A set of differential forms, uniformly bounded with their Laplacians, is precompact in $L^2$.

Question

Let $M$ be a compact Riemannian manifold and let $\Delta$ be a Hodge Laplacian on $k$-forms. How to show that the if the set $\{u_\alpha\} \subset C^2(M,\Lambda^k)$ of $C^2$ $k$-forms is uniformly bounded and the set $\{\Delta u_\alpha\}$ is also uniformly bounded then the set $\{u_\alpha\}$ is precompact in $L^2(M,\Lambda^k)$, i.e. any its sequence contains a Cauchy subsequence?

Answer 1

By Rellich-Kondrachov, it suffices to show $\{u_\alpha\}$ are bounded in some Sobolev space. I think $H^1 = W^{1,2}$ is the space to use here, because integration by parts $\int |\nabla f|^2 = -\int f \Delta f$, gives a uniform bound on Sobolev norm. This is for functions... for forms we get: $$ \|du\|^2 \leq \|du\|^2 + \|\delta u\|^2 = \langle \delta d u,u\rangle + \langle d\delta u,u \rangle = \langle \Delta u, u \rangle. $$

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