On the Restriction of Laplace–Beltrami Eigenfunctions and Cantor-Type Sets

Abstract

Let (M, g) denote a compact Riemannian manifold without boundary. This article is an announcement of Lebesgue norm estimates of Laplace–Beltrami eigenfunctions of M when restricted to certain fractal subsets Γ of M. The proofs in their entirety appear in Eswarathasan and Pramanik (Restriction of Laplace–Beltrami eigenfunctions to random Cantor-type sets on manifolds, 2019). The sets Γ that we consider are random and of Cantor-type. For large Lebesgue exponents p, our estimates give a natural generalization of Lp bounds previously obtained in Hormander (Acta Math 121: 193–218, 1968; Ark Math 11:1–11, 1971; Sogge J Funct Anal 77:123–138, 1988; Burq et al. Duke Math J 138(3):445–487, 2007). The estimates are shown to be sharp in this range. The novelty of our approach is the combination of techniques from geometric measure theory with well-known tools from harmonic and microlocal analysis. Random Cantor sets have appeared in a variety of contexts before, specifically in fractal geometry, multiscale analysis, additive combinatorics and fractal percolation Kahane and Peyriere (Adv Math 22(2):131–145, 1976; Laba and Pramanik, Geom Funct Anal 19:429–456, 2009; Laba and Pramanik, Duke Math J 158(3):347–411, 2011; Shmerkin and Suomala, Birkhauser/Springer, Cham, 2017; Shmerkin and Suomala, Mem Am Math Soc 251:1195, 2018). They play a significant role in the study of optimal decay rates of Fourier transforms of measures, and in the identification of sets with arithmetic and geometric structures. Our methods, though inspired by earlier work, are not Fourier-analytic in nature.