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In this paper, we show that the 2-sphere does not exhibit symmetry of $L^p$norms of eigenfunctions of the Laplacian for $p\geq 6$. In other words, thereexists 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 ofthe positive and negative parts of the eigenfunctions does not tend to $1$ as$n\to\infty$ when $p\geq 6$. Our proof relies on fundamental properties of theLegendre polynomials and Bessel functions of the first kind.

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