Multilinear commutators for fractional integrals in non-homogeneous spaces

Abstract

Under the assumption that $\mu$ is a non-doubling measure on ${\mathbb R}^d$, the authors obtain the $(L^p,L^q)$-boundedness and the weak type endpoint estimate for the multilinear commutators generated by fractional integrals with $\mathrm{RBMO}(\mu)$ functions of Tolsa or with $\mathrm{Osc}_{\exp L^r}(\mu)$ functions for $r\ge 1$, where $\mathrm{Osc}_{\exp L^r}(\mu)$ is a space of Orlicz type satisfying that $\mathrm{Osc}_{\exp L^r}(\mu)=\mathrm{RBMO}(\mu)$ if $r=1$ and $\mathrm{Osc}_{\exp L^r}(\mu)\subset\mathrm{RBMO}(\mu)$ if $r>1$