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 A1 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
Milestones
Received: 31 July 2017
Revised: 10 February 2018
Accepted: 10 April 2018
Published: 6 June 2018
Authors
Fedor Nazarov
Department of Mathematical Sciences
Kent State University
Kent, OH
United States
Alexander Reznikov
Department of Mathematics
Michigan State University
East Lansing, MI
United States
Vasily Vasyunin
V. A. Steklov Mathematical Institute
St. Petersburg
Russia
Alexander Volberg
Department of Mathematics
Michigan State University
East Lansing, MI
United States