Sharp estimates for martingale transforms with unbounded transforming sequences

Suppose that p,q,r≥1 satisfy the condition 1 p=1 q+1 r. The paper contains identification of best constants cp,q,r and Cp,q,r in estimates ‖g‖p,∞≤cp,q,r‖f‖q‖v∗‖ r,‖g‖p≤Cp,q,r‖f‖q‖v∗‖ r where f is an arbitrary Hilbert-space valued martingale, g its transform by a predictable sequence v, v∗ maximal function v. This extended to more general context differential subordination for continuous-time processes.