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 Title: Restricted projection families and weighted Fourier restriction Author(s): Harris, Terence L. J. Director of Research: Erdoğan, Burak Doctoral Committee Chair(s): Tzirakis, Nikolaos Doctoral Committee Member(s): Li, Xiaochun; Albin, Pierre Department / Program: Mathematics Discipline: Mathematics Degree Granting Institution: University of Illinois at Urbana-Champaign Degree: Ph.D. Genre: Dissertation Subject(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}. Issue Date: 2020-04-21 Type: Thesis URI: http://hdl.handle.net/2142/107893 Rights Information: Copyright 2020 Terence Harris Date Available in IDEALS: 2020-08-26 Date Deposited: 2020-05
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