Abstract

Paraproducts can be thought of "parts of a product" of two functions, that isolate particular properties of each of the functions. They have played an essential role in the study of commutators in harmonic analysis, in particular commutators of multiplication by a function and Calder\'{o}n-Zygmund operators. In complex analysis, Hankel and Toeplitz operators can be used to decompose a product of two functions. They play the same role as paraproducts in analyzing commutators of certain operators, so they can be thought of as complex analytic analogues of paraproduct operators. The thesis consists of two parts. In the first part, we study a question motivated by the study of Toeplitz operators in real analysis, and classify the boundedness of a composition of two paraproducts. We also establish weighted bounds for certain compositions of paraproducts. In the second part, motivated by what is known about paraproducts in the real valued setting, we consider the question of two-weight boundedness of Hankel operators, with Muckenhoupt weights. We establish conditions under which a Hankel operator is bounded between two weighted spaces, with possibly different weights.

Committee Chair

Brett Wick

Committee Members

Alan Chang; Cristina Pereyra; Greg Knese; Henri Martikainen

Degree

Doctor of Philosophy (PhD)

Author's Department

Mathematics

Author's School

Graduate School of Arts and Sciences

Document Type

Dissertation

Date of Award

8-5-2025

Language

English (en)

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Mathematics Commons

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