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 Title: A look at T1 and Tb theorems on non-homogeneous spaces through time-frequency analysis Author(s): Klamsakul, Natawat Director of Research: Li, Xiaochun Doctoral Committee Chair(s): Erdogan, M. Burak Doctoral Committee Member(s): Kirr, Eduard-Wilhelm; Tyson, Jeremy T. Department / Program: Mathematics Discipline: Mathematics Degree Granting Institution: University of Illinois at Urbana-Champaign Degree: Ph.D. Genre: Dissertation Subject(s): Calderon-Zygmund operator T1 theorem Tb theorem time-frequency analysis Abstract: One of the widely studied topics in singular integral operators is T1 theorem. More precisely, it asks if one can extend a Calder\'on-Zygmund operator to a bounded operator on $L^p$. In addition, Tb theorem was raised when one asks if the T1 theorem remains true if the function $1$ is substituted by some bounded function $b$. In this dissertation, we apply time-frequency analysis to T1 theorem and Tb theorem. In particular, the theory of tiles and trees is used to prove T1 theorem on non-homogeneous spaces. This provides an alternative and a more visualized point of view to some parts of the proof. We also verify estimates from $L^p\times L^q$ to $L^r$ for the paraproducts appeared in T1 theorem. Although the paraproduct is specific, the method is applicable to this kind of study. Lastly, an extension to the proof of Tb theorem is established via a different tree from T1 theorem. Issue Date: 2016-08-25 Type: Thesis URI: http://hdl.handle.net/2142/95539 Rights Information: Copyright 2016 Natawat Klamsakul Date Available in IDEALS: 2017-03-01 Date Deposited: 2016-12
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