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

Tom 95 / 2011

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

Streszczenie

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.

Autorzy

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

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