Restriction of Laplace-Beltrami eigenfunctions to arbitrary sets on manifolds

2019/01/21 by Eswarathasan, Suresh, Pramanik, Malabika
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1901.07018

Abstract

Given a compact Riemannian manifold (M, g) without boundary, we estimate the Lebesgue norm of Laplace-Beltrami eigenfunctions when restricted to a wide variety of subsets Γ of M. The sets Γ that we consider are Borel measurable, Lebesgue-null but otherwise arbitrary with positive Hausdorff dimension. Our estimates are based on Frostman-type ball growth conditions for measures supported on Γ. For large Lebesgue exponents p, these estimates provide a natural generalization of L^p bounds for eigenfunctions restricted to submanifolds, previously obtained in \citeHo68, Ho71, Sog88, BGT07. Under an additional measure-theoretic assumption on Γ, the estimates are shown to be sharp in this range. As evidence of the genericity of the sharp estimates, we provide a large family of random, Cantor-type sets that are not submanifolds, where the above-mentioned sharp bounds hold almost surely.

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