Weighted inequalities for the martingale square and maximal functions

Let W be a weight, i.e., a uniformly integrable, continuous-path martingale, and let W∗ denote the associated maximal function. We show that if X is an arbitrary càdlàg martingale and X∗, [X] denote its maximal and square functions, then ‖[X]1/2‖Lp(W)≤γp‖X∗‖Lp(W∗),1≤p≤2, where γp2=1+supt>1(2t−1)(1−tp−2)tp−1. The estimate is sharp for p∈{1,2}. Furthermore, it is proved that if p>2, then the above weighted inequality does not hold with any finite constant γp depending only on p.