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

Coherent states extremize L ∞ norms
This is simple and a standard fact about reproducing kernels. The
‘coherent state’ obtained by pinning the spectral projections kernel
Π λ (x, y) for an eigenspace V λ at one point y and dividing by its L 2
norm is always the etremal for pointwise norm at y among
eigenfunctions ϕ λ ∈ V λ
∫
ϕ λ (x) = Π λ (x, y)ϕ λ (y)dy
M
√ ∫
=⇒ |ϕ λ (x)| ≤ |Π λ (x, y)| 2 dy = √ Π λ (x, x)
M
= |Φ x λ (x)|.
In fact, they extremize L p norms for p ≥ p n