Suppose $Q:[0,1]\to [0,1]$ is given by a nondecreasing step function
$$Q(x)=A, if \phantom{0}0\leq x < x^*$$
$$\phantom{0000} = B, if\phantom{0} x^*\leq x\leq 1 $$
s.t. $$A,B\in[0,1] \phantom{0}and \phantom{0} A\leq B$$
where $x^*\in(0,1)$ is given.
Suppose $F:[0,1]\to R_+$ is a c.d.f. What is the necessary and sufficient condition on $(A,B,x^*,F)$ such that there exists $q:[0,1]^2\to [0,1]$ satisfies
$$\int_{[0,1]}q(x,y)dF(y)=Q(x)$$
$$\int_{[0,1]}q(x,y)dF(x)=Q(y)$$
?
Thanks.