NP #P = ∃PP and other remarks about maximized counting

We consider the following decision problem DMAX#SAT, and generalizations thereof: given a quantifier-free propositional formula F (x, y), where x, y are tuples of variables, and a bound B, determine if there is x such that #{y | F (x, y)} ≥ B. This is the decision version of the problem of MAX#SAT: finding x and B for maximal B