An elementary proof of the $A_2$ Bound

A martingale transform $T$, applied to an integrable locally supported function $f$, is pointwise dominated by a positive sparse operator applied to $\lvert f\rvert$, the choice of sparse operator being a function of $T$ and $f$. As a corollary, one derives the sharp $A_p$ bounds for martingale transforms, recently proved by Thiele-Treil-Volberg, as well as a number of new sharp weighted inequalities for martingale transforms. The (very easy) method of proof (a) only depends upon the weak-$L^1$ norm of maximal truncations of martingale transforms, (b) applies in the vector valued setting, and (c) has an extension to the continuous case, giving a new elementary proof of the $A_2$ bounds in that setting.