Schauder estimates for higher order linear elliptic operator on manifold Hi!
Let $(M,g)$ be a smooth compact riemannian manifold without boundary. Let $L$ be a linear elliptic operator on $M$ of order $2k$ with smooth coefficients. Suppose i have $u\in W^{2k,2}(M)$ and $f\in C^{0,\alpha}(M)$ such that
$$L(u)=f$$
Do i have Schauder estimates of type
$$\left\|u\right\|_{C^{2k,\alpha}\left(M\right)}\leq C\left(L\right)\left\|f\right\|_{C^{0,\alpha}\left( M \right)}$$
I can assume also that $L$ (it would be better for $L$ of general type) is self adjoint and $u$ is $L^2$-orthogonal to $\ker\left(L \right)$.
If yes is there a reference for this kind of result?
Thank you in advance.
 A: The result is true with some caveats. Under your assumptions we have the following results.
1. If $u\in W^{2k,2}(M)$ and $Lu\in C^{j,\alpha}(M)$, then $u\in C^{2k+j,\alpha}(M)$.
2.  There exists $C>0$ depending only on $M$, $L$, $j$, and $\alpha$ such that, for any  $u\in C^{2k+j,\alpha}(M)$ we have
$$\Vert u\Vert_{C^{2k+j,\alpha}}  \leq C\Bigl(\; \Vert Lu\Vert_{C^{j,\alpha}}+ \Vert u\Vert\_{C^{0,\alpha}}\;\Bigr) $$
3. There exists $C>0$ depending only on $M$, $L$, $j$, and $\alpha$ such that, for any  $u\in C^{2k+j,\alpha}(M)\cap (\ker L)^\perp $  (the $\perp$ refers to the $L^2$-inner product)   we have
$$\Vert u\Vert_{C^{2k+j,\alpha}}  \leq C \Vert Lu\Vert_{C^{j,\alpha}} $$
For proofs and more details see Chapter 10 of these notes and the references therein.
A: This is probably more of a comment: Section 3.2 in Lunardi's book contains a broad overview on higher order parabolic problems with lots of references, including the most important Hölder estimates. 
On manifolds, you should be able to extend these results using a finite number of coordinate charts (by compactness) as in these notes. But, this is not a reference...
