On a Bernstein inequality for eigenfunctions

Let ${\phi }_{\lambda }$ be an eigenfunction of the Laplace-Beltrami operator on a smooth compact Riemannian manifold $(M,g)$, i.e., ${\mathrm{\Delta }}_g{\phi }_{\lambda }+\lambda {\phi }_{\lambda }=0$. We show that ${\phi }_{\lambda }$ satisfies a local Bernstein inequality, namely for any geodesic ball $B_g(x,r)$ in $M$ there holds: ${\sup }_{B_g(x,r)}\mathrm{\mid }\mathrm{\nabla }{\phi }_{\lambda }\mathrm{\mid }\le C_{\delta }\max \{\frac{\sqrt{\lambda }{\log }^{2+\delta }\lambda }{r},\lambda {\log }^{2+\delta }\lambda \}{\sup }_{B_g(x,r)}\mathrm{\mid }{\phi }_{\lambda }\mathrm{\mid }$. We also prove analogous inequalities for solutions of elliptic PDEs in terms of the frequency function.