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.



Schauder estimates for higher order linear elliptic operator on manifold
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](http://www3.nd.edu/~lnicolae/Lectures.pdf) and the references therein.


