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Local Bernstein inequalities for eigenfunctions
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DOI：10.1016/j.aim.2025.110564.png)
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Abstract
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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: supB(g)(x,r) |del phi lambda| <= C epsilon( lambda 1+epsilon)/(r) supB(g)(x,r) |phi lambda|. We also prove analogous inequalities r for solutions of elliptic PDEs in terms of the frequency function. (c) 2025 The Authors. Published by Elsevier Inc. This is an open access article under the CC BY license (http:// creativecommons.org/licenses/by/4.0/).
Keywords:
Bernstein inequality
Laplace eigenfunctions
Frequency function
Doubling estimates
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