Sharp estimates of the potential kernel for the harmonic oscillator with applications

TL;DR

This paper provides precise estimates for the potential kernel of the harmonic oscillator and demonstrates that existing $L^p-L^q$ bounds for the associated potential operator are optimal, enhancing understanding of harmonic analysis in this context.

Contribution

The paper establishes sharp bounds for the harmonic oscillator's potential kernel and confirms the optimality of recent $L^p-L^q$ estimates for the potential operator.

Findings

Sharp potential kernel estimates for the harmonic oscillator

Validation of the optimality of recent $L^p-L^q$ bounds

Applications to harmonic analysis and operator theory

Abstract

We prove qualitatively sharp estimates of the potential kernel for the harmonic oscillator. These bounds are then used to show that the $L^p-L^q$ estimates of the associated potential operator obtained recently by Bongioanni and Torrea are in fact sharp.