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 $u={\chi }_{\lambda }^{s}f$ of Dirichlet Laplacian ${\mathrm{\Delta }}_M$, $$c_s\lambda \mathrm{\parallel }u{\mathrm{\parallel }}_{L^2(M)}\le \mathrm{\parallel }{\mathrm{\partial }}_{\nu }u{\mathrm{\parallel }}_{L^2(\mathrm{\partial }M)}\le C_s\lambda \mathrm{\parallel }u{\mathrm{\parallel }}_{L^2(M)}$$ 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 \ref{Thm3} for a precise statement), which generalizes the early results for single eigenfunctions by Hassell and Tao.