Pointwise Bounds for Steklov Eigenfunctions

Let \((\Omega ,g)\) be a compact, real-analytic Riemannian manifold with real-analytic boundary \(\partial \Omega .\) The harmonic extensions of the boundary Dirichlet-to-Neumann eigenfunctions are called Steklov eigenfunctions. We show that the Steklov eigenfunctions decay exponentially into the interior in terms of the Dirichlet-to-Neumann eigenvalues and give a sharp rate of decay to first order at the boundary. The proof uses the Poisson representation for the Steklov eigenfunctions combined with sharp h-microlocal concentration estimates for the boundary Dirichlet-to-Neumann eigenfunctions near the cosphere bundle \(S^*\partial \Omega .\) These estimates follow from sharp estimates on the concentration of the FBI transforms of solutions to analytic pseudodifferential equations \(Pu=0\) near the characteristic set \(\{\sigma (P)=0\}\).