Thesis Defense: Haar Multipliers in the Context of Weighted Spaces and Bilinear Operators

Claire Huang, Washington University in St. Louis


This thesis offers a deep dive into Haar multipliers, focusing on their behavior in weighted spaces, two-weight inequalities, and spaces of homogeneous type. Chapter 1 introduces the main themes and summarises subsequent chapters. Chapter 2 provides necessary background information and notation, building a strong foundation for the core discussions that follow.

In Chapter 3, the focus is on a Haar multiplier, $T_w^v$, defined on a pair of weights. The result is a characterization of the conditions under which this Haar multiplier is bounded in Lebesgue spaces.

Chapter 4 extends the previous discussions into the realm of $t$-Haar multipliers, establishing a two-weight theorem characterizing their boundedness. This chapter builds on the work of leading researchers, such as Nazarov, Treil, and Volberg.

Finally, Chapter 5 probes bilinear Haar multipliers in the context of spaces of homogeneous type, highlighting their boundedness properties and classifying them as bilinear Calderón-Zygmund operators. The thesis contributes significantly to the harmonic analysis field and paves the way for future research.

Hosted by Professor Wick, Dean’s Fellow for Digital Transformation and Professor of Mathematics.