NP$^{\#P}$ = $\exists$PP and other remarks about maximized counting

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