I could not answer or find references of this question, even for the following special case: 


On $S^2$ (the two-sphere equiped with the standard Riemannian metric), is every positive smooth function with integral $1$ the Jacobian of some diffeomorphism?


An equivalent formulation of the question is: On $S^2$, is every positive smooth probability measure the translate of the standard one by some diffeomorphism?



On a compact manifold, what kind of function can be the Jacobian of a diffeomorphism?
Here is an "answer-version" of my comment:


Yes, this is true in general. The reference I know is Moser's 1965 paper "On the volume elements on a manifold" (<http://www.jstor.org/stable/1994022>).


Specifically, let $M$ be a compact connected orientable manifold, and let $\sigma$ and $\tau$ be smooth volume forms on $M$ both with integral 1. Then there exists a diffeomorphism $\varphi:M\to M$ such that $\varphi^\*\tau=\sigma$.


The orientability hypothesis isn't really necessary (just use densities rather than volume forms; see Moser's footnote (2)).


