The sharp weighted bound for general Calderón-Zygmund operators

For a general Calderon‐Zygmund operator T on R N , it is shown that kTfkL2(w) C(T) sup Q A Q w Q w 1 a k fkL2(w) for all Muckenhoupt weights w 2 A2. This optimal estimate was known as the A2 conjecture. A recent result of Perez‐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 Calderon‐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.

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