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

Estimate near the self-focal points
This reduces the problem to studying remainder estimates for
points of ⋃ M
j=1 B δ(x j ) − {x j }. We may let δ = λ − 1 2 log λ. We show
that the remainder R(λ, x) at these points is bounded (up to
constants independent of (x, λ)) by the remainder at the center
the corresponding ball. The reason is that the
|R(x, λ, T ) − R(x j , λ, T )| ≤ Ce aT dist(x, x j ).
Hence if the distance is O(λ −1/2 ) we may let T = α log λ to make
this term small.
The ergodic theorem is uniform for a finite set of points, so given
ɛ, we may pick T (i.e. λ) large enough to make |R(x j , λ, T )| ≤ ɛ.