Sharp $L^p$ estimates for generalized Steklov eigenfunctions with an application to nodal sets

Abstract

We study a generalized Steklov problem involving a rough potential on the boundary. We establish sharp $L^p$ estimates for the Steklov eigenfunctions on compact manifolds with boundary, controlled by their $L^2$ norms on the boundary. We first establish sharp boundary estimates by heat kernel bounds and resolvent estimates for the Dirichlet-to-Neumann operator with a rough potential. And then we combine harmonic extension with the Littlewood–Paley decomposition to obtain sharp interior estimates. These results are new even when there is no potential. As an application, we prove the eigenfunctions are $C^1$ if the potential is Lipschitz and refine the previous results by Wang and Zhu (2015) on the lower bound of the size of the boundary nodal sets. A key tool is the commutator estimate for first-order pseudo-differential operators by Calderón (1965), and Coifman and Meyer (1978).