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

We determine the logarithmic growth exponents of the $L^p$ norms, $1\le p\le \mathrm{\infty }$, 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.