  Let $(\Omega,g)$ be a compact, real-analytic Riemannian manifold with
real-analytic boundary $\partial \Omega.$ The harmonic extensions of the
boundary Dirchlet-to-Neumann eigenfunctions are called Steklov eigenfunctions.
We show that the Steklov eigenfuntions 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
$\{\sigma(P)=0\}$.

Analysis of PDEs;Spectral Theory