TITLE: How to solve the $C^\alpha$ Poisson equation on closed Riemannian manifolds?
QUESTION [5 upvotes]: To be specific, suppose $M$ is a closed oriented manifold, $g$ is a Riemannian metric of $M$. 
Let $\Delta_g$ be the Laplace-Beltrami operator w.r.t. $g$. 
Prove: Suppose $f\in C^\alpha(M)$ satisfies $\int_M f\, dVol_g=0$, then there exists a function $u\in C^{2,\alpha}(M)$ such that $\Delta_g u=f$ in $M$, and $u$ is unique up to plus a constant, here $0<\alpha<1$.
My attempt is that, firstly use $D(u):=\int_M(\frac{1}{2} |\nabla u|^2+fu)dVol_g$ is a convex functional with a lower bound on $W_0^{1,2}(M)$ to show that there exists a weak solution $u\in W^{1,2}(M)$, next use the $L^2$-regularity theory to show that $u\in W^{2,2}(M)$, but I don't know how to improve the regularity of $u$ further. (Actually, I can use the method to prove that if $f$ is $C^\infty$, then $u$ is also $C^\infty$, but I cannot extend this result to $C^\alpha$ case.)
Another attempt is Schauder estimate. However, in Gilbarg and Trudinger's book they assume that $u\in C^{2,\alpha}(M)$ already to get some interior derivative norm bound of $u$, while I don't know how to establish $u\in C^{2,\alpha}(M)$. They give a continuity method to ensure that, but it seems their discussion works for domains in Euclidean space, not for manifolds. Therefore, I want to split the question into coordinate charts, but I failed, because I don't know how to use the condition $\int_M f\, dVol_g=0$ and how to give boundary conditions in every coordinate charts.
Since I'm a novice in PDE, my presentation of the problem might have some errors. Please correct them by comments or answers. Also, any comments or answers are welcome.
Thanks for your help.

REPLY [2 votes]: I think I've got an answer to this question.
First, use the functional $D(u)=\int_M(\frac{1}{2}|\nabla u|^2+fu)dVol_g$, we can find a weak solution $u\in W^{1,2}_0(M)$, since $f\in C^\alpha(M)$ hence in $L^2(M)$. Next we shall show that $u\in C^{2,\alpha}(M)$.
We cite theorem 6.14 from Gilbarg-Trudinger's book:

Let $L$ be strictly elliptic in a bounded domain $\Omega$, with $c\leq0$, and let $f$ and the coefficients of $L$ belong to $C^\alpha(\overline{\Omega})$. Suppose that $\Omega$ is a $C^{2,\alpha}$ domain and that $\phi\in C^{2,\alpha}(\overline\Omega)$. Then the Dirichlet problem,
  $$
Lu=f \textrm{ in }\Omega,\quad u=\phi\textrm{ on }\partial\Omega,
$$
  has a unique solution lying in $C^{2,\alpha}(\overline\Omega)$.

The proof relies on the solvability of classical Dirichlet problem of laplacian, continuity method and Schauder estimates.
Suppose $U$ is a coordinate chart s.t. $\overline{U}$ is diffeomorphic to a closed ball, then by the theorem cited above, $\Delta_g v=f$ in $U$, $v = 0$ on $\partial U$ is solvable, and $v\in C^{2,\alpha}(\overline{U})$. Therefore, $u-v$ is a weak solution of $\Delta_g (u-v)=0$. By $L^2$-regularity, we have $u-v\in C^\infty(U)$, thus $u\in C^{2,\alpha}(U)$. Since we can choose finitely many $U$ to cover $M$, finally we have $u\in C^{2,\alpha}(M)$.