On a compact Riemannian maniflod $(M,g)$, for an elliptic complex $\mathcal{C}_0\overset{L_1}{\longrightarrow}\mathcal{C}_1\overset{L_2}{\longrightarrow}\mathcal{C}_2$ where $L_1$ and $L_2$ are partial differential operators of order $l_1$, $l_2$ respectively, and $\mathcal{C}_*$ are vector bundles over $M$. Does the $L^p$ theory hold for elliptic systems of partial differential equations of mixed order? i.e. for any $\xi\in\mathcal{C}$, $\\xi\_{W^{l_1+l_2,p}}\leq C(\\xi\_{L^{p}}+\L_1^*\xi\_{W^{l_2,p}}+\L_2\xi\_{W^{l_1,p}})$
Yes. Check Sections 6.1 and 6.2 of


