Let $E, F \rightarrow M$ be two smooth vector bundles over a closed smooth manifold $M$, and $P: \Gamma(E) \rightarrow \Gamma(F)$ be a linear elliptic differential operator of order 1 with coefficients in $L^p$ for some $p > 2$.
(Actually, in what I have $P$ is smooth outside some set $K\subset M$ of measure zero.)
Is $P$ Fredholm on $W^{k, q}$ for $q\leq p$? If $Pu \in W^{k-1,q}$, does $u$ belong to $W^{k, q}$? Could someone help me with references about this class of operators?
Many thanks in advance.