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
where
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.