An elliptic regularity statement taken from Joyce: Compact Manifolds with Special Holonomy (Theorem 1.4.1) reads:
Suppose $M$ is a compact Riemannian manifold, $V$ and $W$ are vector bundles over $M$ of the same dimension, and $P$ is a smooth, linear, elliptic differential operator of order $k$ from $V$ to $W$. Let $p>1$ and $l \geq 0$ be an integer.
Suppose that $P(v)=w$ holds weakly, with $v \in L^1_p(V)$ and $w \in L^1(W)$. If $w \in L^p_l(W)$ then $v \in L^p_{k+1}(V)$, and $\left| \left| v \right| \right| _{L^p_{k+l}} \leq C ( \left| \left| w \right| \right| _{L^p_{l}} + \left| \left| v \right| \right| _{L^1} ) $ for some $C>0$ independent of $v$, $w$.
Question:
Does this statement hold in the case "$p=\infty$"? I.e., do I have a $C$ such that the following inequality holds? $\left| \left| v \right| \right| _{C^{k+l}} \leq C ( \left| \left| w \right| \right| _{C^l} + \left| \left| v \right| \right| _{C^0} ) $.
The answer to the question is "No", and a counterexample can be found in Section 2.2 of Xavier Fernandez-Real, Xavier Ros-Oton: Regularity Theory for Elliptic PDE. Here is a direct link and an alternative link.
Edit: if the operator is of order 2, then one has $$||u||_{C^1,B_1} \leq ||u||_{C^{1,\alpha},B_1} \leq C(||Pu||_{C^0,B_2}+||u||_{C^0,B_2}),$$ where $B_1$ and $B_2$ denotes the balls of radius 1 and 2 respectively in $\mathbb{R}^n$. This follows from the $L^p$-theory and the Sobolev embedding theorems. It is taken from Theorem 1.4.3 in Joyce: Compact Manifolds with Special Holonomy, who wrongly cites that as Thm 1.5.1 in Morrey: Multiple Integrals in the Calculus of Variations. Correct is Thm 5.5.5 therein.