#### Vol. 11, No. 8, 2018

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On weak weighted estimates of the martingale transform and a dyadic shift

### Fedor Nazarov, Alexander Reznikov, Vasily Vasyunin and Alexander Volberg

Vol. 11 (2018), No. 8, 2089–2109
DOI: 10.2140/apde.2018.11.2089
##### Abstract

We consider weak-type estimates for several singular operators using the Bellman-function approach. In particular, we consider a concrete dyadic shift. We disprove the ${A}_{1}$ conjecture for those operators, which stayed open after Muckenhoupt and Wheeden’s conjecture was disproved by Reguera and Thiele.

##### Keywords
martingale transform, weak weighted estimate
##### Mathematical Subject Classification 2010
Primary: 42A45, 42A61, 42B20, 42B35, 42B37, 47A30
Secondary: 42A50, 49L20, 49L25