Let $(M,g)$ be a closed Riemannnian manifold with dimension $m$, $\Delta$ be the Laplace operator and $\phi_i$ be the $i$-th eigenfunction of $\Delta$ with eigenvalue $\lambda_i$ and $\|\phi_i\|_{L^2}=1$.
My question is why the following estimate holds or how could we obtain a similar estimate: \begin{equation} |\phi_i|+\lambda_i^{\frac12}|\nabla\phi_i|+\lambda_i^{-1}|\nabla^2\phi_i|\leq C\lambda_i^{\frac{m-1}{4}}, \end{equation} where $C$ is independent of $i$.
The above estimate is coming from the paper of Hein and Song about Calabi-Yau conifold. It is equation (2.15) and it said this could be obtained from Moser iteration and Schauder theory. Sorry for previous incomplete question.