# Restriction of Laplace-Beltrami eigenfunctions to arbitrary sets on manifolds

## TL;DR

This paper establishes bounds on the Lebesgue norms of Laplace-Beltrami eigenfunctions when restricted to arbitrary, measure-zero sets on manifolds, extending previous submanifold results and demonstrating sharpness and genericity.

## Contribution

It introduces new Lebesgue norm estimates for eigenfunctions restricted to arbitrary sets with positive Hausdorff dimension, generalizing prior submanifold bounds and proving sharpness under certain conditions.

## Key findings

- Bounds are valid for a wide class of sets with positive Hausdorff dimension.
- Estimates are sharp under additional measure-theoretic assumptions.
- Random Cantor-type sets satisfy the bounds almost surely.

## 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, Lebesgue-null 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 \cite{Ho68, Ho71, Sog88, BGT07}. 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.