Abstract:

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=M^*₀ holds, and characterize the equivalent relations between the weak or strong type weighted inequality and the property of W_∞-weight or W^*_∞-weight for the geometrical maximal operator in the case of two-weight condition. What should be mentioned is that 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.

Key words: Geometric maximal operator, weight, martingale inequality