The sharp weighted bound for general Calderón–Zygmund operators


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.


Tuomas P. Hytönen

P. O. Box 68 (Gustaf Hällströmin katu 2b), FI-00014 University of Helsinki, Finland