$L^{p}$ estimates for joint quasimodes of semiclassical pseudodifferential operators

TL;DR

This paper establishes $L^{p}$ estimates for joint quasimodes of multiple semiclassical pseudodifferential operators, extending previous work from symmetric spaces to more general manifolds and operators.

Contribution

It introduces new $L^{p}$ bounds for joint quasimodes of multiple operators on general manifolds, broadening the scope beyond symmetric spaces.

Findings

Derived $L^{p}$ estimates for joint quasimodes.

Extended previous results from symmetric spaces to general manifolds.

Provides a framework for analyzing approximate eigenfunctions of multiple operators.

Abstract

We develop a set of $L^{p}$ estimates for functions $u$ that are a joint quasimodes (approximate eigenfunctions) of $r$ semiclassical pseudodifferential operators $p_{1}(x,hD),\dots,p_{r}(x,hD)$. This work extends Sarnak and Marshall's work on symmetric space to cover a more general class of manifolds/operators.