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

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.