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

Let $0<t<\mathrm{\infty }$, $0<\alpha <n$, $1<p<r<\mathrm{\infty }$ and $1<q<s<\mathrm{\infty }$. In this paper, we prove that $b\in BMO\left({\mathbb{R}}^n\right)$ if and only if the commutator $[b,T_{\mathrm{\Omega },\alpha }]$ generated by the fractional integral operator with the rough kernel $T_{\mathrm{\Omega },\alpha }$ 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 $b\in Li{p}_{\beta }({\mathbb{R}}^n)$($0<\beta <1)$ if and only if the commutator $[b,T_{\mathrm{\Omega },\alpha }]$ is bounded from $(E_p^q{)}_t({\mathbb{R}}^n)$ to $(E_r^s{)}_t({\mathbb{R}}^n)$.