Weighted integral inequalities for the maximal geometric mean operator in martingales

Under appropriate conditions on Young's functions Φ₁ and Φ₂, we give necessary and sufficient conditions in order that weighted integral inequalities hold for the maximal geometric mean operator G in martingale Orlicz classes. When Φ₁=tp and Φ₂=tp, the inequalities revert to the ones of strong or weak (p,p)-type in martingale spaces.