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 the identification of the best constants c(p,q,r) and C-p,C-q,C-r in the estimates parallel to g parallel to(p,infinity) <= c(p,q,r)parallel to f parallel to(q)parallel to v*parallel to(r), parallel to g parallel to(p) <= C-p,C-q,C-r parallel to f parallel to(q)parallel to v*parallel to(r) where f is an arbitrary Hilbert-space valued martingale, g is its transform by a predictable sequence v, and v* is the maximal function of v. This is extended to the more general context of differential subordination for continuous-time processes.