Boundedness of commutator generated by fractional integral operator and Orlicz-BMO function

TL;DR

This paper proves the boundedness of a commutator involving fractional integral operators and Orlicz-BMO functions between specific function spaces, extending understanding of their behavior in harmonic analysis.

Contribution

It establishes new boundedness results for commutators generated by fractional integrals and Orlicz-BMO functions on Orlicz-Hardy and Lebesgue spaces.

Findings

Boundedness from Orlicz-Hardy space to Lebesgue space for the commutator.

Boundedness from Orlicz-Hardy space to weak Lebesgue space.

Extension of classical results to Orlicz-Hardy and BMO settings.

Abstract

For $\alpha\in(0, n)$ and a growth function $\varphi:[0,\infty)\rightarrow [0,\infty)$, it is proved that the commutator $[b,I_\alpha]$ generated by fractional integral operator $I_\alpha$ and Orlicz $\mathrm{BMO}$ function $b$ is bounded from Orlicz-Hardy space $H_{b}^{\varphi}(\mathbb{R}^{n})$ to Lebesgue space $L^{\frac{n}{n-\alpha}}(\mathbb{R}^{n})$, where $H_{b}^{\varphi}(\mathbb{R}^{n})$ is a suitable Orlicz-Hardy space. Moreover, the authors also establish that the boundedness of commutator $[b,I_\alpha]$ from Orlicz-Hardy space $H^{\varphi}(\mathbb{R}^{n})$ to weak Lebesgue space $L^{\frac{n}{n-\alpha},\infty}(\mathbb{R}^{n})$.