Let $(M^{m},g)$ be a closed Riemannian manifold of dimension $m\geq 2$. For any $p\in\mathbb{N}$, denote by $\Lambda^{(p)}\equiv\{\lambda_{j}\}_{j=0}^{\infty}$ with $\lambda_{0}=0$ the spectrum of the Hodge Laplacian $\Delta$ acting on the $p$-forms. For any $k\in\mathbb{N}$, there is some constant $C>0$ depending only on $(M,g)$ and $k$, $p$ such that for all $\phi_{j}\in\Omega^{p}(M^{m})$ satisfying

(3.350)

$\displaystyle\begin{cases}\Delta\phi_{j}=\lambda_{j}\phi_{j},\\ \|\phi_{j}\|_{L^{2}(M^{m})}=1,\end{cases}$

we have

(3.351)

$$\|\nabla^{k}\phi_{j}\|_{C^{k}(M^{m})}\leq C\cdot(\lambda_{j})^{\frac{1}{2}[\frac{m}{2}]+\frac{k+1}{2}}.$$