Description
- Abstract:
- Harmonic analysis is an artform of understanding how mathematical objects behave. In this thesis, we develop a framework for understanding Calder\'on-Zygmund Singular Integral Operators (CZOs), complicated objects that have intrigued mathematicians since even before the 1950s, by much simpler operators, in a very general setting. CZOs include a multitude of operators present in applications, such as the Hilbert transform, which arises in connections to Fourier series and cosmology. Specifically, in a more general situation than the Euclidean space $\R^n$ called a \emph{space of homogeneous type}, we show that any CZO is bounded above in norm by a supremum of \emph{sparse operators}, which are sums of simple averaging operators over a distinguished collection of cubes called a \emph{sparse family}. This has many applications to the area of weighted norm inequalities, that is, an area involving boundedness questions of weighted extensions of Lebesgue space. In adapting the proof strategy of Andrei Lerner which was used in his simple proof of a long standing conjecture in this area, called the $A_2$ theorem, we are able to not only extend the $A_2$ theorem to spaces of homogeneous type, but to obtain many extensions of results in one and two-weighted norm inequalities. The techniques developed in this thesis allow for great simplification of many existing proofs by using the innovative technology developed first by Lerner, and later in this thesis. Sparse operators and weighted norm inequalities are an active area of research, with new progress disseminated frequently, and our results have aided the rapid development of this field over the past few years.
- Notes:
- Thesis (Ph.D. -- Brown University (2015)
Access Conditions
- Rights
- In Copyright
- Restrictions on Use
- Collection is open for research.
Citation
Anderson, Theresa C.,
"A Framework for Calderon-Zygmund Singular Integral Operators on Spaces of Homogeneous Type"
(2015).
Mathematics Theses and Dissertations.
Brown Digital Repository. Brown University Library.
https://doi.org/10.7301/Z0JS9NT0
