Thesis Defense: Some problems in Harmonic Analysis

Abstract: 

This PhD thesis presentation will move across three different axes.

First, we will talk about modulation invariant operators, and explain how the study of such objects connects to the pointwise convergence of Fourier series as well as the importance of their near $L^1$ behavior. We will also report on recent progress in this topic based on joint work with Francesco Di Plinio.

Second, we will discuss about the compactness of multilinear Calder\’{o}n-Zygmund operators. In particular, we will provide a characterization of compactness in terms of wavelet type testing conditions. Our approach yields a representation theorem for compact CZ forms in terms of wavelet and paraproduct forms that reflect the compact nature of the operator. This part of the presentation stems from joint work with Walton Green and Brett Wick.

Finally, we will be concerned with estimates of commutators of singular integral operators. In further detail, we will revisit Bloom’s original inequality as well as all known one-weight off-diagonal results for cancellative Calder\’{o}n-Zygmund operators via the wavelet representation theorem devised by Di Plinio, Wick and Williams.

Advisor: Brett Wick