On a Bernstein inequality for eigenfunctions

Abstract

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

Replaces the previous version which contained a mistake in the proof of Theorem 2. The main result is almost unchanged except for logarithmic terms, the new proof is substantially different from the previous version. 22 pages, comments welcome!