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. Geom. Funct. Anal. 8 (1998), 273–282. (MathSciNet review.)

[2] G. Csató, B. Dacorogna, O. Kneuss, The pullback equation for differential forms. Progress in Nonlinear Differential Equations and their Applications, 83. Birkhäuser/Springer, New York, 2012. (MathSciNet review.)

[3] B. Dacorogna, J. Moser, On a partial differential equation involving the Jacobian determinant. Ann. Inst. H. Poincaré Anal. Non Linéaire 7 (1990), 1–26. (MathSciNet review.)

I think that the usual proof goes through in this case, although, obviously, you don't get a diffeomorphism (i.e., $C^\infty$ invertible map) identifying the two volume forms, just a $C^{1+\alpha}$ map with a $C^{1+\alpha}$ inverse.

Look at the steps: First, you need to find an $(n-1)$-form $\phi$ such that $\omega_2-\omega_1 = \mathrm{d}\phi$, and you should make sure that it is at least $C^{1+\alpha}$. To do this, you note that $\omega_2-\omega_1$ is zero in deRham cohomology (this is the 'equal mass' hypothesis; of course, you need to assume that $M$ is connected for this to work, but that's part of the hypothesis anyway), and then use, say, a Green's operator (which, at least, doesn't decrease regularity) for some smooth metric to write $$ \omega_2-\omega_1 = \mathrm{d}\left(G(\omega_2{-}\omega_1)\right), $$ then take $\phi = G(\omega_2{-}\omega_1)$. Second, on $M\times [0,1]$ (with $t$ as the coordinate on the second factor), consider the $n$-form (which is $C^{1+\alpha}$) $$ \omega = (1{-}t)\,\omega_1 + t\,\omega_2 + \mathrm{d}t\wedge\phi. $$ This form satisfies $\mathrm{d}\omega = 0$ by construction, and it is never vanishing since $\omega_1$ and $\omega_2$ determine the same orientation of $M$. Third, there is a unique vector field $X$ on $M\times[0,1]$ that satisfies $$ \iota_X\left(\mathrm{d}t\wedge\omega\right) = \omega, $$ where $\iota_X$ means interior product with $X$. This vector field satisfies $\mathrm{d}t(X) \equiv 1$, so we can look at the time $1$ flow of this vector field, which carries $M\times\{0\}$ to $M\times\{1\}$. Fourth, since $\omega$ is closed and since $\iota_X(\omega) = 0$, it follows from Cartan's formula that the Lie derivative of $\omega$ with respect to $X$ is zero, i.e., that the flow of $X$ preserves $\omega$.

But now, the time $1$ flow of $X$ (which is a $C^{1+\alpha}$ vector field) is then a $C^{1+\alpha}$ map (with $C^{1+\alpha}$ inverse) from $M$ to $M$ that pulls back $\omega_2$ to $\omega_1$. This is because $\omega$ pulls back to $M\times\{0\}$ to be $\omega_1$ and it pulls back to $M\times\{1\}$ to be $\omega_2$.