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Some results in harmonic analysis related to pointwise convergence and maximal operators

Bailey, Andrew David (2012)
Ph.D. thesis, University of Birmingham.

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Abstract

Pointwise convergence problems are of fundamental importance in harmonic analysis and studying the boundedness of associated maximal operators is the natural viewpoint from which to consider them. The first part of this two-part thesis pertains to Lennart Carleson’s
landmark theorem of 1966 establishing almost everywhere convergence of Fourier series for functions in L\(^2\)(\(\char{bbold10}{0x54}\)). Here, partial progress is made towards adapting the time-frequency analytic proof of Carleson’s result by Michael Lacey and Christoph Thiele to bound an almost
periodic analogue of Carleson’s maximal operator for functions in the Besicovitch space B\(^2\). A model operator of the type of Lacey and Thiele is formed and shown to relate to Carleson’s operator in a natural way and be susceptible to a similar kind of analysis.

In the second part of this thesis, recent work of Per Sjölin and Fernando Soria is improved, with precise boundedness properties determined for the Schrödinger maximal operator with complex-valued time as a special case of more general estimates for a family of maximal operators associated to dispersive partial differential equations. Boundedness properties of other maximal operators naturally related to the Schrödinger maximal operator are also established using similar techniques.

Type of Work:Ph.D. thesis.
Supervisor(s):Bennett, Jonathan and Bez, Neal and Cowling, Michael
School/Faculty:Colleges (2008 onwards) > College of Engineering & Physical Sciences
Department:School of Mathematics
Additional Information:

Andrew Bailey's MPhil Thesis can be found at
http://etheses.bham.ac.uk/268/

Subjects:QA Mathematics
Institution:University of Birmingham
ID Code:3373
This unpublished thesis/dissertation is copyright of the author and/or third parties. The intellectual property rights of the author or third parties in respect of this work are as defined by The Copyright Designs and Patents Act 1988 or as modified by any successor legislation. Any use made of information contained in this thesis/dissertation must be in accordance with that legislation and must be properly acknowledged. Further distribution or reproduction in any format is prohibited without the permission of the copyright holder.
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