Abstract
For a general Calderón–Zygmund operator T on ℝN, it is shown that
∥Tf∥L2(w)≤C(T)⋅supQ(∫Qw⋅∫Qw−1)⋅∥f∥L2(w)
for all Muckenhoupt weights w∈A2. This optimal estimate was known as the A2 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.

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