Sparse Domination of the Martingale Transform

Michael Scott Kutzler, University of New Mexico

Abstract

Linear operators are of huge importance in modern harmonic analysis. Many operators can be dominated by finitely many sparse operators [5]. The main result in this thesis is showing a toy operator, namely the Martingale Transform is dominated by a single sparse operator [6]. 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 [3].