A theorem of Moser, published in "On the Volume Elements of a Manifold" (Transactions of the Americal Mathematical Society 120, 1965), shows that if a $C^\infty$ compact manifold $M$ has two $C^\infty$ volume forms $\omega_1$ and $\omega_2$ with the same total mass, then there is a diffeomorphism of $M$ sending one to the other.

I am interested in what is known if the manifold and volume forms have lower regularity (in particular, I really want to know about the $C^{1+\alpha}$ case.

Thanks for any reference suggestions.