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Li, Xiaochun Erdogan, M. Burak Kirr, Eduard-Wilhelm Tyson, Jeremy T. Klamsakul, Natawat 2017-03-01T17:00:50Z 2017-03-01T17:00:50Z 2019-03-02T10:15:27Z 2016-08-25 2016-12 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. Submission published under a 24 month embargo labeled 'Closed Access', the embargo will last until 2018-12-01 The student, Natawat Klamsakul, accepted the attached license on 2016-08-24 at 19:45. The student, Natawat Klamsakul, submitted this Dissertation for approval on 2016-08-24 at 20:13. This Dissertation was approved for publication on 2016-08-25 at 14:16. DSpace SAF Submission Ingestion Package generated from Vireo submission #10125 on 2017-02-28 at 14:40:44 Made available in DSpace on 2017-03-01T17:00:50Z (GMT). No. of bitstreams: 2 KLAMSAKUL-DISSERTATION-2016.pdf: 571204 bytes, checksum: fb134ed373ab251de34ca98aac039de7 (MD5) LICENSE.txt: 4214 bytes, checksum: de0f0165484f93045a6d16eba84d3a9c (MD5) Previous issue date: 2016-08-25 Embargo set by: Seth Robbins for item 98655 Lift date: 2019-03-01T17:02:22Z Reason: Author requested closed access (OA after 2yrs) in Vireo ETD system Embargo set by: Seth Robbins for item 98655 Lift date: 2019-03-01T17:03:32Z Reason: Author requested closed access (OA after 2yrs) in Vireo ETD system Embargo set by: Seth Robbins for item 98655 Lift date: 2019-03-01T17:05:02Z Reason: Author requested closed access (OA after 2yrs) in Vireo ETD system Embargo set by: Seth Robbins for item 98655 Lift date: 2019-03-01T17:06:55Z Reason: Author requested closed access (OA after 2yrs) in Vireo ETD system Limited Restriction Lifted for Item 98655 on 2019-03-02T10:15:27Z. application/pdf http://hdl.handle.net/2142/95539 en Copyright 2016 Natawat Klamsakul Calderon-Zygmund operator T1 theorem Tb theorem time-frequency analysis A look at T1 and Tb theorems on non-homogeneous spaces through time-frequency analysis text text Mathematics Mathematics University of Illinois at Urbana-Champaign Dissertation Ph.D.