Abstract

One the most central questions in harmonic analysis of whether the Fourier series of a square integrable function on the torus $\mathbb T$ converges Lebesgue a.e.\ $ x\in\mathbb T$ was answered positively by L.\ Carleson in 1966 \cite{C}, by means of a weak-$L^2$ inequality for the maximal operator \begin{equation} \label{carleson0} \mathcal C f(x) =\sup_{N\in \mathbb Z} \Bigg| \sum_{|\xi|\leq N} \widehat f(\xi) \exp(ix\xi) \Bigg|, \qquad x\in \mathbb T. \end{equation} The argument of \cite{C} estimates $\mathcal C $ pointwise as a maximal modulated Hilbert transform, outside appropriately constructed exceptional sets whose mass is controlled by almost-orthogonality. The implicit distributional estimate in \cite{C} was later exploited by Hunt \cite{H} to deduce the family of restricted weak-type $L^p$ bounds \begin{equation} \label{e:ch} \left\|\mathcal{C}f\right\|_{L^{p,\infty}(\mathbb T)} \leq \frac{Cp^2}{p-1}|F|^{\frac1p}, \qquad F\subset \mathbb T,\; |f|\leq \cic{1}_F, \qquad 1

Committee Chair

Brett Wick

Degree

Doctor of Philosophy (PhD)

Author's Department

Mathematics

Author's School

Graduate School of Arts and Sciences

Document Type

Dissertation

Date of Award

6-26-2024

Language

English (en)

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Mathematics Commons

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