Failure of $L^p$ Symmetry of Zonal Spherical Harmonics

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