$L^{p}$ estimates for joint quasimodes of semiclassical pseudodifferential operators whose characteristic sets have $k$th order contact

TL;DR

This paper develops $L^{p}$ estimates for joint quasimodes of semiclassical pseudodifferential operators with characteristic sets having $k$th order contact, using Fourier integral operators to analyze curved level sets.

Contribution

It introduces new $L^{p}$ estimates for joint quasimodes with characteristic sets meeting with $k$th order contact, employing Fourier integral operators for curved level set analysis.

Findings

Derived $L^{p}$ bounds for joint quasimodes with higher order contact

Extended wavelet analysis to curved characteristic sets

Utilized Fourier integral operators for technical development

Abstract

In this paper we develop $L^{p}$ estimates for functions $u$ which are joint quasimodes of semiclassical pseudodifferential operators $p_{1}(x,hD)$ and $p_{2}(x,hD)$ whose characteristic sets meet with $k$th order contact, $k\geq{}1$. As part of the technical development we use Fourier integral operators to adapt a flat wavelet analysis to the curved level sets of $p_{1}(x,\xi)$.