Lp and weak L1 estimates for the maximal Riesz transform and the maximal   Beurling transform
  We estimate in Lp the maximal Riesz transform in terms of the Riesz transform itself for p greater than 1. In the limiting case p=1 the weak L1 inequality is shown to fail. Surprisingly, the weak L1 inequality for the maximal Beurling transform in terms of the Beurling transform does hold. 
Joan Mateu and Joan Verdera