Weighted bounds for the Variational Carleson Operator - capturing locality using embedding maps into time-frequency space

We will introduce the Variational Carleson Operator and present a novel proof of its boundedness using embedding maps. The correct functional framework for dealing with embedding maps seems to be given by the iterated outer measure L^p spaces.
We will show how iterated outer measure L^p spaces capture the locality of the operator and thus allow to obtain sparse domination results. Sharp Muckenhoupt weighted bounds then follow by well established methods and thus solve a long open problem.