Suppose \((M,g)\) is a compact Riemannian manifold. There exist a constant \(C>0\), which solely depends on \(M\), such that a solution of

\[ -\Delta_g u = \lambda u \]

satisfies, for a sufficient small \(r>0\), a local Bernstein-type inequality

\[ \lVert \nabla u\rVert_{L^2(B_r(x))} \le \frac{C \sqrt{\lambda}}{r} \lVert u\rVert_{L^2(B_r(x))} \]

[1].

Links Link to heading

• eigenfunctions of the Laplace-Beltrami operator on compact manifolds
• Bernstein inequality
• Bernstein inequality on the sphere with covariant derivatives

Questions Link to heading

• Do I get the same estimate on good geodesic balls on the sphere?

References Link to heading

1. H. Donnelly and C. Fefferman, Growth and geometry of eigenfunctions of the Laplacian, in Analysis and partial differential equations, Dekker, New York, 1990, vol. 122, p. 635–655.