Semiclassical Lp estimates

TL;DR

This paper employs semiclassical analysis to unify and extend Lp estimates for high energy eigenfunctions and spectral clusters, applicable to a broader class of operators without relying on ellipticity.

Contribution

It introduces a generalized approach to Lp estimates that do not depend on ellipticity or order, applicable to operators only selfadjoint at the principal level.

Findings

Estimates are valid for weakly approximate solutions to semiclassical pseudodifferential equations.

Corrected an exponent in the main theorems for accuracy.

Establishes a unified framework for high energy spectral analysis.

Abstract

The purpose of this paper is to use semiclassical analysis to unify and generalize Lp 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. The revision corrects an exponent in the main theorems.