We characterize strong type and weak type inequalities with two weights for positive operators on filtered measure spaces. These estimates are probabilistic analogues of two-weight inequalities for positive operators associated to the dyadic cubes in $\mathbb R^n$ due to Lacey, Sawyer and Uriarte-Tuero . Several mixed bounds for the Doob maximal operator on filtered measure spaces are also obtained. In fact, Hytönen–Pérez type and Lerner–Moen type norm estimates for Doob maximal operator are established. Our approaches are mainly based on the construction of principal sets.
The research of the first author is supported by the National Natural Science Foundation of China (11971419, 11771379), the Natural Science Foundation of Jiangsu Province (BK20161326), and the Jiangsu Government Scholarship for Overseas Studies (JS-2017-228). The research of the fourth author is supported by the National Natural Science Foundation of China (Grant No. 11471337).
"Two-weighted estimates for positive operators and Doob maximal operators on filtered measure spaces." J. Math. Soc. Japan 72 (3) 795 - 817, July, 2020. https://doi.org/10.2969/jmsj/80058005