Abstract

Let $(X,d,\mu )$ be a space of homogeneous type in the sense of Coifman andWeiss, i.e. $d$ is a quasi metric on $X$ and $\mu $ is a nonzero measure satisfying the doubling condition. Suppose that $u$ and $v$ are two locally finite positive Borel measures on $(X,d,\mu )$, the two weight inequality for the Calder\'on-Zygmund operator is of the form \begin{align*} & \|T(f\cdot u)\|_{L^2(v)}\lesssim \|f\|_{L^2(u)}. \end{align*} Subject to the pair of weights satisfying a side condition, we have given a characterization of the boundedness of a Calder\'{o}n-Zygmund operator $T$ from $L^{2}(u)$ to $L^{2}(v)$ in terms of the $A_{2}$ condition and two testing conditions. The proof uses stopping intervals and corona decompositions originating in work of Nazarov, Treil and Volberg, along with the pivotal side condition. \vspace{0.2 cm}We also give the two weight quantitative estimates for the commutator of maximal functions and the maximal commutators with respect to the symbol in weighted BMO space on spaces of homogeneous type. These commutators turn out to be controlled by the sparse operators in the setting of space of homogeneous type. The lower bound of the maximal commutator is also obtained.\vspace{0.2 cm}In continuation we also obtain the boundedness and compactness characterizations of the commutator of Calder\'{o}n-Zygmund operators $T$ on spaces of homogeneous type $(X,d,\mu)$ in the sense of Coifman and Weiss. More precisely, we show that the commutator $[b, T]$ is bounded on weighted Morrey space $L_{\omega}^{p,\kappa}(X)$ ($\kappa\in(0,1), \omega\in A_{p}(X), 1

Committee Chair

Brett D. Wick

Committee Members

Francesco Di Plinio, Gregory Knese, John E. McCarthy, Elodie Pozzi,

Degree

Doctor of Philosophy (PhD)

Author's Department

Mathematics

Author's School

Graduate School of Arts and Sciences

Document Type

Dissertation

Date of Award

Spring 5-15-2021

Language

English (en)

Author's ORCID

http://orcid.org/0000-0001-9194-5250

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Mathematics Commons

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