Exact $L^p$ growth rates of Laplace eigenfunctions on the unit disk

Abstract

We determine the logarithmic growth exponents of the $L^p$ norms, $1\le p\le\infty$, of $L^2$-normalized Laplace eigenfunctions on the unit disk, for both Dirichlet and Neumann boundary conditions. We also prove sharp uniform $L^p$ upper and lower bounds for every $L^2$-normalized Dirichlet eigenfunction and every non-constant Neumann eigenfunction $u_{\lambda}$ on the disk. The proof uses stationary phase estimates and integral estimates for Bessel functions.