Abstract
The main result of this paper is an estimate as in the title, near a sufficiently smooth part of the boundary of a compact n-dimensional Riemannian manifold Q, for either Dirichlet or Neumann boundary conditions. This yields a sharp estimate for the remainder in the Weyl asymptotic formula, for any boundary conditions. For manifolds whose boundaries have singularities such as edges or corners, the estimate remains valid with Dirichlet conditions.
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