Park City Lectures on Eigenfunctions, Lecture 5: Lp norms of ...

Local Weyl law with remainder
The eigenfunction bounds are based on the following estimates for
the local Weyl remainder.
Theorem
Let R(λ, x) denote the remainder for the local Weyl law at x. Then
R(λ, x) = o(λ n−1 ) if |L x | = 0. (10)
Additionally, if |L x | = 0 then, given ε > 0, there is a neighborhood
N of x and a Λ =< ∞, both depending on ε so that
|R(λ, y)| ≤ ελ n−1 , y ∈ N , λ ≥ Λ. (11)