Assuming the bilinear reverse Hölder’s condition, we character weighted inequalities for the bilinear maximal operator on filtered measure spaces. We also obtain Hytönen–Pérez 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.

中文翻译：

---

滤波度量空间上双线性极大算子的加权估计

假设双线性逆Hölder条件，我们对滤波后的度量空间上双线性最大算子的加权不等式进行刻画。我们还获得了双线性最大算子的Hytönen-Pérez类型加权估计。我们的方法主要基于主体集的双线性版本的新构造和经过过滤的度量空间上的新的Carleson嵌入定理。特别是，我们发现了结构的新属性，我们称其为条件集的条件稀疏性。