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

Abstract

In this paper, we prove the upper and lower bounds for normal derivatives of spectral clusters $u=χ_λ^s f$ of Dirichlet Laplacian $Δ_M$, $$c_s λ\|u\|_{L^2(M)} \leq \| \partial_νu \|_{L^2(\partial M)} \leq C_s λ\|u\|_{L^2(M)} $$ 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 \ref{Thm3} for a precise statement), which generalizes the early results for single eigenfunctions by Hassell and Tao.