The aim of this paper is to establish the boundednes of the commutator $[b,T_{\alpha }]$ generated by θ-type generalized fractional integral $T_{\alpha }$ and $b\in \widetilde{\mathrm{RBMO}}(\mu )$ over a non-homogeneous metric measure space. Under the assumption that the dominating function λ satisfies the ϵ-weak reverse doubling condition, the author proves that the commutator $[b,T_{\alpha }]$ is bounded from the Lebesgue space $L^{p}(\mu )$ into the space $L^{q}(\mu )$ for $\frac{1}{q}=\frac{1}{p}-\alpha $ and $\alpha \in (0,1)$ , and bounded from the atomic Hardy space $\widetilde{H}^{1}_{b}(\mu )$ with discrete coefficient into the space $L^{\frac{1}{1-\alpha },\infty }(\mu )$ . Furthermore, the boundedness of the commutator $[b,T_{\alpha }]$ on a generalized Morrey space and a Morrey space is also got.

中文翻译：

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非齐次空间上θ型广义分数积分的交换子

本文的目的是建立由θ型广义分数积分$ T _ {\ alpha} $和$ b \在\ widetilde {\ mathrm {中生成的换向器$ [b，T _ {\ alpha}] $的有界RBMO}}（\ mu）$在非均匀度量度量空间上。在主导函数λ满足ϵ-弱反向加倍条件的前提下，作者证明了换向子$ [b，T _ {\ alpha}] $受勒贝格空间$ L ^ {p}（\ mu）的限制将$$放入$ L ^ {q}（\ mu）$空间中，以获得$ \ frac {1} {q} = \ frac {1} {p}-\ alpha $和$ \ alpha \ in（0,1）$ ，并从具有离散系数的原子Hardy空间$ \ widetilde {H} ^ {1} _ {b}（\ mu）$到空间$ L ^ {\ frac {1} {1- \ alpha}，\ infty}（\ mu）$。此外，还得到了换向子$ [b，T _ {\ alpha}] $在广义Morrey空间和Morrey空间上的有界性。