## Arts & Sciences Electronic Theses and Dissertations

#### ORCID

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

Spring 5-15-2021

#### Author's School

Graduate School of Arts and Sciences

Mathematics

#### Degree Name

Doctor of Philosophy (PhD)

Dissertation

#### 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

English (en)

Brett D. Wick

#### Committee Members

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

COinS

#### DOI

https://doi.org/10.7936/f5d0-7128