Positive operators and maximal operators in a filtered measure space

In a filtered measure space, a characterization of weights for which the trace inequality of a positive operator holds is given by the use of discrete Wolffʼs potential. A refinement of the Carleson embedding theorem is also introduced. Sawyer type characterization of weights for which a two-weight norm inequality for a generalized Doobʼs maximal operator holds is established by an application of our Carleson embedding theorem. Moreover, Hytönen–Pérez type one-weight norm estimate for Doobʼs maximal operator is obtained by the use of our two-weight characterization.