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## The linear bound in $A_2$ for Calderón–Zygmund operators: a survey

### Volume 95 / 2011

Banach Center Publications 95 (2011), 97-114 MSC: Primary 42B20, 42B35; Secondary 47B38. DOI: 10.4064/bc95-0-7

#### 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.

#### Authors

• Michael LaceySchool of Mathematics
Georgia Institute of Technology
Atlanta GA 30332, USA
e-mail

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