A look at T1 and Tb theorems on non-homogeneous spaces through time-frequency analysis

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.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2017

Author dc:creator

• Klamsakul, Natawat

Contributors dc:contributor

• Li, Xiaochun
• Erdogan, M. Burak
• Kirr, Eduard-Wilhelm
• Tyson, Jeremy T.

• Calderon-Zygmund operator
• T1 theorem
• Tb theorem
• time-frequency analysis

Statement dc:rights

• Copyright 2016 Natawat Klamsakul

Language dc:language

en

Handle dc:identifier

http://hdl.handle.net/2142/95539

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/95539