In harmonic analysis, maximal operators are a helpful tool to allow us to prove convergence results for nicer dense subspaces. A famous and reasonably simple example is the maximal function operator, used to prove the famous Lebesgue Differentiation Theorem, and likely to appear in any analysis class. In this talk, we will introduce one of its many cousins - the dyadic maximal function - which behaves very similarly but allows us to decompose our spaces into a "good part" and neatly organized "bad parts".

This is a technique inherent to the study of Calderòn-Zygmund operators (the Hilbert Transform cousins, for those who have heard of this operator) and a crucial piece of the reasonably recent method of Sparse Domination. We hope to introduce a "baby version" of this technique to prove a weighted result for the dyadic maximal function, an argument that can be adapted without much hurdle into a proof of the A2 conjecture for Calderòn-Zygmund operators - a conjecture on the dependence of long-established weighted bounds on the "size" of the weights - in a well-defined A2 characteristic sense.