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

Abstract: 

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.