Weighted estimates for the bilinear maximal operator on filtered measure spaces

Assuming the bilinear reverse Holder's condition, we character weighted inequalities for the bilinear maximal operator on filtered measure spaces. We also obtain Hytonen-Perez type weighted estimates for the bilinear maximal operator. Our approaches are mainly based on the new construction of bilinear versions of principal sets and the new Carleson embedding theorem on filtered measure spaces. In particular, we find a new property of the construction and we call it the conditional sparsity of principal sets.