The sharp weighted bound for general Calderón-Zygmund operators. (English) Zbl 1250.42036

Authors’ abstract: “For a general Calderón-Zygmund operator \(T\) on \(\mathbb{R}^N\), it is shown that \[ \|{Tf}\|_{L^2(w)}\leq C(T)\cdot\sup_Q\Big(\int_Q w\cdot \int_Q w^{-1}\Big)\cdot\|{f}\|_{L^2(w)} \] 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.”


42B10 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
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