  We derive explicit bounds for the remainder term in the local Weyl law for
locally hyperbolic manifolds, we also give the estimates of the derivative of
this remainder. We use these to obtain explicit bounds for the C^k-norms of the
L^2-normalised eigenfunctions in the case spectrum of the Laplacian is
discrete, e.g. 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.
