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

Abstract

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

Keywords

• Non-homogeneous metric measure space
• θ-type generalized fractional integral
• Commutator
• RBMO ˜ ( μ ) $\widetilde{\mathrm{RBMO}}(\mu )$
• Hardy space H ˜ b 1 ( μ ) $\widetilde{H}^{1}_{b}(\mu )$