Weighted Inequalities for the Geometric Maximal Operator on Martingale Spaces

In this article, the authors introduce two operators—geometrical maximal operator M0 and the closely related limiting operator M*0, then they give sufficient conditions under which the equality M0 = M0holds, and characterize the equivalent relations between the weak or strong type weighted inequality and the property ofW∞-weight or W*∞-weight for the geometrical maximal operator in the case of two-weight condition. What should be mentioned isthat the new operator—the geometrical minimal operator is equal to the limitation of the minimal operator sequence, and the results for the minimal operator proved in [12] makes the proof elegant and evident.