Sharp weak-type inequalities for differentially subordinated martingales

Let M, N be real-valued martingales such that N is differentially subordinate to M. The paper contains the proofs of the following weak-type inequalities: (i) If M >= 0 and 0 < p <= 1, then parallel to N parallel to(p,infinity) <= 2 parallel to M parallel to(p) and the constant is the best possible. (ii) If M >= 0 and p >= 2, then parallel to N parallel to(p,infinity) <= p/2(p-1)(-1/p)parallel to M parallel to(p) and the constant is the best possible. (iii) If 1 <= p <= 2 and M and N are orthogonal, then parallel to N parallel to(p,infinity) <= K-p parallel to M parallel to(p), where K-p(p) = 1/Gamma(p+1).(pi/1)(p-1).1 + 1/3(2) + 1/5(2) + 1/7(2) + .../1 - 1/3(p+1) + 1/5(p+1) - 1/7(p+1) + ... The constant is the best possible. We also provide related estimates for harmonic functions on Euclidean domains.