On a Bernstein inequality for eigenfunctions

2022/08/22 by Stefano Decio, Eugenia Malinnikova, Decio, Stefano +1
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Analytic and geometric function theory #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.2208.10541

openalex publication_date 2022/08/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let φ_λ be an eigenfunction of the Laplace-Beltrami operator on a smooth compact Riemannian manifold (M,g), i.e., Δ_g φ_λ + λφ_λ=0. We show that φ_λ satisfies a local Bernstein inequality, namely for any geodesic ball B_g(x,r) in M there holds: sup_Bg(x,r)|∇φ_λ|≤ C_δmax\\frac√λlog^2+δλr,λlog^2+δλ\sup_Bg(x,r)|φ_λ|. We also prove analogous inequalities for solutions of elliptic PDEs in terms of the frequency function.

Related

• Almost sharp local Bernstein estimates for Laplace eigenfunctions on compact Riemannian manifolds
• Restrictions of the Laplace-Beltrami eigenfunctions to submanifolds
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• Norms of geodesic restrictions for eigenfunctions on hyperbolic surfaces and representation theory
• Bernstein inequalities via the heat semigroup