Geodesic
The sharp weighted bound for general Calderón–Zygmund operators
Tuomas P. Hytönen1
1 P. O. Box 68 (Gustaf Hällströmin katu 2b), FI-00014 University of Helsinki, Finland
Annals of mathematics, Tome 175 (2012) no. 3, pp. 1473-1506.

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For a general Calderón–Zygmund operator $T$ on $\Bbb{R}^N$, it is shown that $$ \Vert{Tf}\Vert{L^2(w)}\leq C(T)\cdot\sup_Q\Big(∫_Q w\cdot ∫_Q w^{-1}\Big)\cdot\Vert{f}\Vert{L^2(w)} \end{equation*} for all Muckenhoupt weights $w\in A_2$. This optimal estimate was known as the $A_2$ conjecture. A recent result of Pérez–Treil–Volberg reduced the problem to a testing condition on indicator functions, which is verified in this paper.
The proof consists of the following elements: (i) a variant of the Nazarov–Treil–Volberg method of random dyadic systems with just one random system and completely without “bad” parts; (ii) a resulting representation of a general Calderón–Zygmund operator as an average of “dyadic shifts;” and (iii) improvements of the Lacey–Petermichl–Reguera estimates for these dyadic shifts, which allow summing up the series in the obtained representation.
DOI : 10.4007/annals.2012.175.3.9

Tuomas P. Hytönen 1

1 P. O. Box 68 (Gustaf Hällströmin katu 2b), FI-00014 University of Helsinki, Finland 
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Tuomas P. Hytönen. The sharp weighted bound for general Calderón–Zygmund operators. Annals of mathematics, Tome 175 (2012) no. 3, pp. 1473-1506. doi : 10.4007/annals.2012.175.3.9. http://geodesic.mathdoc.fr/articles/10.4007/annals.2012.175.3.9/

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