Restriction of eigenfunctions on products of spheres to submanifolds of maximal flats

TL;DR

This paper establishes sharp $L^p$ bounds for Laplace eigenfunctions restricted to submanifolds within maximal flats on products of rank-one symmetric spaces, using advanced asymptotic and harmonic analysis techniques.

Contribution

It provides the first sharp restriction estimates for eigenfunctions on these symmetric spaces, combining Jacobi polynomial asymptotics and spherical function Fourier analysis.

Findings

Sharp $L^p$ bounds for eigenfunction restrictions

Applicable to submanifolds in maximal flats

Utilizes Jacobi polynomial asymptotics and Fourier positivity

Abstract

Let $M$ be a product of rank-one symmetric spaces of compact type, each of dimension at least $3$. We establish sharp $L^p$ bounds for the restriction of Laplace--Beltrami eigenfunctions on $M$ to arbitrary submanifolds contained in a maximal flat, for all $p \ge 2$. The proof combines precise asymptotics of Jacobi polynomials and positivity of Fourier coefficients of spherical functions.