A theorem of Moser, published in "On the Volume Elements of a Manifold" (Transactions of the Americal Mathematical Society 120, 1965; [doi: 10.1090/S0002-9947-1965-0182927-5](https://doi.org/10.1090/S0002-9947-1965-0182927-5), [jstor](http://www.jstor.org/stable/1994022)), 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:



> 
> 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?
> 


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



> 
> 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 to the other?
> 


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



Lower regularity version of Moser's theorem on volume elements

> 
> 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 to the other?
> 


**The answer is no.** The equivalent problem is: given a positive and continuous function $g$, can we find a diffeomorphism with the Jacobian equal to $g$, $\det Df=g$? A counterexample is Theorem 1.2 in [1].



> 
> 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?
> 


**The answer is yes, at least locally** (see also a comment of alvarezpaiva). Whether the result has a global version on manifolds, I do not know. This is Theorem 1 in [3]. For a comprehensive treatment of related results, see [2].



> 
> **Theorem.** *Let $k\geq 0$ and $\alpha\in (0,1)$. If $\Omega\subset\mathbb{R}^n$ is abounded domain with $C^{k+3,\alpha}$
>  boundary and $\omega\in C^{k,\alpha}$ is a volume form such that
>  $\int\_\Omega\omega=|\Omega|$, then there is a diffeomorphism
>  $\varphi:\Omega\to\Omega$ that is identity on the boundary,
>  $\varphi,\varphi^{-1}\in C^{k+1,\alpha}(\bar{\Omega})$ and such that
>  $\varphi^\*\omega=dx\_1\wedge\ldots\wedge dx\_n$.*
> 
> 
> 


[1] **D. Burago, B. Kleiner,** [Separated nets in Euclidean space and Jacobians of bi-Lipschitz maps](https://link.springer.com/article/10.1007%2Fs000390050056).
*Geom. Funct. Anal.* 8 (1998), 273–282. 
([MathSciNet review](https://mathscinet.ams.org/mathscinet/search/publdoc.html?arg3=&co4=AND&co5=AND&co6=AND&co7=AND&dr=all&pg4=AUCN&pg5=TI&pg6=RT&pg7=JOUR&pg8=ET&review_format=html&s4=burago%20and%20kleiner&s5=&s6=&s7=&s8=All&sort=Newest&vfpref=html&yearRangeFirst=&yearRangeSecond=&yrop=eq&r=3&mx-pid=1616135).)


[2] **G. Csató, B. Dacorogna, O. Kneuss,** [The pullback equation for differential forms](https://link.springer.com/book/10.1007/978-0-8176-8313-9). Progress in Nonlinear Differential Equations and their Applications, 83. Birkhäuser/Springer, New York, 2012. 
([MathSciNet review](https://mathscinet.ams.org/mathscinet-getitem?mr=2883631%0A).)


[3] **B. Dacorogna, J. Moser,**
[On a partial differential equation involving the Jacobian determinant](http://caa.epfl.ch/publications/22-Dacorogna-Moser1990.pdf). 
*Ann. Inst. H. Poincaré Anal. Non Linéaire* 7 (1990), 1–26. 
([MathSciNet review](https://mathscinet.ams.org/mathscinet/search/publdoc.html?arg3=&co4=AND&co5=AND&co6=AND&co7=AND&dr=all&pg4=AUCN&pg5=TI&pg6=RT&pg7=JOUR&pg8=ET&r=1&review_format=html&s4=dacorogna%20and%20moser&s5=&s6=&s7=&s8=All&sort=Newest&vfpref=html&yearRangeFirst=&yearRangeSecond=&yrop=eq).)


