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

Abstract

We develop a set of $L^{p}$ estimates for functions $u$ that are a joint quasimode (approximate eigenfunction) of $r$ 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