Detecting and Estimating

The purpose of this paper is to use semiclassical analysis to unify and generalize L estimates on high energy eigenfunctions and spectral clusters. In our approach these estimates do not depend on ellipticity and order, and apply to operators which are selfadjoint only at the principal level. They are estimates on weakly approximate solutions to semiclassical pseudodifferential equations. To motivate our results let us first recall Sogge’s L estimate [18] on spectral clusters, Π[λ,λ+1], of the Laplace-Beltrami operator, −∆g, on a compact Riemannian manifold, (M, g): (1.1) Π[λ,λ+1] = O(λ 1 p ) : L(M, d volg) −→ L(M, d volg) , p = 2(n+ 1) n− 1 ,