Sharp estimates of martingale transforms in higher dimensions and   applications to the Ahlfors-Beurling operator
  The main aspiration of this note is to construct several different Haar-type systems in euclidean spaces of higher dimensions and prove sharp Lp bounds for the corresponding martingale transforms. In dimension one this was a result of Burkholder. The motivation for working in this direction is the search for Lp estimates of the Ahlfors-Beurling operator. 
Oliver Dragicevic, Stefanie Petermichl, Alexander Volberg