Restricted projection families and weighted Fourier restriction

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.~θ \in [0,2π) the projection π_θ(A) of $A$ onto the 2-dimensional plane orthogonal to \frac{1}{\sqrt{2}}(\cos θ, \sin θ, 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}.

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

2020

Author dc:creator

• Harris, Terence L. J.

Contributors dc:contributor

• Erdoğan, Burak
• Tzirakis, Nikolaos
• Li, Xiaochun
• Albin, Pierre

• Hausdorff dimension
• Orthogonal projections

Statement dc:rights

• Copyright 2020 Terence Harris

Language dc:language

en

Handle dc:identifier

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

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/107893