Necessary and sufficient conditions for boundedness of commutators of fractional integral operators on slice spaces

TL;DR

This paper establishes necessary and sufficient conditions for the boundedness of commutators of fractional integral operators with rough kernels on slice spaces, linking boundedness to BMO and Lipschitz space membership.

Contribution

It provides a complete characterization of when these commutators are bounded on slice spaces, connecting function space membership to operator boundedness.

Findings

Boundedness of commutators characterized by BMO space membership.

Boundedness characterized by Lipschitz space membership.

Provides necessary and sufficient conditions for boundedness on slice spaces.

Abstract

Let $0<t<\infty$, $0<\alpha<n$, $1<p<r<\infty$ and $1<q<s<\infty$. In this paper, we prove that $b\in B M O\left(\mathbb{R}^{n}\right)$ if and only if the commutator $[b, T_{\Omega,\alpha}]$ generated by the fractional integral operator with the rough kernel $T_{\Omega,\alpha}$ and the locally integrable function $b$ is bounded from the slice space $(E_{p}^{q})_{t}(\mathbb{R}^{n})$ to $(E_{r}^{s})_{t}(\mathbb{R}^{n})$. Meanwhile, we also show that $b\in Lip_\beta(\mathbb{R}^{n}) $($0<\beta<1)$ if and only if the commutator $\left[b, T_{\Omega,\alpha}\right]$ is bounded from $(E_{p}^{q})_{t}(\mathbb{R}^{n})$ to $(E_{r}^{s})_{t}(\mathbb{R}^{n})$.