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.

EDITED:
So having had an answer from Robert Bryant, I realized I should have been more precise about the specific question(s) that I was asking:

<blockquote>
If $M$ is a $C^{1+\alpha}$ manifold and $\omega_i$, $i=1,2$ are two $C^\alpha$
volume forms with the same mass, does there exist a $C^{1+\alpha}$ diffeomorphism sending one the other?
</blockquote>

The comment below from AlvarezPaiva suggests the answer to the above might be yes, but the context there appears to be bounded subsets of $\mathbb R^n$.

<blockquote>
If $M$ is a $C^{1}$ manifold and $\omega_i$, $i=1,2$ are two continuous
volume forms with the same mass, does there exist a $C^1$ diffeomorphism sending one the other?
</blockquote>

Final full disclosure in case this drastically simplifies things: my manifold is topologically a two-dimensional torus.