Upper and lower bounds for normal derivatives of spectral clusters of Dirichlet Laplacian

In this paper, we prove the upper and lower bounds for normal derivatives of spectral clusters u = χ λ s f of Dirichlet Laplacian Δ M , c s λ ‖ u ‖ L 2 ( M ) ⩽ ‖ ∂ ν u ‖ L 2 ( ∂ M ) ⩽ C s λ ‖ u ‖ L 2 ( M ) where the upper bound is true for any Riemannian manifold, and the lower bound is true for some small 0 < s < s M , where s M depends on the manifold only, provided that M has no trapped geodesics (see Theorem 1.3 for a precise statement), which generalizes the early results for single eigenfunctions by Hassell and Tao in 2002.