Weighted Inequalities for the Generalized Maximal Operator in Martingale Spaces

The generalized maximal operator M in martingale spaces is considered. For 1 < p ≤ q < ∞, the authors give a necessary and sufficient condition on the pair (^(m)]\hat \mu , v) for M to be a bounded operator from martingale space L ^p (v) into L ^q (^(m)]\hat \mu ) or weak-L ^q (^(m)]\hat \mu ), where ^(m)]\hat \mu is a measure on Ω × ℕ and v a weight on Ω. Moreover, the similar inequalities for usual maximal operator are discussed.