Commutators of θ-type generalized fractional integrals on non-homogeneous spaces

Abstract

The aim of this paper is to establish the boundednes of the commutator $[b,T_{\alpha }]$ [ b , T α ] generated by θ-type generalized fractional integral $T_{\alpha }$ T α and $b\in \widetilde{\mathrm{RBMO}}(\mu )$ b ∈ RBMO ˜ ( μ ) 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 $[b,T_{\alpha }]$ [ b , T α ] is bounded from the Lebesgue space $L^{p}(\mu )$ L p ( μ ) into the space $L^{q}(\mu )$ L q ( μ ) for $\frac{1}{q}=\frac{1}{p}-\alpha $ 1 q = 1 p − α and $\alpha \in (0,1)$ α ∈ ( 0 , 1 ) , and bounded from the atomic Hardy space $\widetilde{H}^{1}_{b}(\mu )$ H ˜ b 1 ( μ ) with discrete coefficient into the space $L^{\frac{1}{1-\alpha },\infty }(\mu )$ L 1 1 − α , ∞ ( μ ) . Furthermore, the boundedness of the commutator $[b,T_{\alpha }]$ [ b , T α ] on a generalized Morrey space and a Morrey space is also got.