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=χλsf of Dirichlet Laplacian ΔMΔM,csλ‖u‖L2(M)⩽‖∂νu‖L2(∂M)⩽Csλ‖u‖L2(M)csλ‖u‖L2(M)⩽‖∂νu‖L2(∂M)⩽Csλ‖u‖L2(M) where the upper bound is true for any Riemannian manifold, and the lower bound is true for some small 0