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?

P.S. I'm sorry but for some mysterious reason the estimate doesn't appear, if someone is able to fix it i'd be very grateful.

Thank you in advance.

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.