Restriction of Laplace–Beltrami Eigenfunctions to Arbitrary Sets on Manifolds. (7th July 2020)

• Record Type: Journal Article
• Authors: Eswarathasan, Suresh; Pramanik, Malabika
• Is Part Of: International mathematics research notices. Volume 2022:Number 2(2022)
• Page Start: 1538
• Page End: 1600
• Publication Date: 2020-07-07
• Subjects: Mathematics -- Periodicals; 510
• DOI: 10.1093/imrn/rnaa167
• Languages: English
• ISSNs: 1073-7928

Abstract

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 $\Gamma $ of $M$ . The sets $\Gamma $ that we consider are Borel measurable, Lebesguenull but otherwise arbitrary with positive Hausdorff dimension. Our estimates are based on Frostman-type ball growth conditions for measures supported on $\Gamma $ . For large Lebesgue exponents $p$, these estimates provide a natural generalization of $L^p$ bounds for eigenfunctions restricted to submanifolds, previously obtained in [ 8, 18, 19, 32 ]. Under an additional measure-theoretic assumption on $\Gamma $, 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.