Refined L^p restriction estimate for eigenfunctions on Riemannian surfaces

2024/11/03 by Changxing Miao, Gao, Chuanwei, Miao, Changxing +2 · 4 citations
Mathematics · #Advanced Harmonic Analysis Research #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Spectral Theory (math.SP) #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2411.01577

openalex publication_date 2024/11/03 · openalex created_date 2024/11/14 · openalex updated_date 2026/07/28

Abstract

We refine the L^p restriction estimates for Laplace eigenfunctions on a Riemannian surface, originally established by Burq, Gérard, and Tzvetkov. First, we establish estimates for the restriction of eigenfunctions to arbitrary Borel sets on the surface, following the formulation of Eswarathasan and Pramanik. We achieve this by proving a variable coefficient version of a weighted Fourier extension estimate of Du and Zhang. Our results naturally unify the L^p(M) estimates of Sogge and the L^p(γ) restriction bounds of Burq, Gérard, and Tzvetkov, and are sharp for all p ≥ 2, up to a λ^ε loss. Second, we derive sharp estimates for the restriction of eigenfunctions to tubular neighborhoods of a curve with nonvanishing geodesic curvature. These estimates are closely related to a variable-coefficient version of the Mizohata--Takeuchi conjecture, providing new insights into eigenfunction concentration phenomena.

Cited by

• Kuznecov formulae for fractal measures
• Weighted geodesic restrictions of arithmetic eigenfunctions
• Curved Kakeya sets for generic phases in odd dimensions
• Sharp microlocal Kakeya--Nikodym estimates for eigenfunctions with applications

Related

• Eigenfunctions restriction estimates for curves with nonvanishing geodesic curvatures in compact Riemannian surfaces with nonpositive curvature
• Restrictions of the Laplace-Beltrami eigenfunctions to submanifolds
• A few endpoint geodesic restriction estimates for eigenfunctions
• The L^p restriction bounds for Neumann data on surface
• Sharp endpoint estimates for eigenfunctions restricted to submanifolds of codimension 2