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
$\xi\in\mathcal{C}$,
$\|\xi\|\_{W^{l\_1+l\_2,p}}\leq C(\|\xi\|\_{L^{p}}+\|L\_1^\*\xi\|\_{W^{l\_2,p}}+\|L\_2\xi\|\_{W^{l\_1,p}})$



A priori estimate of elliptic complex
Yes. Check Sections 6.1 and 6.2 of



> 
> Charles B. Morrey: Multiple Integrals in the Calculus of Variations, Springer Verlag.
> 
> 
> 


