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

Abstract

Let $0<t<\infty$, $0<α<n$, $1<p<r<\infty$ 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 $[b, T_{Ω,α}]$ generated by the fractional integral operator with the rough kernel $T_{Ω,α}$ 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_β(\mathbb{R}^{n}) $($0<β<1)$ if and only if the commutator $\left[b, T_{Ω,α}\right]$ is bounded from $(E_{p}^{q})_{t}(\mathbb{R}^{n})$ to $(E_{r}^{s})_{t}(\mathbb{R}^{n})$.

Theorems & Definitions (16)

• Definition 1.1
• Definition 1.2
• Definition 1.3
• Theorem 1.4
• Theorem 1.5
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Lemma 2.5