Failure of $L^p$ symmetry of zonal spherical harmonics

Abstract

We show that the 2-sphere does not exhibit symmetry of $L^p$ norms of eigenfunctions of the Laplacian for $p\ge 6$, which answers a question of Jakobson and Nadirashvili. In other words, there exists a sequence of spherical eigenfunctions ${\psi }_n$, with eigenvalues ${\lambda }_n\to \infty$ as $n\to \infty$, such that the ratio of the $L^p$ norms of the positive and negative parts of the eigenfunctions does not tend to $1$ as $n\to \infty$ when $p\ge 6$. Our proof relies on fundamental properties of the Legendre polynomials and Bessel functions of the first kind.