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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 UrbanaChampaign 
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 2dimensional 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:  20200421 
Type:  Thesis 
URI:  http://hdl.handle.net/2142/107893 
Rights Information:  Copyright 2020 Terence Harris 
Date Available in IDEALS:  20200826 
Date Deposited:  202005 
This item appears in the following Collection(s)

Graduate Dissertations and Theses at Illinois
Graduate Theses and Dissertations at Illinois