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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 $\xi\in\mathcal{C}$, $\|\xi\|_{W^{l_1+l_2,p}}\leq C(\|\xi\|_{L^{p}}+\|L_1^*\xi\|_{W^{l_2,p}}+\|L_2\xi\|_{W^{l_1,p}})$

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up vote 3 down vote accepted

Yes. Check Sections 6.1 and 6.2 of

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

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