Local Bernstein inequalities for eigenfunctions

Let ${\phi }_{\lambda }$ be an eigenfunction of the Laplace-Beltrami operator on a smooth compact Riemannian manifold, meaning that ${\Delta }_g{\phi }_{\lambda }+\lambda {\phi }_{\lambda }=0$. We show that ${\phi }_{\lambda }$ satisfies a local Bernstein inequality; namely for any geodesic ball $B(x,r)$ and any $\epsilon >0$ the following inequality holds: ${\sup }_{B_g(x,r)}\left|\nabla {\phi }_{\lambda }\right|\le C_{\epsilon }\frac{{\lambda }^{1+\epsilon }}{r}{\sup }_{B_g(x,r)}\left|{\phi }_{\lambda }\right|$. We also prove analogous inequalities for solutions of elliptic PDEs in terms of the frequency function.