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 $u={\chi }_{\lambda }^{s}f$ of Dirichlet Laplacian ${\Delta }_M$, $$c_s\lambda \Vert u{\Vert }_{L^2(M)}\le \Vert {\partial }_{\nu }u{\Vert }_{L^2(\partial M)}\le C_s\lambda \Vert u{\Vert }_{L^2(M)}$$ 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.

14 pages, Rewrite section 6 (the appendix) and correct a mistake in section 6

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