{"paper":{"title":"Upper and lower bounds for normal derivatives of spectral clusters of Dirichlet Laplacian","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.SP"],"primary_cat":"math.AP","authors_text":"Xiangjin Xu","submitted_at":"2010-04-14T22:18:08Z","abstract_excerpt":"In this paper, we prove the upper and lower bounds for normal derivatives of spectral clusters $u=\\chi_{\\lambda}^s f$ of Dirichlet Laplacian $\\Delta_M$, $$c_s \\lambda\\|u\\|_{L^2(M)} \\leq \\| \\partial_{\\nu}u \\|_{L^2(\\partial M)} \\leq C_s \\lambda \\|u\\|_{L^2(M)} $$ 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."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1004.2517","kind":"arxiv","version":4},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}