论文标题
$ l^p $ Zonal球形谐波对称性的故障
Failure of $L^p$ Symmetry of Zonal Spherical Harmonics
论文作者
论文摘要
在本文中，我们表明，2个速度没有显示$ l^p $ n of th laplacian的$ l^p $规范的对称性，以$ P \ geq 6 $。 In other words, there exists a sequence of spherical eigenfunctions $ψ_n$, with eigenvalues $λ_n\to\infty$ 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 $p\geq 6$.我们的证明依赖于第一种Legendre多项式和Bessel功能的基本属性。
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, there exists a sequence of spherical eigenfunctions $ψ_n$, with eigenvalues $λ_n\to\infty$ 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 $p\geq 6$. Our proof relies on fundamental properties of the Legendre polynomials and Bessel functions of the first kind.
