Upper and lower bounds for normal derivatives of Dirichlet eigenfunctions

Suppose that $M$ is a compact Riemannian manifold with boundary and $u$ is an $L^2$-normalized Dirichlet eigenfunction with eigenvalue $\lambda$. Let $\psi$ be its normal derivative at the boundary. Scaling considerations lead one to expect that the $L^2$ norm of $\psi$ will grow as $\lambda^{1/2}$ as $\lambda \to \infty$. We prove an upper bound of the form $\|\psi \|_2^2 \leq C\lambda$ for any Riemannian manifold, and a lower bound $c \lambda \leq \|\psi \|_2^2$ provided that $M$ has no trapped geodesics (see the main Theorem for a precise statement). Here $c$ and $C$ are positive constants that depend on $M$, but not on $\lambda$. The proof of the upper bound is via a Rellich-type estimate and is rather simple, while the lower bound is proved via a positive commutator estimate.