$L^\infty$-$BMO$ bounds for pseudo-multipliers associated to the harmonic oscillator

TL;DR

This paper establishes conditions under which pseudo-multipliers related to the harmonic oscillator are bounded from $L^ abla$ to BMO, and also explores Hermite multipliers' continuity from $ extnormal{H}^1$ to $L^1$, advancing harmonic analysis techniques.

Contribution

It introduces new H"ormander-Mihlin type conditions ensuring $L^ abla$-BMO boundedness for harmonic oscillator pseudo-multipliers and investigates Hermite multipliers' $ extnormal{H}^1$-$L^1$ continuity.

Findings

Established $L^ abla$-${ extnormal{BMO}}$ bounds under new conditions.

Proved $ extnormal{H}^1$-$L^1$ continuity for Hermite multipliers.

Extended harmonic analysis tools for pseudo-multipliers.

Abstract

In this note we investigate some conditions of H\"ormander-Mihlin type in order to assure the $L^\infty$-${\textnormal{BMO}}$ boundedness for pseudo-multipliers of the harmonic oscillator. The $\textnormal{H}^1$-$L^1$ continuity for Hermite multipliers also is investigated. The final version of this paper will appear in Rev. Colombiana Mat.