A Mikhlin--H\"ormander multiplier theorem for the partial harmonic oscillator

TL;DR

This paper establishes a Mikhlin--H"ormander multiplier theorem for the partial harmonic oscillator using Littlewood--Paley functions and heat kernel estimates, extending harmonic analysis tools to this operator.

Contribution

It introduces a new multiplier theorem for the partial harmonic oscillator, expanding the scope of harmonic analysis techniques to this class of operators.

Findings

Proved a Mikhlin--H"ormander multiplier theorem for the partial harmonic oscillator.

Developed heat kernel estimates for the associated operators.

Applied Littlewood--Paley functions to analyze multipliers.

Abstract

We prove a Mikhlin--H\"ormander multiplier theorem for the partial harmonic oscillator $H_{\textup{par}}=-\pa_\rho^2-\Delta_x+|x|^2$ for $(\rho, x)\in\R\times\R^d$ by using the Littlewood--Paley $g$ and $g^\ast$ functions and the associated heat kernel estimate. The multiplier we have investigated is defined on $\mathbb R \times \mathbb N$.