H\"ormander condition for pseudo-multipliers associated to the harmonic oscillator

TL;DR

This paper establishes Hörmander-Mihlin multiplier theorems for pseudo-multipliers related to the harmonic oscillator, extending to multilinear cases and analyzing boundedness and compactness properties in various $L^p$ spaces.

Contribution

It introduces new Hörmander-Mihlin type theorems for pseudo-multipliers of the harmonic oscillator, including multilinear and spectral estimates.

Findings

Proves $L^p$-boundedness for pseudo-multipliers of the harmonic oscillator.

Establishes $L^p$-compactness properties for these multipliers.

Provides $(L^p,L^q)$-estimates for spectral pseudo-multipliers.

Abstract

In this paper we prove H\"ormander-Mihlin multiplier theorems for pseudo-multipliers associated to the harmonic oscillator (also called the Hermite operator). Our approach can be extended to also obtain the $L^p$-boundedness results for multilinear pseudo-multipliers. By using the Littlewood-Paley theorem associated to the harmonic oscillator we also give $L^p$-boundedness and $L^p$-compactness properties for multipliers. $(L^p,L^q)$-estimates for spectral pseudo-multipliers also are investigated.