Let $(M,g)$ be a smooth oriented closed Riemannian manifold, $E\to M$ a smooth vector bundle, and $C^{k,\alpha}(E)$ the Banach space of sections of $E$ that are $k$-times differentiable (with respect to the Levi-Civita connection for $g$) and whose $k^{th}$ derivatives are all $\alpha$-Holder, $0<\alpha<1$. Let $L: C^\infty(E)\to C^\infty(E)$ be a linear elliptic partial differential operator of order $m$. Is it true that $L$ extends to a Fredholm operator from $C^{k+m,\alpha}\to C^{k,\alpha}$?
I'm under the impression the answer is yes, and I came up with a proof. It's just a slight modification of the proof of Fredholmness for the Sobolev spaces $W^{k,p}$. We want to show the (obviously continuous) extension has (1) finite dimensional kernel , (2) closed range, and (3) finite dimensional cokernel.
The key observation is that for any $r\leq k, \beta> \alpha$, the identity map $C^{k,\alpha}\to C^{r,\beta}$ is compact.
(1) If $u_n$ is a $C^{k+m,\alpha}$-bounded sequence contained in $\ker L$, then by passing to $C^{k,\alpha}$ we extract a subsequence along which it converges in $C^{k,\alpha}$ to a $C^{k,\alpha}$ section $u\in \ker L$. By the Schauder estimates we can control the $C^{k+m,\alpha}$-norm of $u_n$ uniformly by the $C^{k,\alpha}$ norm. This gives convergence in $C^{k+m,\alpha}$. This implies the unit ball of the kernel is compact, and hence this kernel is finite dimensional.
(2) First we prove a Poincare type inequality: let $(\ker L)^\perp$ be the set of $C^{m+k,\alpha}$-sections that are $L^2$-orthogonal to the kernel. Then for any $u\in (\ker L)^\perp$, $$\|u\|_{C^{k,\alpha}}\leq C\|Lu\|_{C^{k,\alpha}}$$ Indeed, if not, we can find a sequence of sections $u_n$ with $\|u_n\|_{k,\alpha}=1$ and $\|Lu_n\|_{k,\alpha}\to 0$. The Schauder estimates then give you a uniform upper bound on $\|u_n\|_{m+k,\alpha}$. Passing to $C^{m+k,\beta}$, for $\beta<\alpha$, the $u_n$ converge along a subsequence in $C^{m+k,\beta}$ to a section that is $L^2$-orthogonal to $\ker L$. However, $Lu=0$, so $u=0$, which is impossible.
With this in mind, let $y_n = Lx_n \in C^{k,\alpha}$ converge to $y\in C^{k,\alpha}$. We may assume $x_n$ is $L^2$ orthogonal to the kernel. By the Schauder estimates and the inequality proved above, $$\|x_n-x_m\|_{m+k,\alpha}\leq C\|y_n-y_m\|_{k,\alpha}$$ and hence the $x_n$ are Cauchy. The result follows.
(3) By compactness, $C^{k,\alpha}$ embeds into $W^{k,p}$ for any $p\geq 1$. $L$ is Fredholm on $W^{k,p}$, so there is a splitting $$W^{k,p}=\ker L^* \oplus \textrm{Im}(L)$$ where the second summand denotes the image of $L: W^{k+m,p} \to W^{k,p}$. We write $u$ uniquely as $$u=u_1+Lu_2$$ with $u_1\in \ker L^*$, $u_2\in W^{k+m,p}$. If $M$ has dimension $n$, then Morrey's inequality gives $u_2\in C^{r,\beta}$ if $$k+ m - \frac{n}{p}> r + \beta$$ Since we can choose $p$ as large as we like, we see $u_2\in C^{k+m-1, \alpha}$. Also, $Lu_2 = u-u_1$ implies $Lu_2\in C^{k,\alpha}$, because $u_1$ is smooth by the Weyl lemma. Appealing to the Schauder estimates once more we see $u_2\in C^{k+m,\alpha}$. This implies we have a splitting $$C^{k,\alpha} = \ker L^* \oplus \textrm{Im}(L)$$ where now the second summand is the image of $L: C^{k+m,\alpha}\to C^{k,\alpha}$. Hence, the cokernel identifies isomorphically with $\ker L^*$, and this is finite dimensional.
Is something in my work incorrect? Is there some obstruction I've missed? I can't see anything wrong with it--I'm just fairly agitated that I've spent a long time looking in texts and online and haven't found what should be a basic fact in any source at all. Thanks.
