Time-Frequency Analysis of the Variational Carleson Operator using outer-measure <em>L^p</em> spaces

Abstract

In this thesis we are concerned with developing a systematic framework for dealing with the Carleson and the Variational Carleson Operators. We extend the framework of outer measure Lp spaces of Do and Thiele to an iterated version. The first chapter contains an intuitive introduction and a simplified dyadic model for the results of the rest of the thesis.<br /> We formulate embedding maps into time-frequency space and we obtain bounds for these embedding maps in terms of iterated outer Lp spaces, the main novelty of this thesis.<br /> Finally we show how the above embedding bounds can be used to easily obtain sharp weighted estimates for the Carleson operator and its variational counterpart.