A sharp estimate of weighted dyadic shifts of complexity 0 and 1
  A simple shortcut to proving sharp weighted estimates for the Martingale Transform and for the dyadic shift of order 1 (and so for the Hilbert transform) is presented. It is a unified proof for these both transforms.   Key words: Calder\on--Zygmund operators, $A_2$ weights, $A_1$ weights, Carleson embedding theorem, Bellman function, dyadic shifts, nonhomogeneous Harmonic Analysis. 
Alexander Reznikov, Sergei Treil, Alexander Volberg
math.CA math.AP