Let
W be a weight, i.e., a uniformly integ
rable, continuous-path martingale, and let
W∗ denote the associated maximal function. We show that if
X is an arbit
rary c&ag
rave;dl&ag
rave;g 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.