Terence L. J.

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https://hdl.handle.net/2142/107893 Copy

Description

Title

Restricted projection families and weighted Fourier restriction

Author(s)

Harris, Terence L. J.

Issue Date

2020-04-21

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

Erdoğan, Burak

Doctoral Committee Chair(s)

Tzirakis, Nikolaos

Committee Member(s)

• Li, Xiaochun
• 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)

• Hausdorff dimension
• Orthogonal projections

Abstract

In the first part of this thesis, it is shown that if $A \subseteq \mathbb{R}^3$ is a Borel set of Hausdorff dimension $\dim A > 3/2$, then for a.e.~$\theta \in [0,2\pi)$ the projection $\pi_{\theta}(A)$ of $A$ onto the 2-dimensional plane orthogonal to $\frac{1}{\sqrt{2}}(\cos \theta, \sin \theta, 1)$ satisfies \[ \dim \pi_{\theta}(A) \geq \min\left\{\frac{4\dim A}{9} + \frac{5}{6},2 \right\}. \] This improves the bound of Oberlin and Oberlin \cite{oberlin}, and of Orponen and Venieri \cite{venieri}, for $\dim A \in (1.5,2.4)$. In the second part, an improved lower bound is given for the decay of conical averages of Fourier transforms of measures, for cones of dimension $d \geq 4$. The proof uses a weighted version of the broad restriction inequality, a narrow decoupling inequality for the cone, and some techniques of Du and Zhang \cite{zhang} originally developed for the Schrödinger equation. Most of the work in this thesis was published by the author in different forms in \cite{THarris1} and \cite{THarris3}.

Graduation Semester

2020-05

Type of Resource

Thesis

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http://hdl.handle.net/2142/107893

Copyright and License Information

Copyright 2020 Terence Harris

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