Failure of LP symmetry of zonal spherical harmonics

We show that the 2-sphere does not exhibit symmetry of Lp norms of eigenfunctions of the Laplacian for p > 6, which answers a question of Jakobson and Nadirashvili. In other words, there exists a sequence of spherical eigenfunctions *n, with eigenvalues An oo as n oo, such that the ratio of the Lp norms of the positive and negative parts of the eigenfunctions does not tend to 1 as n oo when p > 6. Our proof relies on fundamental properties of the Legendre polynomials and Bessel functions of the first kind.

Automated summary

We prove that the 2-sphere does not have symmetry of Lp norms of eigenfunctions of the Laplacian for p > 6, which addresses a question posed by Jakobson and Nadirashvili. In other words, there exists a sequence of spherical eigenfunctions *n, with eigenvalues An approaching infinity as n approaches infinity, such that the ratio of the Lp norms of the positive and negative parts of the eigenfunctions does not converge to 1 when p > 6. Our proof relies on fundamental properties of the Legendre polynomials and Bessel functions of the first kind.