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

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