A look at T1 and Tb theorems on non-homogeneous spaces through time-frequency analysis
Klamsakul, Natawat
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https://hdl.handle.net/2142/95539
Description
Title
A look at T1 and Tb theorems on non-homogeneous spaces through time-frequency analysis
Author(s)
Klamsakul, Natawat
Issue Date
2016-08-25
Director of Research (if dissertation) or Advisor (if thesis)
Li, Xiaochun
Doctoral Committee Chair(s)
Erdogan, M. Burak
Committee Member(s)
Kirr, Eduard-Wilhelm
Tyson, Jeremy T.
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(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.
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