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Abstract

For an $ L ^2 $-bounded Calderón–Zygmund Operator $ T$ acting on $ L ^2 (\mathbb R ^{d})$, and a weight $ w \in A_2$, the norm of $ T$ on $ L ^2 (w)$ is dominated by $ C _T \Vert w\Vert_{A_2}$. The recent theorem completes a line of investigation initiated by Hunt–Muckenhoupt–Wheeden in 1973 (MR0312139), has been established in different levels of generality by a number of authors over the last few years. It has a subtle proof, whose full implications will unfold over the next few years. This sharp estimate requires that the $ A_2$ character of the weight can be exactly once in the proof. Accordingly, a large part of the proof uses two-weight techniques, is based on novel decomposition methods for operators and weights, and yields new insights into the Calderón–Zygmund theory. We survey the proof of this Theorem in this paper.