Restricted projection families and weighted Fourier restriction

Hausdorff dimension;Orthogonal projections

Harris, Terence L. J.

关键词: Hausdorff dimension;    Orthogonal projections;   

Others  :  https://www.ideals.illinois.edu/bitstream/handle/2142/107893/HARRIS-DISSERTATION-2020.pdf?sequence=1&isAllowed=y

美国|英语

来源: The Illinois Digital Environment for Access to Learning and Scholarship

【 摘 要 】

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}.