Sparse Domination of the Martingale Transform
Linear operators are of huge importance in modern harmonic analysis. Many operators can be dominated by finitely many sparse operators . The main result in this thesis is showing a toy operator, namely the Martingale Transform is dominated by a single sparse operator . Sparse operators are based on a sparse family which is simply a subset of a dyadic grid. We also show the $A_2$ conjecture for the Martingale Transform which follows from the sparse domination of the Martingale Transform and the $A_2$ conjecture for sparse operators .