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We derive explicit bounds for the remainder term in the local Weyl law for locally hyperbolic manifolds; we also give estimates of the derivative of this remainder. We use these to obtain explicit bounds for the    C  l    norms of the    L  2   ‐normalized eigenfunctions in the case that the spectrum of the Laplacian is discrete, for example, for closed Riemannian manifolds. We also derive bounds for the local heat trace. Our estimates are purely local and therefore also hold for any manifold at points near which the metric is locally hyperbolic.

Explicit bounds on eigenfunctions and spectral functions on manifolds hyperbolic near a point