Restriction of eigenfunctions on products of spheres to submanifolds of maximal flats
Yunfeng Zhang
TL;DR
The paper proves sharp $L^p$ restriction bounds for Laplace--Beltrami eigenfunctions on a product of compact rank-one symmetric spaces, restricted to submanifolds inside a maximal flat. The authors combine Jacobi-polynomial asymptotics with the positivity of spherical-function Fourier coefficients to reduce restriction questions to convolution bounds on tori, analyzed via a TT$^*$ framework and Jacobi operator bounds. They obtain explicit polynomial growth bounds $\|f\|_{L^p(S)} \lesssim N^{\frac{d-2}{2}+\sum_{i=1}^k\tau(d_i,p)} \|f\|_{L^2(M)}$ (up to an $\varepsilon$-loss) for all $p\ge2$, with the $\varepsilon$-loss removable when there are at least five factors; the result is sharp in that regime. The work extends sharp restriction phenomena to products of CROSSs and clarifies the role of maximal flats and Jacobi-convolution theory in spectral restriction problems.
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.