The sharp weighted bound for general Calderón–Zygmund operators

Abstract

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.