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 $D(u):=\frac{1}{2}\int\_M( |\nabla u|^2+fu)dVol\_g$ 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.


*Remark: I've already asked this question on math.stackexchange.com, but nobody replied. Maybe this question is not so suitable for MO, but I really want to get an answer.*



How to solve the $C^\alpha$ Poisson equation on closed Riemannian manifolds?
Careful. C^\infty is not dense in C^{\alpha}. Nonetheless you can first produce an L^2 solution then use Schauder estimates to show that the solution is C^{2,\alpha}. This last step is localizable -- you can multiply by a smooth cutoff function (and use some preliminary
estimates to say that the solution is C^2, for example) and transfer to a coordinate chart, then apply the estimates and continuity method in Gilbarg&Trudinger. There are other cleaner ways to proceed: the solution u = G f is given by a pseudodifferential operator of order -2 and it is not hard to prove (see Taylor, Vol. III) that such a psido is bounded from
C^\alpha to C^{2,\alpha}.


